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300000
- R1
- R2
- P-
- R3
- P0
- U1
- U2
- P0U
This guide will introduce you to the subject of Knot Theory. The first section gives an introduction to important math
theory concepts and terminology.
If you are only looking for quick tips to solve the unknotting puzzles, skip ahead to the second section. You can always
consult the first section if the
meaning of some words is unclear.
Introduction with Definitions
Introduction
First things first : what are knots? Knots in mathematics are different from knots in your everyday life.
What kinds of knots do you use every day?
How are these different from mathematical knots?
Are there any mathematical knots in everyday life?
How does the Unknotting game work?
Definitions
From here on, we only talk about mathematical knots. To avoid misunderstandings and to be able to decide whether any statement is true or false we should start with defining the meaning of several words for the rest of this page.
knot line:
knot diagram:
mathematical knot (or simply, knot):
ambient isotopy:
step:
crossing:
pass:
switch:
orientation:
crossing handedness:
writhe number:
knot invariant:
crossing number:
arc:
hole:
over-strand:
under-strand:
Reidemeister moves:
Reidemeister 1 move:
Reidemeister 2 move:
Reidemeister 3 move:
pass move:
P- move:
P0 move:
P+ move:
unknotting number:
How to Simplify Diagrams
Finding R1 Moves
Finding R2 Moves
About the Interface (1)
Finding P- Moves
Finding R3 Moves
About the Interface (2)
Finding P0 Moves
Finding U1 Moves
Finding U2 Moves
Finding P0U Moves
More References about Knots
Introduction with Definitions
Introduction
First things first : what are knots? Knots in mathematics are different from knots in your everyday life.
What kinds of knots do you use every day?
Most of us use knots to tie our shoelaces, put on a necktie or scarf, to close up a bag, and so on... You might
know many more knots if you go sailing, camping, fishing, or if you sew, knit, or style hair.
However, none of these are mathematical knots!
How are these different from mathematical knots?
Have a look at the following two drawings of knots. Confusingly, these are both known as 'figure-eight' knots,
because of the figure 8 they contain.
Everyday Figure-eight Knot Mathematical Figure-eight Knot
What big difference can you see? We're sure you can figure it out!
Everyday Figure-eight Knot Mathematical Figure-eight Knot
What big difference can you see? We're sure you can figure it out!
The biggest difference is that the mathematical knot is a closed curve − that is, there are no loose
ends, it's a closed loop. What we call 'knots' in everyday life are known as 'braids' in
mathematics.
Also, while everyday knots might include more than one strand of material, in mathematics knots are a single, closed, continuous strand. Objects that involve more than one knot woven together are known as 'links'.
Also, while everyday knots might include more than one strand of material, in mathematics knots are a single, closed, continuous strand. Objects that involve more than one knot woven together are known as 'links'.
Are there any mathematical knots in everyday life?
Of course! You now know that in mathematics, a knot is a single, closed, continuous strand.
With this definition in mind, what is the simplest mathematical knot?
How can you make a mathematical knot from a piece of string?
With this definition in mind, what is the simplest mathematical knot?
The simplest mathematical knot is just a single loop or circle, like this:
We will talk more about this knot later, but there are plenty of examples of simple loops like this in real life.
We will talk more about this knot later, but there are plenty of examples of simple loops like this in real life.
How can you make a mathematical knot from a piece of string?
The simplest mathematical knot is a circle. To make it, simply glue the ends of your string together.
What happens if you twist your circle once?
Is this a different knot?
Can all knots be deformed to make a circle?
How different can a knot diagram look from the simplest form?
What happens if you twist your circle once?
If you take your loop of string, twist it, and lay it flat, you might get something like this:
Is this a different knot?
Of course knot! All you did was twist it.
This might seem easy, but finding ways to tell whether two diagrams (pictures) show the same knot is a very important and difficult question for mathematicians who study Knot Theory.
One way to tell that two pictures represent the same knot is whether you can deform one of them to look like the other. For example, here you just need to twist it back to get the loop.
This might seem easy, but finding ways to tell whether two diagrams (pictures) show the same knot is a very important and difficult question for mathematicians who study Knot Theory.
One way to tell that two pictures represent the same knot is whether you can deform one of them to look like the other. For example, here you just need to twist it back to get the loop.
Can all knots be deformed to make a circle?
To answer this question, try this out:
Can you deform this to get a circle?
- take a strand of string
- twist it to form a loop
- pass one end through the loop
Can you deform this to get a circle?
Try as you might, there is no way to deform this knot into a circle. At least, not without cutting
the string and gluing it back together.
This is actually a different mathematical knot, called the trefoil knot because it looks like a three-leaf clover.
This is actually a different mathematical knot, called the trefoil knot because it looks like a three-leaf clover.
How different can a knot diagram look from the simplest form?
Another important property of mathematical knots is that they can be arbirtrarily stretched and bent.
For example, our diagram of the simplest knot looks more like a square than a circle − we could
draw it as a perfect circle, and it would be the same knot. You could take the simplest knot, a circle,
and stretch it out into a long thin ellipse, then use it as a string to tie it into the 'everyday
figure-8 knot'. Mathematically, it is still a circle.
As you might expect, then, the different diagrams of the same underlying knot can look drastically different! For example, the Gordian Knot and this diagram can both be deformed into a circle, with enough patience. They are the diagrams of the same knot as the circle.
The circle is just the easiest way of drawing this knot, but the knot isn't really a circle, it's an abstract mathematical object that we can represent in many different ways. In the same way that "1 car" and "1 apple" is not the number 1, a circle is just one way to represent this knot.
As you might expect, then, the different diagrams of the same underlying knot can look drastically different! For example, the Gordian Knot and this diagram can both be deformed into a circle, with enough patience. They are the diagrams of the same knot as the circle.
The circle is just the easiest way of drawing this knot, but the knot isn't really a circle, it's an abstract mathematical object that we can represent in many different ways. In the same way that "1 car" and "1 apple" is not the number 1, a circle is just one way to represent this knot.
How does the Unknotting game work?
In this game, you deform mathematical knot diagrams to reduce the number of
crossings as much as possible.
While you are allowed to 'cut' the strands by clicking on them, you are only modifying the diagram, not the underlying knot. This is why there is a restriction on how you can reattach the ends you have 'cut'. This restriction guarantees that the mathematical knot is unchanged in any re-routing of strands even if the knot changes its appearance.
When do you undo everyday knots in real life?
Try some other mathematical knot games!
While you are allowed to 'cut' the strands by clicking on them, you are only modifying the diagram, not the underlying knot. This is why there is a restriction on how you can reattach the ends you have 'cut'. This restriction guarantees that the mathematical knot is unchanged in any re-routing of strands even if the knot changes its appearance.
When do you undo everyday knots in real life?
You might be familiar with the struggle of disentangling electronics cables like for earbuds. If you tie
your shoes, you have to untie the laces.
The same property of real knots is what makes them useful, but also harder to undo.
What makes knots in real life so hard to undo?
The same property of real knots is what makes them useful, but also harder to undo.
What makes knots in real life so hard to undo?
The answer is friction! However, mathematical knots have no friction. You can think of them as
'infinitely slippery'.
Try some other mathematical knot games!
Our unknotting game is one way to have fun with knots on a screen, but here are some other games for you to
try:
- 1 player : Eiffel Tower and other string tricks
- 2 players : Cat's cradle
- Group : Human Knot
Definitions
From here on, we only talk about mathematical knots. To avoid misunderstandings and to be able to decide whether any statement is true or false we should start with defining the meaning of several words for the rest of this page.
knot line:
a closed curve in 3-dimensional space which does not
intersect itself and which has a finite thickness (to avoid
infinitely many smaller and smaller knots along the line), Example:
the figure-8 knot.
knot diagram:
a projection of a knot line into 2 dimensions where
different parts of the knot line can cross each other (on this website
lines cross under an angle of 90°), but do not lie on top of each other.
Examples for knot diagrams:
Unknot Trefoil Figure-eight Cinquefoil Three-twist
The 'unknot' is also called the 'trivial knot'.
Which knot does each of these 3 diagrams represent?
Which knot does this diagram represent? 
Do you see how to deform this into the earlier trefoil diagram?
Examples for knot diagrams:
Unknot Trefoil Figure-eight Cinquefoil Three-twist
The 'unknot' is also called the 'trivial knot'.
Which knot does each of these 3 diagrams represent?
The unknot. Do you see how they can be deformed to a rectangle?
Which knot does this diagram represent?
The trefoil.
Do you see how to deform this into the earlier trefoil diagram?
You can deform this diagram into the other by flipping down the top arc.
→
→
mathematical knot (or simply, knot):
the abstract object behind
a set of (infinitely many) knot diagrams that can all be deformed,
stretched, and shifted into each other without being cut. Example:
the knot 31 also called 'trefoil' which is the simplest
non-trivial knot.
ambient isotopy:
the mathematical term
when one knot line can be continuously distorted to another one.
step:
On this website knot diagrams are drawn using only 6 tiles which we call steps:
crossing:
the place in a diagram where two steps cross, one on top of
the other:
pass:
a step that is part of a crossing, there are over-passes (fully
visible) and under-passes (partially covered).
switch:
swapping over- and under-pass of a crossing, i.e. switching between these two crossings:
If a crossing is switched, the old and new diagram in general
represent different knots. Switching all crossings is equivalent to
changing a knot to its mirror image.
Some knots are identical to their mirrored version, that means there is an
ambient isotopy between them. These are called 'achiral'. For example, the figure-8 knot is achiral.
Others can not be deformed to their mirrored version, like the
trefoil. They are called 'chiral'.
Can this knot be deformed into its mirror image?
Can this knot be deformed into its mirror image?
Yes! This diagram represents the figure-eight knot which is achiral and can be deformed into its mirror
image as follows:
Notice that the only difference between the first diagram and the last one is that all of the crossings are switched!
Initial position
A 180° rotation
Moving a strand
The mirror image
Notice that the only difference between the first diagram and the last one is that all of the crossings are switched!
orientation:
This is not a property of the knot line, nor of the knot.
It is a question of how to move along the knot line. One can move in 2
directions, also called 2 orientations.
A knot that can be deformed via an ambient isotopy into itself but with the
orientation reversed is called 'invertible' otherwise it is called
noninvertible. The smallest noninvertible knot is 817 which is achiral but
if an orientation is added it becomes chiral (find more on
the Invertible Knot Wikipedia page). Adding more
structure (here
an orientation) causes it to lose symmetry (not identical anymore to its mirror
image).
A chiral knot (a knot that can not be deformed to its mirror image) can still be invertible (symmetric against change of orientation). Such knots are called 'reversible'.
A chiral knot (a knot that can not be deformed to its mirror image) can still be invertible (symmetric against change of orientation). Such knots are called 'reversible'.
crossing handedness:
For a given knot diagram, crossings are either right- or left-handed.
In the following we will explore two types of crossings and determine which is right- or left-handed.
How many different crossings are there if we consider
which pass is an over-/under-pass and consider both orientations?
If one keeps the diagram unchanged and only switches one crossing,
does the handedness of that crossing change?
By using one's hands, how can one remember whether a crossing is
right- or left-handed?
In the following we will explore two types of crossings and determine which is right- or left-handed.
How many different crossings are there if we consider
which pass is an over-/under-pass and consider both orientations?
In total there are 8 cases:
If the horizontal pass is the over-pass then there are 4 options:
Similarly if the vertical pass is the over-pass then there are 4 more options:
The crossings in group 1, 7, 4, 6 are called right-handed crossings and
the crossings in group 2, 5, 3, 8 are called left-handed crossings.
If the horizontal pass is the over-pass then there are 4 options:
1
2
3
4
5
6
7
8
- Principle: Handedness should not depend on the orientation (direction of stepping through the knot line), so reversing both arrows we identify 4 pairs of crossings: 1 = 4, 2 = 3, 5 = 8, 6 = 7. Therefore, whichever groups we end up with, crossings 1 and 4 should be in the same group and so on.
- Principle: The group that a crossing belongs to should not change if we rotate the whole knot. We therefore identify crossings 1 = 7 = 4 = 6 and 2 = 5 = 3 = 8.
The crossings in group 1, 7, 4, 6 are called right-handed crossings and
the crossings in group 2, 5, 3, 8 are called left-handed crossings.
If one keeps the diagram unchanged and only switches one crossing,
does the handedness of that crossing change?
Yes. Try switching any of the eight crossings,
then check which one it has become and check whether it is
still in the same handedness group. For example, switching crossing 1
gives crossing 5, both are in different handedness groups.
Statement: The question whether a crossing is right-handed or left-handed does not only depend on the crossing itself but on the diagram around it.
Proof: Whether the horizontal pass is above or below does not alone decide the handedness. Both cases can be right- and left-handed (see the 8 crossings above). If one rotates the knot so that the over-pass is horizontal and if one then leaves the crossing through the overhead pass to the right (to the East) then it depends on the rest of the knot whether one returns to the crossing from the South (then the crossing is right-handed) or from the North (then the crossing is left-handed).
The fact that the two groups of crossings are called left- and right-handed gives a hint that one can distinguish crossings with the left and right hands.
Statement: The question whether a crossing is right-handed or left-handed does not only depend on the crossing itself but on the diagram around it.
Proof: Whether the horizontal pass is above or below does not alone decide the handedness. Both cases can be right- and left-handed (see the 8 crossings above). If one rotates the knot so that the over-pass is horizontal and if one then leaves the crossing through the overhead pass to the right (to the East) then it depends on the rest of the knot whether one returns to the crossing from the South (then the crossing is right-handed) or from the North (then the crossing is left-handed).
The fact that the two groups of crossings are called left- and right-handed gives a hint that one can distinguish crossings with the left and right hands.
By using one's hands, how can one remember whether a crossing is
right- or left-handed?
Stretch out your fingers so that all are in one plane, and your
thumb is at a right angle to all others which are parallel to
each other. Rotate your hand so that you can see your palm and
your thumb points towards the outgoing direction of the
over-pass and your fingers point towards the outgoing direction of
the under-pass. The hand that can do that decides the
handedness.
For example, for the crossing below, you would stretch your hand out like this:
Since you can only do this with your left hand, this is a left-handed crossing.
Right- and left-handed crossings are also referred to as positive or negative crossings.
Since you can only do this with your left hand, this is a left-handed crossing.
Right- and left-handed crossings are also referred to as positive or negative crossings.
writhe number:
the difference between the number of left- and right-handed
crossings in one diagram. The writhe number characterizes a diagram, not a knot as there can be 2
different diagrams of the same knot with different writhe numbers.
What is the writhe number of this diagram?
What is the writhe number of this diagram?
The right-handed crossings in the diagram are highlighted in red and the left-handed crossings in green.
To get the writhe number, we can count the number of left- and right-handed crossings, then subtract the
number of right-handed crossings
from the number of left-handed crossings.
This diagram has 2 left-handed crossings and 4 right-handed crossings, so its writhe number is 2 − 4 =
−2.
knot invariant:
a number or a polynomial or a feasibility statement that is characteristic for all (infinitely many)
diagrams of a knot. The properties of a knot being chiral/achiral, invertible/noninvertible,
reversible are knot invariants.
crossing number:
the minimal number of crossings that any diagram of
this knot can have after deformation, this is a characteristic of each knot and therefore a knot
invariant.
How many crossings does this diagram have? 
What is the crossing number of the knot represented through the above diagram?
What are the two lowest crossing numbers that a knot can have?
How many crossings does this diagram have?
This diagram has 5 crossings.
What is the crossing number of the knot represented through the above diagram?
Zero! The crossing number is a property of the abstract
mathematical knot, it is not the property of a diagram. The
diagram above can be deformed to get the unknot
which has zero crossings.
Do you see how?
Because the crossing number is the minimal number of crossings of any diagram of a knot and because one cannot have less than zero crossings, the crossing number of the knot represented by the above diagram is zero.
which has zero crossings. Do you see how?
Because the crossing number is the minimal number of crossings of any diagram of a knot and because one cannot have less than zero crossings, the crossing number of the knot represented by the above diagram is zero.
What are the two lowest crossing numbers that a knot can have?
The lowest crossing number belongs to the unknot which is 0.
A knot diagram with 1 crossing would look like:
and could be deformed to the unknot. A knot diagram with 2 crossings would look like
and could also be deformed to the unknot.
The trefoil diagram shown further above has 3 crossings and can not be deformed to the unknot, so the lowest crossing numbers are 0 and 3.
and could be deformed to the unknot. A knot diagram with 2 crossings would look like
and could also be deformed to the unknot.The trefoil diagram shown further above has 3 crossings and can not be deformed to the unknot, so the lowest crossing numbers are 0 and 3.
arc:
The part of the knot line in a diagram from one crossing to the next.
How many arcs does a diagram with N crossings have?
How many arcs does a diagram with N crossings have?
Each crossing has 4 ends of arcs. Each arc has 2 ends, so there are 4/2 = 2
times as many arcs as crossings, so 2N arcs.
hole:
empty space in a diagram that is surrounded by arcs. The whole
empty space outside the diagram is also one hole.
How many holes does a diagram with N crossings have?
How many holes does a diagram with N crossings have?
One could draw several knots and guess a formula but one can derive
it too. Euler's formula says that for any drawing in the plane where
m lines (here m=2N arcs) each connect 2 out of n points (here n=N
crossings)
then the number f of faces (here holes) is f = 2 + m − n. That gives
for the number of holes of a knot: 2 + 2N − N = N + 2.
over-strand:
A sequence of consecutive arcs in a diagram
(i.e. arcs following one another) which starts and ends at an
under-pass and otherwise involves 0, 1 or more over-passes.
under-strand:
A sequence of consecutive arcs in a diagram
(i.e. arcs following one another) which starts and ends at an
over-pass and otherwise involves 0, 1 or more under-passes.
(In literature, a 'strand' is often used for what we call an over-strand. For us the number of over-passes of an over-strand matters as well as the number of under-passes of an under-strand. We therefore consider under-strands as well as over-strands.)
What kind of strand is the shown horizontal line consisting of five arcs?
How many over-strands does a diagram with N crossings have?
(In literature, a 'strand' is often used for what we call an over-strand. For us the number of over-passes of an over-strand matters as well as the number of under-passes of an under-strand. We therefore consider under-strands as well as over-strands.)
What kind of strand is the shown horizontal line consisting of five arcs?
This is an under-strand with 4 under-passes.
How many over-strands does a diagram with N crossings have?
At each crossing there are 2 ends of strands (either one end of
2 different strands or both ends from one strand). On the other hand,
each strand has 2 ends which are at a crossing. Therefore the number
of crossings is equal to the number of over-strands and because of
symmetry also equal to the number of under-strands, so there are
N of each.
Reidemeister moves:
In 1927 the German mathematician Kurt
Reidemeister and, independently, James Waddell Alexander and Garland
Baird Briggs (1926), proved that any two diagrams that represent the
same knot can be deformed into each other through a sequence of only 3
different types of moves. The problem is that during the deformation
the number of crossings may temporarily rise and a sharp upper bound for
this increase is unknown as well as the number of needed moves.
Reidemeister 1 move:
removes or adds a hole surrounded by one arc:
Which diagram shows a left-handed crossing and which
shows a right-handed crossing?
Which diagram shows a left-handed crossing and which
shows a right-handed crossing?
The left diagram shows a right-handed crossing and the right diagram shows a
left-handed crossing. A Reidemeister 1 move therefore changes the number
of right- or left-handed crossings by 1 and thus changes the
writhe number of the diagram.
Reidemeister 2 move:
removes or adds a hole surrounded by 2 arcs:
What can one say about the handedness of the two crossings that are
added or removed in a Reidemeister 2 move?
What can one say about the handedness of the two crossings that are
added or removed in a Reidemeister 2 move?
One of the two crossings is right-handed and one is left-handed.
A Reidemeister 2 move therefore does not change the writhe
number of a diagram.
Reidemeister 3 move:
removes and adds a hole surrounded by 3 arcs.
Which 2 types of holes surrounded by 3 arcs can you think of?
Which 2 types of holes surrounded by 3 arcs can you think of?
Either:
Verify that the result of the 3 moves is always the same.
Does the handedness of the 3 crossings change in a Reidemeister 3 move?
What have we learned?
- 1) each arc has 1 over-pass and 1 under-pass:
then none of the 3 arcs can be moved over/under/through the other crossing
- 2) one arc has 2 over-passes, one has 1 over- and 1
under-pass, and one has 2 under-passes:
then there are 3 moves. One can move the over-over-strand which stays an over-over-strand:
or move the under-over-strand which becomes an over-under-strand:
or move the under-under-strand which stays an under-under-strand:
Verify that the result of the 3 moves is always the same.
When comparing the right-hand sides of the above moves it is easy to see that
all 3 moves produce identical results. Therefore, if there is a Reidemeister 3 move
then there is only one. All that changes is that for all 3 arcs the other two arcs are
now crossed in the reverse order. This means that for the middle arc the order of over-pass
and under-pass is reversed.
Does the handedness of the 3 crossings change in a Reidemeister 3 move?
No. To see that, pick any orientation for each strand and use the
hand rule above.
What have we learned?
We learned:
- how to spot holes with 3 arcs that allow a Reidemeister 3 move,
- that for such a hole it does not matter which arc is moved,
- that the handedness of the 3 crossings does not change,
- that the order of over- and under-pass is reversed for the middle arc.
pass move:
This has nothing to do with a 'pass' defined above.
A pass move replaces an over-(under-)strand with another
over-(under-)strand where both strands have the same ends. For examples, please see P-, P0 and P+ moves below.
P- move:
a pass move where the new strand has less passes than the
old strand.
Find a P- move replacing the green strand in this diagram:
Find a P- move replacing the green strand in this diagram:
In this diagram the new red strand has fewer passes than the old green strand. Therefore this diagram shows
a P- move.
P0 move:
a pass move where the new strand has the same number of passes
as the old strand.
Find a P0 move replacing the green strand in this diagram:
Find a P0 move replacing the green strand in this diagram:
In this diagram the new red strand has the same number of passes as the old green strand. Therefore this
diagram shows a P0 move.
P+ move:
a pass move where the new strand has more passes
than the old strand.
Find a P+ move replacing the green strand in this diagram:
P+ moves become necessary if one wants to change the writhe number of a diagram. More about that is described further below under 'Finding P0 Moves'.
Find a P+ move replacing the green strand in this diagram:
In this diagram the new red strand has one more pass than the old green strand, therefore this is a P+ move.
P+ moves become necessary if one wants to change the writhe number of a diagram. More about that is described further below under 'Finding P0 Moves'.
unknotting number:
The unknotting number is the property of a knot, not the property of a diagram and is therefore
a knot invariant.
Starting with a knot diagram it is the minimal number of times that one or more crossings need to be switched in order to obtain the unknot. Before the first switch and inbetween switches the diagram can be arbitrarily deformed. The unknotting number is therefore not an easy number to determine because any deformation is allowed.
Why does the trefoil have unknotting number 1?
Starting with a knot diagram it is the minimal number of times that one or more crossings need to be switched in order to obtain the unknot. Before the first switch and inbetween switches the diagram can be arbitrarily deformed. The unknotting number is therefore not an easy number to determine because any deformation is allowed.
Why does the trefoil have unknotting number 1?
The trefoil cannot have the unknotting number 0 because it cannot
be deformed into the unknot (this needs to be and can be proven). The unknot has unknotting number 0. So the
trefoil has unknotting number ≥1. On the other hand one can easily
see that switching any one crossing of the trefoil diagram
shown further above produces the unknot, so the unknotting
number of the trefoil is ≤1. If it is ≥1 and ≤1 then it must be =1.
How to Simplify Diagrams
Finding R1 Moves
Simple cases of R1 moves, like here:
where one can flip a loop 4 times and instantly get the unknot
are easy to spot by following the knot line and looking for an arc
with both ends at the same crossing. The order of performing R1 moves
does not matter.
But there are more general cases of applying Reidemeister 1 moves. If an over-strand starts at a crossing of which it is an overpass but otherwise lies completely on top of other arcs (therefore being called 'over-strand') then this loop can surely be shortcut and thus removed. For example, at first the loop on top in the center can be removed and then the other ones, one by one:
This loop can also be removed when lying entirely underneath:
where one can flip a loop 4 times and instantly get the unknot
are easy to spot by following the knot line and looking for an arc
with both ends at the same crossing. The order of performing R1 moves
does not matter. But there are more general cases of applying Reidemeister 1 moves. If an over-strand starts at a crossing of which it is an overpass but otherwise lies completely on top of other arcs (therefore being called 'over-strand') then this loop can surely be shortcut and thus removed. For example, at first the loop on top in the center can be removed and then the other ones, one by one:
This loop can also be removed when lying entirely underneath:
Finding R2 Moves
Similarly to R1 moves it is easy to spot prototype R2 moves like
here where two R2 moves need to be done before an R1 move yields
the unknot:
In the following example one has to perform an R2 move with the same strand twice, once pulling this strand from underneath and once from above:
The last step is not an R2 move. It is only added to show that the knot is the sum of two trefoil knots.
In the following example one has to perform an R2 move with the same strand twice, once pulling this strand from underneath and once from above:
The last step is not an R2 move. It is only added to show that the knot is the sum of two trefoil knots.
About the Interface (1)
The example above is suitable to demonstrate the optimal use of the
interface. After intercepting the knot line:
One can NOT retreat one end all the way; and then retreat the other end all the way because one would first remove an under-pass and have both ends in the under-pass status and in this status it is not possible to remove an over-pass. Instead, one removes one under-pass, jumps to the other end, removes the other under-pass which brings both ends into the same hole, which changes the ends into neutral status and then allows to remove over-passes and then to re-unite the ends. In short, one jumps between ends to remove all under-passes then all over-passes and so on.
This implementation of a status of ends is not a weakness of the program but it guarantees that interactive modification of the diagram does not change the mathematical knot.
One can NOT retreat one end all the way; and then retreat the other end all the way because one would first remove an under-pass and have both ends in the under-pass status and in this status it is not possible to remove an over-pass. Instead, one removes one under-pass, jumps to the other end, removes the other under-pass which brings both ends into the same hole, which changes the ends into neutral status and then allows to remove over-passes and then to re-unite the ends. In short, one jumps between ends to remove all under-passes then all over-passes and so on.
This implementation of a status of ends is not a weakness of the program but it guarantees that interactive modification of the diagram does not change the mathematical knot.
Finding P- Moves
P- moves replace an over-strand with one having less over-passes
or an under-strand with one having less under-passes. In both
cases the number of crossings is reduced.
To find such moves one steps through the knot line and looks for as many consecutive over-passes as possible or for as many consecutive under-passes as possible, at least two. If one found such a strand, for example, an over-strand then one tries to find an alternative route with less over-passes that links the same two under-pass ends.
In the following example a strand with 4 consecutive under-passes is replaced by a strand without underpasses and then this strand with 3 consecutive over-passes is replaced by a strand with one over-pass. Two more P- moves remove each 2 more crossings. The resulting diagram can be simplified further by two more R1 moves, as seen below.
The more consecutive passes of one sort one finds, the higher the
chance to find a different route that needs less passes.
To find such moves one steps through the knot line and looks for as many consecutive over-passes as possible or for as many consecutive under-passes as possible, at least two. If one found such a strand, for example, an over-strand then one tries to find an alternative route with less over-passes that links the same two under-pass ends.
In the following example a strand with 4 consecutive under-passes is replaced by a strand without underpasses and then this strand with 3 consecutive over-passes is replaced by a strand with one over-pass. Two more P- moves remove each 2 more crossings. The resulting diagram can be simplified further by two more R1 moves, as seen below.
Finding R3 Moves
R1, R2 and P- moves change the number of crossings. An R3
move does not change the number of crossings, therefore we place
its description after the P- move. The following example shows how
R3 moves can still be useful by making P- moves possible.
As described in the first section, an R3 hole is surrounded by a top arc
with 2 over-pass ends (here A,B), a middle arc with 1 over-pass end (C) and 1
under-pass end (B) and a bottom arc with 2 under-pass ends (here
A,C).
(unknot taken from the Unknot Wikipedia page)
We found in the above section that an R3 move reverses for the middle strand (here through B,C) the order of under- and over-pass. An R3 move is beneficial if the R3 move leads for the continuation of the middle arc to an increase of consecutive over-passes and/or consecutive under-passes and thus increases the chance to find a P- move. This is the case if in the continuation of the middle arc after the under-pass B comes an over-pass (D,E are over-passes) and/or after the over-pass (C) follows an under-pass (F,G,H are under-passes). The R3 move slides the middle arc BC between the top arc and bottom arc at A:
Before, there were only 2 consecutive over-passes at D and E,
now there are 3 at C, D and E. This longer over-strand can now be
re-routed in a P- move:
reducing the number of crossings by 2 from 13 to 11.
Also, the other side of the strand can be re-routed in a P- move. Before, there were 3 consecutive under-passes at F,G and H, now there are 4 at B, F, G, and H. This P- move leads to:
and is also reducing the number of crossings by 2. Both diagrams can be
simplified further through P- and R1 moves resulting finally in the unknot.
Can you see how? Just follow the hints on how to spot P- moves given above.
Let us practise that with an example.
How many R3 moves are possible in this diagram:
We found in the above section that an R3 move reverses for the middle strand (here through B,C) the order of under- and over-pass. An R3 move is beneficial if the R3 move leads for the continuation of the middle arc to an increase of consecutive over-passes and/or consecutive under-passes and thus increases the chance to find a P- move. This is the case if in the continuation of the middle arc after the under-pass B comes an over-pass (D,E are over-passes) and/or after the over-pass (C) follows an under-pass (F,G,H are under-passes). The R3 move slides the middle arc BC between the top arc and bottom arc at A:
Also, the other side of the strand can be re-routed in a P- move. Before, there were 3 consecutive under-passes at F,G and H, now there are 4 at B, F, G, and H. This P- move leads to:
Let us practise that with an example.
How many R3 moves are possible in this diagram:
Three R3 moves are possible. For each one we show in light blue the three arcs that are
involved. What is easily overlooked is the third one where the hole
is the whole outside space which is 'surrounded' by only 3 arcs.
Perform the 1. R3 move and find out whether it is beneficial:
Perform the 2. R3 move and find out whether it is beneficial:
Perform the 3. R3 move and find out whether it is beneficial:
1. R3 move
2. R3 move
3. R3 move
Perform the 1. R3 move and find out whether it is beneficial:
This R3 move is beneficial. It allows afterwards a P- moves as shown in a sequence of moves further
below.
Our definition for an R3 move to be beneficial it is not
neccessarily to allow a P- move but to increase the number of consecutive
over- or under-passes and that is easy to see even without
performing all these moves. In the following diagram the middle arc
of the R3 hole has an
over-pass at A, an under-pass at B followed by two over-passes at C and D.
In an R3 move the order of over- and under-pass is reversed for the middle arc
as shown in the Diagram 3 with now 3 consecutive over-passes. This is enough
to find a P- move needing less than 3 passes in Diagram 5.
About the sequence of diagrams below: In Dia 1 we make space to prepare the R3 move in Dia 2 (here by
moving
the top arc) with the result in Dia 3. In Dia 4 we make space to prepare
the P- move in Dia 5 where the green strand with 3 over-passes is replaced
by the red strand with only 1 over-pass in Dia 6. In Dia 7 we shift a strand
to make space for the next P- move in Dia 9 with result in Dia 10 and Dia 11
after shrinking which is easily identified as knot 51.
1. Widening
2. The R3 move
3. After the R3 move
4. Widening
5. A P- move
6. After the P- move
7. For the next P- move
8. Before the P- move
9. The 2nd P- move
10. Afterwards
11. Contracted
Perform the 2. R3 move and find out whether it is beneficial:
The sequence shows that the R3 move is beneficial.
1. Widening
2. The R3 move
3. After the R3 move
4. A P- move
5. After the P- move
6. Another P- move
7. After the P- move
8. Contracted
Perform the 3. R3 move and find out whether it is beneficial:
The 3rd R3 move is also beneficial. To execute this move, one follows the same principle: the middle
strand cuts the two other strands, which this time
'surround' the outside hole, in reverse order.
The result is that travelling upwards in the Dia 2 along the light blue strand one first gets to an overpass and then an underpass. If after the R3 move one travels along the red strand then one passes the other 2 strands in reverse order and therefore first gets to an underpass and then an overpass. As one can see in Dia 5, the sequence of now 3 consecutive underpasses of the light blue strand there enables a P- move.
The result is that travelling upwards in the Dia 2 along the light blue strand one first gets to an overpass and then an underpass. If after the R3 move one travels along the red strand then one passes the other 2 strands in reverse order and therefore first gets to an underpass and then an overpass. As one can see in Dia 5, the sequence of now 3 consecutive underpasses of the light blue strand there enables a P- move.
1. Widening
2. An R3 move
3. After the R3 move
4. Widening
5. A P- move
6. Widening
7. A 2nd P- move
8. After the P- move
9. Shortening
10. After shortening
11. Shortening
12. Straightening
13. ↻90° rotation
About the Interface (2)
The example above is suitable to demonstrate how to perform an R3
move with our interface.
As described in the 'Introduction with Definitions' > 'Reidemeister 3 move' there are 3 ways to perform a Reidemeister 3 move: moving the bottom strand, moving the middle strand, or, moving the top strand. As shown there, all 3 ways have the same result, they perform the same Reidemeister 3 move.
Our interface only allows one to complete an R3 move by moving the bottom strand or the top strand but not the middle strand. The reason is the feature of our interface that BOTH ends can only add/remove over-passes or BOTH ends can only add/remove under-passes at a time. But that does not prevent us from performing R3 moves, as moving any one of the 3 strands gives the same result.
As described in the 'Introduction with Definitions' > 'Reidemeister 3 move' there are 3 ways to perform a Reidemeister 3 move: moving the bottom strand, moving the middle strand, or, moving the top strand. As shown there, all 3 ways have the same result, they perform the same Reidemeister 3 move.
Our interface only allows one to complete an R3 move by moving the bottom strand or the top strand but not the middle strand. The reason is the feature of our interface that BOTH ends can only add/remove over-passes or BOTH ends can only add/remove under-passes at a time. But that does not prevent us from performing R3 moves, as moving any one of the 3 strands gives the same result.
Finding P0 Moves
P0 moves are pass moves that do not change the number of
crossings, just like R3 moves which are special versions of
P0 moves. Like R3 moves, a P0 move may be beneficial and
enable a P- move. Because P0 moves are less useful on
average, they occur more frequently but it is more
challenging to see whether they enable a P- move.
To find a P0 move one looks for an over-strand or an under-strand just like for a P- move. To check whether a P0 move may be beneficial and enable a P- move one proceeds like in the case of the R3 move. One looks whether the removal of the strand increases the number of consecutive over- or under-passes of strands that were crossed and one checks whether after re-routing the new strand increases the number of consecutive over- or under-passes of strands that are crossed now. In either one of these cases one checks whether the strands with an increased number of consecutive over- or under-passes can be re-routed with less crossings.
Let us look at this example:
We label the crossings:
and find a beneficial P0 move step by step.
How many over-strands with at least 2 over-passes
and how many under-strands with at least 2
under-passes do you see?
It is not difficult to see that the IE strand has a P0 move that re-locates it to cross over the GC and BH strand:
but the question is whether this P0 move is beneficial.
Did moving the IE strand increase the number of consecutive over-/under-passes
of the previously crossed DF or HJ strands?
Did more consecutive over-/under-passes get created when placing the strand on top of the
2 strands GC and BH?
Can this BH strand be re-routed in a P- move to reduce the number of crossings?
In the original diagram this P- move could have been done first as a P0 move and the P0 move mentioned above afterwards as P- move with the same total saving of 2 crossings.
In difficult problems it may be necessary to perform several P0 moves before a P- move becomes possible.
A diagram is maximally simplified if the number of crossings is the crossing number (see first section). In that case P0 moves will never enable a P- move.
To find a P0 move one looks for an over-strand or an under-strand just like for a P- move. To check whether a P0 move may be beneficial and enable a P- move one proceeds like in the case of the R3 move. One looks whether the removal of the strand increases the number of consecutive over- or under-passes of strands that were crossed and one checks whether after re-routing the new strand increases the number of consecutive over- or under-passes of strands that are crossed now. In either one of these cases one checks whether the strands with an increased number of consecutive over- or under-passes can be re-routed with less crossings.
Let us look at this example:
How many over-strands with at least 2 over-passes
and how many under-strands with at least 2
under-passes do you see?
We get three over-strands with at least 2 over-passes:
AB, GC, IE and three under-strands with at least 2
under-passes: EF, BH, DJ.
It is not difficult to see that the IE strand has a P0 move that re-locates it to cross over the GC and BH strand:
Did moving the IE strand increase the number of consecutive over-/under-passes
of the previously crossed DF or HJ strands?
Yes, the HJ strand now has 2 over-passes but this strand can not be re-routed
to link the same two holes with less over-passes.
Did more consecutive over-/under-passes get created when placing the strand on top of the
2 strands GC and BH?
Yes, the BH under-strand had 2 under-passes and now has 3 under-passes.
Can this BH strand be re-routed in a P- move to reduce the number of crossings?
Yes: The new route of the BH strand links the same holes but with only 1 under-pass instead of
3 under-passes.
The fact that the new strand is longer (involves more steps) in this
diagram than the replaced strand does not matter. All that matters
is the reduction of the number of crossings from 10 to 8 which now
allows to identify this knot as knot 817. In the original diagram this P- move could have been done first as a P0 move and the P0 move mentioned above afterwards as P- move with the same total saving of 2 crossings.
In difficult problems it may be necessary to perform several P0 moves before a P- move becomes possible.
A diagram is maximally simplified if the number of crossings is the crossing number (see first section). In that case P0 moves will never enable a P- move.
Finding U1 Moves
A U1 move switches a crossing which afterwards allows to simplify
the diagram to remove all crossings and show that the switch
produced the unknot.
In general nice puzzles should have unique solutions so our U1 and U2 puzzles show diagrams where the switch of only one crossing results in the unknot. This hint allows you to cut down the search.
If the diagram includes a twist of the knot line like this:
would it matter which one of the crossings is switched?
If there are still several crossing candidates, one should try to imagine how many R1, R2 moves will become available through the switch and try that switch first which seems to allow the most simplifications afterwards.
Another hint to minimize trying switches is to imagine whether after the switch there will definitely be a knot remaining, like a trefoil. If so then that switch is not the correct one in U1 puzzles.
On the worksheet the number of available switches is limited to the minimum that is needed to get the unknot. If you did a switch one can not switch it back because the diagram may have been changed, so one would have to reset the diagram.
In general nice puzzles should have unique solutions so our U1 and U2 puzzles show diagrams where the switch of only one crossing results in the unknot. This hint allows you to cut down the search.
If the diagram includes a twist of the knot line like this:
It will not matter which crossing is switched. Both results are equivalent:
Therefore either both of these crossings are unknotting switches or
none of them. Because our puzzles have only one unknotting switch,
these two crossings can be ignored.
=
=
If there are still several crossing candidates, one should try to imagine how many R1, R2 moves will become available through the switch and try that switch first which seems to allow the most simplifications afterwards.
Another hint to minimize trying switches is to imagine whether after the switch there will definitely be a knot remaining, like a trefoil. If so then that switch is not the correct one in U1 puzzles.
On the worksheet the number of available switches is limited to the minimum that is needed to get the unknot. If you did a switch one can not switch it back because the diagram may have been changed, so one would have to reset the diagram.
Finding U2 Moves
For U2 puzzles the same hint about equivalent switches applies as
for U1 puzzles. Also for U2 puzzles there is just one crossing that
reduces the unknotting number, i.e. makes progress towards the
unknot. Once that unique first crossing is switched and the
resulting diagram is simplified, there may be more than one switch
possible that creates the unknot.
Finding P0U Moves
Research performed by Caribou Contests on unknotting numbers showed
that there exists maximally simplified knot diagrams (with the
minimal number of crossings), which do not have a simplifying
switch. In other words, there are diagrams where switching any
crossings will not make progress to reach the unknot. In that case
one first has to perform one or more P0 moves that change the puzzle
into a U2 puzzle. The good news is that diagrams requiring P0 moves
first are rare and therefore it will be likely that any P0 moves
will change it into a U2 puzzle.
More References about Knots
Mathematical Knot Theory is an old research subject so there
exists a vast amount of literature for it. However, it also is a young
subject as several milestones have only been reached in recent
decades. For example, there is a scientific "Journal of Knot
Theory and Its Ramifications" dedicated to knots which has
a new issue every month.
A great book that we recommend is: Adams, Colin (2004), The Knot Book: An Elementary Introduction to the Mathematical Theory of Knots, American Mathematical Society, ISBN 978-0-8218-3678-1
There are also many websites about knots. A good place to start is the Knot Theory page on Wikipedia.
For videos, check out Numberphile's Playlist of Knot videos on YouTube. They also have a great explanation of Colouring Knots, which is another way to help identify diagrams of the same knot.
Caribou produced two posters on Unknotting and colouring knots.
A great book that we recommend is: Adams, Colin (2004), The Knot Book: An Elementary Introduction to the Mathematical Theory of Knots, American Mathematical Society, ISBN 978-0-8218-3678-1
There are also many websites about knots. A good place to start is the Knot Theory page on Wikipedia.
For videos, check out Numberphile's Playlist of Knot videos on YouTube. They also have a great explanation of Colouring Knots, which is another way to help identify diagrams of the same knot.
Caribou produced two posters on Unknotting and colouring knots.
Ce guide introduit le sujet de la Théorie des Noeuds. La première section donne une introduction à quelques concepts clés de la Théorie des Noeuds ainsi que son lexique.
Si vous cherchez uniquement de l'aide pour résoudre des casse-têtes, sautez la première section pour la consulter lorsque vous ne comprenez pas certains termes.
Introduction et Définitions
Introduction
Tout d'abord : Que sont des noeuds? Les noeuds mathématiques diffèrent des noeuds du quotidien.
Utilisez-vous des noeuds tous les jours? Quand ça?
Comment ces noeuds diffèrent-ils des noeuds mathématiques?
Du coup, est-ce qu'on retrouve des noeuds mathématiques dans la vie de tous les jours?
Comment joue-t-on à « Défaire le Noeud »?
Définitions
Désormais, on ne parle que des noeuds mathématiques. Afin d'éviter de la confusion et de pouvoir déterminer si un énoncé est vrai ou faux, commençons par identifier ce que signifient certains mots sur cette page.
ligne de noeud :
projection ou diagramme de noeud :
noeud mathématique ( ou noeud, tout simplement) :
isotopie ambiante :
étape :
croisement :
passage :
inversion :
orientation :
chiralité ou polarité de croisement :
entortillement :
invariant de noeud :
Nombre minimal de croisements :
arc :
trou :
brin par-dessus :
brin par-dessous :
Mouvements de Reidemeister :
Mouvement Reidemeister de type 1 (R1) :
Mouvement Reidemeister de type 2 (R2) :
Mouvement Reidemeister de type 3 (R3) :
Un mouvement à passage :
mouvement P- :
mouvement P0 :
mouvement P+ :
nombre de dénouement:
Comment Simplifier les Diagrammes de Noeud
Trouver des Mouvements R1
Trouver des Mouvements R2
À Propos de l'Interface (1)
Trouver des Mouvements P-
Trouver des Mouvements R3
À Propos de l'Interface (2)
Trouver des Mouvements P0
Trouver des Mouvements U1
Trouver des Mouvements U2
Trouver des Mouvements P0U
Des Références Supplémentaires sur les Noeuds Mathématiques
Introduction et Définitions
Introduction
Tout d'abord : Que sont des noeuds? Les noeuds mathématiques diffèrent des noeuds du quotidien.
Utilisez-vous des noeuds tous les jours? Quand ça?
La plupart d'entre nous utilise des noeuds au quotidien pour se nouer les lacets, pour mettre une cravate ou une écharpe, pour refermer un sac, et ainsi de suite... Selon vos passe-temps, vous connaissez peut-être bien plus de noeuds, par exemple si vous faites de la voile, du camping, de la pêche, si vous faites de la couture, du tricot, ou même si vous aimez coiffer des cheveux.
Cependant... aucun d'entre ces derniers n'est un noeud mathématique!
Cependant... aucun d'entre ces derniers n'est un noeud mathématique!
Comment ces noeuds diffèrent-ils des noeuds mathématiques?
Jetez un coup d'oeil aux deux schémas de noeud ci-dessous. Figurez-vous qu'ils s'appellent tous les deux « Noeud en huit »! Oui, chacun contient un 8, mais ils ne sont pas tout à fait pareils...
Noeud en huit de tous les jours Noeud en huit mathématique
Quelle grande différence discernez-vous?
Noeud en huit de tous les jours Noeud en huit mathématique
Quelle grande différence discernez-vous?
Ne coupons pas les cheveux en huit! La plus grande différence entre les deux noeuds, c'est que le noeud mathématique est une courbe fermée. Ce qu'on appelle un « noeud » dans la vie de tous les jours est en mathématiques appelé une « tresse ».
Et aussi, alors que les noeuds au quotidien peut lier plusieurs ficelles ou cordes, les noeuds mathématiques comprennent un seul brin continu et fermé. On appelle « entrelacs » les objets comprenant plus d'un noeud.
Et aussi, alors que les noeuds au quotidien peut lier plusieurs ficelles ou cordes, les noeuds mathématiques comprennent un seul brin continu et fermé. On appelle « entrelacs » les objets comprenant plus d'un noeud.
Du coup, est-ce qu'on retrouve des noeuds mathématiques dans la vie de tous les jours?
Mais bien sûr! Vous savez maintenant qu'en mathématiques, un noeud est une seule courbe continue et fermée.
Quel est le noeud mathématique le plus simple qui colle à cette définition?
Comment peut-on former un noeud mathématique à partir d'une ficelle?
Quel est le noeud mathématique le plus simple qui colle à cette définition?
Le noeud mathématique le plus simple est, tout simplement, une boucle ou cercle, comme dans le schéma suivant:
On en reparlera plus tard. Il est facile de trouver des exemples d'une boucle simple comme celle-ci dans la réalité.
On en reparlera plus tard. Il est facile de trouver des exemples d'une boucle simple comme celle-ci dans la réalité.
Comment peut-on former un noeud mathématique à partir d'une ficelle?
Le noeud mathématique le plus simple est un cercle. Pour le faire, coller les deux bouts de votre ficelle pour en faire une boucle.
Prenez votre boucle et retournez un côté. Que devient-elle?
Est-ce qu'il s'agit d'un noeud différent?
Peut-on déformer tous les noeuds pour obtenir une boucle?
Le diagramme d'un noeud peut-il beaucoup différer de sa représentation la plus simple?
Prenez votre boucle et retournez un côté. Que devient-elle?
If you take your loop of string, twist it, and lay it flat, you might get something like this:
Est-ce qu'il s'agit d'un noeud différent?
Bien sûr que non. Vous l'avez tordu, c'est tout.
Cette question peut paraître banale, mais pour les mathématiciens qui étudient la Théorie des Noeuds, de déterminer si deux diagrammes représentent le même noeud est une question non seulement importante mais très difficile.
Pour déterminer si deux images représentent le même noeud, une technique consiste à déformer un noeud pour voir si on peut obtenir l'autre. Par exemple, ici il suffit de tordre la ficelle dans l'autre sens pour revenir à la boucle.
Cette question peut paraître banale, mais pour les mathématiciens qui étudient la Théorie des Noeuds, de déterminer si deux diagrammes représentent le même noeud est une question non seulement importante mais très difficile.
Pour déterminer si deux images représentent le même noeud, une technique consiste à déformer un noeud pour voir si on peut obtenir l'autre. Par exemple, ici il suffit de tordre la ficelle dans l'autre sens pour revenir à la boucle.
Peut-on déformer tous les noeuds pour obtenir une boucle?
Pour répondre à cette question, essayez ce qui suit :
Est-ce qu'on peut déformer ce noeud pour obtenir un cercle?
- prenez un bout de ficelle
- tordez-le pour former une boucle
- passez une extrémité de la ficelle à travers la boucle
Est-ce qu'on peut déformer ce noeud pour obtenir un cercle?
Vous aurez beau essayé, vous ne pourrez pas déformer ce noeud en un cercle. Au moins, pas sans tricher en coupant la ficelle pour la recoller après.
Alors, ceci est un noeud mathématique différent. Il s'appelle le noeud de trèfle car il ressemble à un trèfle à trois feuilles.
Alors, ceci est un noeud mathématique différent. Il s'appelle le noeud de trèfle car il ressemble à un trèfle à trois feuilles.
Le diagramme d'un noeud peut-il beaucoup différer de sa représentation la plus simple?
Une autre propriété importante des noeuds mathématiques, c'est qu'on peut les étirer et les plier de manière arbitraire. Par exemple, notre diagramme du noeud le plus simple ressemble plus à un carré qu'un cercle. On pourrait le dessiner comme un cercle parfait, et ce serait tout de même le même noeud. On pourrait étirer ce noeud, un cercle, jusqu'à obtenir une longue et fine éllipse, et utiliser cette éllipse comme d'une ficelle pour la nouer en le noeud en huit de tous les jours. Mathématiquement parlant, ce serait toujours un cercle.
Alors, comme vous pouvez vous l'imaginer, les divers diagrammes d'un même noeud peuvent être très différents. Par exemple, le noeud gordien et ce diagramme peuvent tous les deux se déformer en un cercle, si vous avez assez de patience! Quoiqu'ils paraissent compliqués, ces dessins représentent le même noeud qu'un cercle.
Le cercle est le diagramme de ce noeud le plus facile à dessiner, mais il est important de noter que le noeud n'est pas vraiement un cercle. C'est un objet mathématique abstrait qu'on peut représenter de maintes manières. Tout comme « 1 pomme » et « 1 voiture » ne sont pas le nombre 1, un cercle n'est qu'une façon de représenter ce noeud.
Alors, comme vous pouvez vous l'imaginer, les divers diagrammes d'un même noeud peuvent être très différents. Par exemple, le noeud gordien et ce diagramme peuvent tous les deux se déformer en un cercle, si vous avez assez de patience! Quoiqu'ils paraissent compliqués, ces dessins représentent le même noeud qu'un cercle.
Le cercle est le diagramme de ce noeud le plus facile à dessiner, mais il est important de noter que le noeud n'est pas vraiement un cercle. C'est un objet mathématique abstrait qu'on peut représenter de maintes manières. Tout comme « 1 pomme » et « 1 voiture » ne sont pas le nombre 1, un cercle n'est qu'une façon de représenter ce noeud.
Comment joue-t-on à « Défaire le Noeud »?
Le défi s'agit de déformer le noeud mathématique afin de réduire le nombre de croisements autant que possible.
Bien qu'on vous permette de « couper » les brins en les cliquant dessus, vous ne modifiez que le diagramme et non le noeud sous-jacent. C'est pourquoi vous êtes limité par rapport au raboutissage des extrémités coupées. Cette contrainte guarantit que le noeud mathématique reste inchangé par la manipulation des brins même si son apparence est modifiée.
Est-ce que vous avez l'habitude de défaire des noeuds?
Essayez d'autres jeux avec les noeuds mathématiques!
Bien qu'on vous permette de « couper » les brins en les cliquant dessus, vous ne modifiez que le diagramme et non le noeud sous-jacent. C'est pourquoi vous êtes limité par rapport au raboutissage des extrémités coupées. Cette contrainte guarantit que le noeud mathématique reste inchangé par la manipulation des brins même si son apparence est modifiée.
Est-ce que vous avez l'habitude de défaire des noeuds?
Dans la vie de tous les jours, vous avez peut-être l'habitude de devoir démêler les fils des vos écouteurs, par exemple (Vive les écouteurs sans-fils!). Si vous nouez vos lacets de chaussures, vous devez les dénouer.
Les noeuds de tous les jours ont une caractéristique qui n'est pas partagée par les noeuds mathématiques. C'est elle qui les rend utiles, mais aussi plus difficile à défaire.
Pourquoi le noeuds sont-ils difficiles à défaire? Autrement dit, pourquoi les noeuds ne se défont pas tout seuls?
Les noeuds de tous les jours ont une caractéristique qui n'est pas partagée par les noeuds mathématiques. C'est elle qui les rend utiles, mais aussi plus difficile à défaire.
Pourquoi le noeuds sont-ils difficiles à défaire? Autrement dit, pourquoi les noeuds ne se défont pas tout seuls?
La réponse : la friction! Cependant, les noeuds mathématiques n'ont pas de friction. Ils sont effectivement « infiniment glissants ».
Essayez d'autres jeux avec les noeuds mathématiques!
Vous pouvez vous amusez à manipuler les noeuds sur un écran en jouant à « Défaire le Noeud ». Voici quelques autres jeux à essayer :
- 1 joueur : la Tour Eiffel et d'autres figures formées à partir d'une ficelle
- 2 joueurs : Le Berceau du Chat
- En Groupe : Le Noeud Humain
Définitions
Désormais, on ne parle que des noeuds mathématiques. Afin d'éviter de la confusion et de pouvoir déterminer si un énoncé est vrai ou faux, commençons par identifier ce que signifient certains mots sur cette page.
ligne de noeud :
une courbe dans
l'espace trois-dimensionnel, fermée, continue (elle ne s'intersecte pas), et d'épaisseur fini (pour éviter des noeuds de plus en plus petits le long de la ligne). Exemple : le noeud en 8
projection ou diagramme de noeud :
c'est la projection bidimensionnelle d'un noeud
où deux parties de la courbe peuvent se croiser (sur notre site, les lignes se croisent toujours à 90°) mais ne peuvent pas être superposées.
Quelques exemples de projections de noeuds :
Noeud Trivial Noeud de Trèfle Noeud en huit Noeud de Potentille Noeud à Trois Torsions
Le Noeud Trivial, parfois appelé le Noeud Non-Noué est tout simplement un cercle, ou sur notre site un rectangle (puisque toutes les lignes se croisent à 90°).
Les trois projections suivantes représentent le même noeud. Lequel?
Quel noeud cette projection représente-t-elle?
Est-ce que vous voyez comment la manipuler pour obtenir la projection de tout à l'heure?
Quelques exemples de projections de noeuds :
Noeud Trivial Noeud de Trèfle Noeud en huit Noeud de Potentille Noeud à Trois Torsions
Le Noeud Trivial, parfois appelé le Noeud Non-Noué est tout simplement un cercle, ou sur notre site un rectangle (puisque toutes les lignes se croisent à 90°).
Les trois projections suivantes représentent le même noeud. Lequel?
Le noeud trivial! Arrivez-vous à voir comment on peut les déformer pour obtenir un rectangle?
Quel noeud cette projection représente-t-elle?
Le noeud de trèfle!
Est-ce que vous voyez comment la manipuler pour obtenir la projection de tout à l'heure?
Vous pouvez déformer cette projection en l'autre en retournant tout simplement l'arc supérieur.
→
→
noeud mathématique ( ou noeud, tout simplement) :
the abstract object behind
l'objet abstrait que représentent un nombre infini de projections de noeud qu'on peut déformer et étirer pour les transformer les unes en les autres sans les couper. Exemple : le noeud à trèfles, aussi appelé le noeud 31, le plus simple après le noeud trivial.
isotopie ambiante :
le terme mathématique pour décrire l'équivalence entre deux projections de noeud lorsqu'on peut déformer
une ligne de noeud en une autre de manière continue.
étape :
Notre site dessine les noeuds à l'aide de seulement 6 tuiles que l'on appelle des étapes.
croisement :
Là où deux étapes se croisent
l'une par-dessus l'autre dans une projection. Par exemple :
passage :
une étape qui fait partie d'une croisement.
Il y a des passages par-dessus (la ligne est totalement visible) et par-dessous (la ligne est partiellement recouverte).
inversion :
Inverser des passages dessus-dessous d'un croisement dans une projection.
En générale lorsqu'un croisement est inversé, la projection résultante représente un noeud différent que l'originale.
Le fait d'inverser tous les croisements d'un noeud le change en son image miroir. Certains noeuds et leurs images miroir sont identiques,
tel le noeud en 8. On appelle « achiral » un tel noeud, qui est relié à son image miroir par une isotopie ambiante. On appelle donc « chiral » un noeud qui ne peut pas se transformer ainsi en sont image miroir, dont le noeud de trèfle.
Ce noeud peut-il se transformer en son image miroir?
Ce noeud peut-il se transformer en son image miroir?
Oui! Ce diagramme montre le noeud en 8 qui est achiral et qui peut être déformé
pour obtenir son image miroir par les manipulations suivantes:
Remarquez bien, la seule différence entre le premier diagramme et le deuxième, c'est qu'on a inversé tous les croisements!
Position Initiale
Rotation de 180°
Déplacer un brin
L'image miroir
Remarquez bien, la seule différence entre le premier diagramme et le deuxième, c'est qu'on a inversé tous les croisements!
orientation :
Ce n'est pas une propriété de la ligne du noeud, ni du noeud lui-même. C'est le sens dans lequel on se déplace sur la ligne du noeud : il y en a 2, et donc il y a 2 orientations possibles.
Un noeud qui peut se changer via une isotopie ambiante en lui-même mais avec l'orientation inversée s'appelle un noeud « inversible », sinon il est non-inversible.
Le plus petit noeud non-inversible est 817, un noeud achiral mais qui devient chiral lorsqu'on ajoute une orientation (pour plus d'informations, consultez
la page Wikipédia sur les Noeuds Inversibles). Le fait d'ajouter plus de structure, à l'occurence une orientation, entraîne une perte de symmétrie (il n'est plus identique à son image miroir).
Un noeud chiral, c.-à-d. un noeud ayant la capacité de se tranformer en son image miroir, peut quand même être inversible (symmétrique quant au changement d'orientation). De tels noeuds sont dits « reversibles ».
Un noeud chiral, c.-à-d. un noeud ayant la capacité de se tranformer en son image miroir, peut quand même être inversible (symmétrique quant au changement d'orientation). De tels noeuds sont dits « reversibles ».
chiralité ou polarité de croisement :
Pour toute projection d'un noeud, chaque croisement est soit « à main droite », soit « à main gauche ».
Nous présentons en ce qui suit les deux types de croisements pour ensuite déterminer lequel est à main droite et lequel à main gauche.
Lorsqu'on considère tous les passages par-dessus/par-dessous et les deux orientations,
combien y a-t-il de croisements différents?
Si on ne change rien dans une projection à part une seule inversion de croisement, est-ce que la chiralité de ce croisement change? C'est à dire, est-ce que ce croisement change de groupe?
Comment peut-on se servir de ses mains pour faire la différence entre les croisements à main droite et à main gauche?
Nous présentons en ce qui suit les deux types de croisements pour ensuite déterminer lequel est à main droite et lequel à main gauche.
Lorsqu'on considère tous les passages par-dessus/par-dessous et les deux orientations,
combien y a-t-il de croisements différents?
Au total on énumère 8 possibilités :
Si le passage horizontal est par-dessus alors il y a 4 options :
Pareillement, si le passage verticale est par-dessus, il y a 4 options de plus :
Les croisements 1, 7, 4, 6 appartiennent au groupe des croisements à main droite.
Les croisements 2, 5, 3, 8 appartiennent au groupe des croisements à main gauche.
Si le passage horizontal est par-dessus alors il y a 4 options :
1
2
3
4
5
6
7
8
- Principe n°1 : La chiralité de devra pas dépendre de l'orientation (le sens dans lequel on avance sur la ligne du noeud), alors en inversant les flèches on identifie 4 paires de croisements équivalents: 1 = 4, 2 = 3, 5 = 8, 6 = 7. Donc les groupes qu'on identifie à la fin devraient répertorier les croisements 1 et 4 dans le même group, et ainsi de suite.
- Principe n°2 : Le groupe auquel appartient un croisement ne devrait pas changer après la rotation du noeud entier. Alors, les croisements 1 = 7 = 4 = 6 et 2 = 5 = 3 = 8.
Les croisements 1, 7, 4, 6 appartiennent au groupe des croisements à main droite.
Les croisements 2, 5, 3, 8 appartiennent au groupe des croisements à main gauche.
Si on ne change rien dans une projection à part une seule inversion de croisement, est-ce que la chiralité de ce croisement change? C'est à dire, est-ce que ce croisement change de groupe?
Oui! Essayez-le avec n'importe lequel des 8 types de croisement. Par exemple, si on inverse le croisement 1 on obtient le croisement 5, c.-à-d. un croisement dans l'autre groupe de chiralité.
Une Assertion : Le fait qu'un croisement soit à main droite ou gauche ne dépend pas uniquement du croisement en lui-même mais la projection qui l'entoure.
Démonstration : Il n'importe pas à la chiralité si le passage horizontal se trouve dessus ou dessous. Les deux cas peuvent être à main droite comme à main gauche (voir les 8 croisements ci-dessus). Si on fait tourner le noeud pour que le passage par-dessus soit horizontal et si on avance en partant de ce passage vers la droite (dans le sens est) alors il dépend du reste du noeud si on retourne à ce croisement du sud (dans ce cas le croisement est à main droite) ou du nord (dans ce cas, le croisement est à main gauche).
Il y a une raison pour laquelle on appelle ces deux groupes à main droite et à main gauche. Indice : vous pouvez distinguer entre les deux types de croisement avec vos mains!
Une Assertion : Le fait qu'un croisement soit à main droite ou gauche ne dépend pas uniquement du croisement en lui-même mais la projection qui l'entoure.
Démonstration : Il n'importe pas à la chiralité si le passage horizontal se trouve dessus ou dessous. Les deux cas peuvent être à main droite comme à main gauche (voir les 8 croisements ci-dessus). Si on fait tourner le noeud pour que le passage par-dessus soit horizontal et si on avance en partant de ce passage vers la droite (dans le sens est) alors il dépend du reste du noeud si on retourne à ce croisement du sud (dans ce cas le croisement est à main droite) ou du nord (dans ce cas, le croisement est à main gauche).
Il y a une raison pour laquelle on appelle ces deux groupes à main droite et à main gauche. Indice : vous pouvez distinguer entre les deux types de croisement avec vos mains!
Comment peut-on se servir de ses mains pour faire la différence entre les croisements à main droite et à main gauche?
Tenez vos doigts ensemble pour qu'ils soient parallèles, et votre pouce à 90° pour qu'il soit perpendiculaire aux doigts. Avec votre paume face à vous, pointez votre pouce dans le sens où se dirige le passage par-dessus, et vos doigts dans le sens où se dirige le passage par-dessous. Pour chaque croisement, ce n'est possible qu'avec une main, d'où le groupe à main droite ou à main gauche!
Par exemple, pour le croisement ci-dessous, vous vous tiendriez la main ainsi :
Vous ne pouvez faire ceci qu'avec la main gauche alors c'est un croisement à main gauche.
La chiralité des croisements est également connue en tant que polarité des croisements, où les croisements à main droite sont « positifs » et ceux à main gauche « négatifs ».
Vous ne pouvez faire ceci qu'avec la main gauche alors c'est un croisement à main gauche.
La chiralité des croisements est également connue en tant que polarité des croisements, où les croisements à main droite sont « positifs » et ceux à main gauche « négatifs ».
entortillement :
aussi appelé « la vrille », c'est une propriété des projections de noeud qui décrit la différence entre le nombre de croisements à main gauche et à main droite dans la projection. Ce nombre peut différer entre deux projections d'un même noeud.
Quel est l'entortillement de la projection ci-dessus?
Quel est l'entortillement de la projection ci-dessus?
On a colorié les croisements à main droite en rouge et ceux à main gauche en vert. Alors, il suffit de compter le nombre de croisements de chaque groupe et soustraire pour trouver la différence.
Cette projection a 2 croisements à main gauche et 4 à main droite, alors l'entortillement est 2 − 4 =
−2.
invariant de noeud :
il peut s'agir d'un nombre, d'un polynôme, ou d'une assertion de faisabilité qui caractérise toutes les projections (infiniment nombreuses) d'un même noeud. Par exemple, les propriétés invariantes d'un noeud comprennent s'il est chiral ou achiral, inversible ou non-inversible, et s'il est reversible.
Nombre minimal de croisements :
C'est une caractéristique invariable de chaque noeud qui dénote le nombre de croisements minimum nécessaires pour le dessiner.
Combien cette projection a-t-elle de croisements?
Quel est le nombre minimal de croisements du noeud représenté par la projection?
Quels sont les deux plus petits nombres de croisement que peut avoir un noeud?
Combien cette projection a-t-elle de croisements?
Elle a 5 croisements.
Quel est le nombre minimal de croisements du noeud représenté par la projection?
Zéro! Le nombre minimal de croisements est une propriété du noeud abstrait et non de telle ou telle projection de ce noeud.
On peut déformer la projection ci-haut pour obtenir le noeud trivial,
,
qui n'a aucun croisement.
Est-ce que vous voyez pourquoi?
Puisque le nombre minimum de croisements est le plus petit nombre de croisements que peut avoir la projection d'un certain noeud, lorsqu'on ne peut plus le décomposer, le nombre minimum de croisements du noeud représenté par cette projection est zéro.
,
qui n'a aucun croisement. Est-ce que vous voyez pourquoi?
Puisque le nombre minimum de croisements est le plus petit nombre de croisements que peut avoir la projection d'un certain noeud, lorsqu'on ne peut plus le décomposer, le nombre minimum de croisements du noeud représenté par cette projection est zéro.
Quels sont les deux plus petits nombres de croisement que peut avoir un noeud?
Le plus petit nombre minimum de croisements possible est 0, ce qui caractérise le noeud trivial.
Un noeud à 1 croisement, est-ce possible? Voici une projection de noeud à 1 croisement :
Cependant, elle peut se déformer en le noeud trivial. Et à 2 croisements? Une projection à 2 croisements, telle que
peut également se faire décomposer pour obtenir le noeud trivial.
Et donc, un noeud à 3 croisements, est-ce possible? Oui! Vous en avez déjà vu ci-haut : la projection du noeud de trèfle a 3 croisements. Il est impossible de le déformer en le noeud trivial, alors les plus petits nombre de croisement possible sont 0 et 3.
Cependant, elle peut se déformer en le noeud trivial. Et à 2 croisements? Une projection à 2 croisements, telle que
peut également se faire décomposer pour obtenir le noeud trivial.Et donc, un noeud à 3 croisements, est-ce possible? Oui! Vous en avez déjà vu ci-haut : la projection du noeud de trèfle a 3 croisements. Il est impossible de le déformer en le noeud trivial, alors les plus petits nombre de croisement possible sont 0 et 3.
arc :
La partie de la ligne du noeud dans une projection qui relie un croisement au prochain.
Combien d'arcs y a-t-il dans une projection à N croisements?
Combien d'arcs y a-t-il dans une projection à N croisements?
Chaque croisement implique 4 bouts d'arc. Chaque arc ayant 2 bouts, il y a donc 4/2 = 2 fois plus d'arcs que de croisements, alors 2N arcs.
trou :
L'espace vide entouré par des arcs dans une projection. L'espace extérieur à la projection est aussi un trou.
Combien y a-t-il de trous dans une projection à N croisements?
Combien y a-t-il de trous dans une projection à N croisements?
On pourrait dessiner quelques noeuds pour ensuite deviner une formule, mais on peut également la dériver. Selon la formule d'Euler, pour tout dessin sur un plan où m lignes (ici 2N arcs) relient 2 sur n points (ici N croisemens) alors le nombre f de faces (ici trous) est égal à f = 2 + m − n. Alors pour le nombre de trous d'un noeud : 2 + 2N − N = N + 2.
brin par-dessus :
Une suite d'arcs consécutifs dans une projecton (c.-à-d. d'arcs à la queue leu leu) qui commence et termine à un passage par-dessous et qui comprend 0, 1, ou plusieurs passages par-dessus.
brin par-dessous :
Une suite d'arcs consécutifs dans une projecton (c.-à-d. des arcs les uns après les autres) qui commence et termine à un passage par-dessus et qui comprend 0, 1, ou plusieurs passages par-dessous.
Dans la litérature sur la théorie des noeuds, on appelle souvent « un brin » ce qu'ici on distingue en tant que « brin par-dessus ». Pour notre discussion, il est aussi important de considérer le nombre de passages par-dessus qu'a un brin par-dessus que le nombre de passages par-dessous qu'a un brin par-dessous. Alors on fait la différence entre les brins par-dessus et par-dessous.
À droite il y a une ligne horizontale comprenant cinq arcs : est-ce un brin par-dessus ou par-dessous?
Combien de brins par-dessous y a-t-il dans une projection à N croisements?
Dans la litérature sur la théorie des noeuds, on appelle souvent « un brin » ce qu'ici on distingue en tant que « brin par-dessus ». Pour notre discussion, il est aussi important de considérer le nombre de passages par-dessus qu'a un brin par-dessus que le nombre de passages par-dessous qu'a un brin par-dessous. Alors on fait la différence entre les brins par-dessus et par-dessous.
À droite il y a une ligne horizontale comprenant cinq arcs : est-ce un brin par-dessus ou par-dessous?
C'est un brin par-dessous à 4 passages par-dessous.
Combien de brins par-dessous y a-t-il dans une projection à N croisements?
À chaque croisement il y a 2 bouts de brin (soit un bout de 2 brins différents, soit les deux bouts d'un seul brin).
D'autre part, chaque brin a 2 bouts, chacun se terminant à un croisement. Alors le nombre de croisements est égal au nombre de brins par-dessus et, vue la symétrie, au nombre de brins par-dessous aussi. Il y a donc N brins de chaque type.
Mouvements de Reidemeister :
En 1927 le mathématicien allemand Kurt Reidemeister, et, indépendamment, James Waddell Alexander et Garland Baird Briggs en 1926, ont démontré que deux diagrammes de noeuds représentant le même noeud peuvent se déformer l'un en l'autre par une suite de 3 types de mouvements.
Or, il reste plusieurs lacunes, par exemple durant cette déformation le nombre de croisements peut augmenter temporairement, mais la limite supérieure de cette augmentation reste inconnue ainsi que le nombre nécessaire de mouvements. Les 3 mouvements sont numérotés selon le nombre de brins impliqués.
Mouvement Reidemeister de type 1 (R1) :
création ou suppression d'une boucle (un trou renfermé par un seul arc).
Quelle projection montre un croisement à main gauche? Et à main droite?
Quelle projection montre un croisement à main gauche? Et à main droite?
La projection à gauche montre un croisement à main gauche, et celle à droite montre un croisement à main droite. Un mouvement Reidemeister de type 1 change par 1 le nombre de croisements à main gauche ou droite et change donc l'entortillement du diagramme.
Mouvement Reidemeister de type 2 (R2) :
création ou suppression de deux croisements « jumeaux » dessus-dessus ou dessous-dessous (un trou renfermé par deux arcs).
Que remarquez-vous par rapport à la chiralité/polarité des deux croisements créés/supprimés par un mouvement R2?
Que remarquez-vous par rapport à la chiralité/polarité des deux croisements créés/supprimés par un mouvement R2?
Un des deux croisements est à main droite, et l'autre à main gauche. Alors, ce mouvement n'affecte pas l'entortillement du diagramme.
Mouvement Reidemeister de type 3 (R3) :
création et suppression d'un trou renfermé par 3 arcs (déplacement d'un brin passant au-dessus ou en-dessous d'un croisement).
Quels sont les 2 types de trous renfermés par 3 arcs?
Quels sont les 2 types de trous renfermés par 3 arcs?
Soit :
Vérifiez que le résultat des 3 mouvements est toujours pareil.
Est-ce qu'un mouvement R3 change la chiralité/polarité des 3 croisements?
Qu'est-ce qu'on a appris sur les mouvements Reidemeister de type 3?
- 1) chaque arc a un passage par-dessus et un passage par-dessous.
Alors on ne peut déplacer aucun des 3 arcs par-dessus/par-dessous/à travers l'autre croisement.
- 2) e premier arc a 2 passages par-dessus, le deuxième a 1 passage par-dessus et 1 passage par-dessous, et le troisième a 2 passages par-dessous.
Alors il y a 3 mouvements de type R3 possible. On peut (1) déplacer le brin dessus-dessus qui reste un brin dessus-dessus :
On peut (2) déplacer le brin dessous-dessus qui devient un brin dessus-dessous :
Ou bien on peut (3) déplacer le brin dessous-dessous qui reste un brin dessous-dessous :
Vérifiez que le résultat des 3 mouvements est toujours pareil.
Lorsqu'on compare les diagrammes sur le côté droit des mouvements ci-haut, il est évident que les trois produisent des résultats identiques. Alors, s'il y a un mouvement de type R3, il y a un seul résultat. La seule chose qui change, c'est le fait que pour chacun des trois arcs, les deux autres se croisent en l'ordre inverse. Ceci veut dire que pour l'arc au milieu des deux, l'ordre des passages par-dessus et par-dessous est inversé.
Est-ce qu'un mouvement R3 change la chiralité/polarité des 3 croisements?
Non. Attribuez aux brins ci-haut une orientation quelconque et comparez ensuite la chiralité des croisements pour confirmer ceci.
Qu'est-ce qu'on a appris sur les mouvements Reidemeister de type 3?
On a appris ::
- comment répérer des trous à 3 arcs qui permettent un mouvement R3,
- que pour un tel trou, on peut effectuer au plus UN mouvement R3,
- que la chiralité/polarité des 3 croisements reste inchangée,
- que l'ordre des passages par-dessus/par-dessous est inversé pour l'arc au milieu.
Un mouvement à passage :
Ceci n'a rien à voir avec les 'passages' définis ci-haut. Un mouvement à passage (pass move en anglais) remplace un brin par-dessus/dessous avec un autre brin par-dessus/dessous où les deux brins ont les mêmes bouts. Pour des exemples, voir les définitions de P-, P0 et P+ ci-dessous.
mouvement P- :
un mouvement à passage où le brin résultant a moins de passages que l'original.
Trouvez un mouvement P- remplaçant le brin vert dans le diagramme ci-dessous.
Trouvez un mouvement P- remplaçant le brin vert dans le diagramme ci-dessous.
Dans ce diagramme le nouveau brin rouge a moins de passages que le brin vert original. Alors ce diagramme montre un mouvement P-.
mouvement P0 :
un mouvement à passage où le brin résultant a le même nombre de passages que l'original.
Trouvez un mouvement P0 remplaçant le brin vert dans le diagramme ci-dessous.
Trouvez un mouvement P0 remplaçant le brin vert dans le diagramme ci-dessous.
Dans ce diagramme le nouveau brin rouge a autant de passages que le brin vert original. Alors ce diagramme montre un mouvement P0.
mouvement P+ :
un mouvement à passage où le brin résultant a plus de passages que l'original.
Trouvez un mouvement P+ remplaçant le brin vert dans le diagramme ci-dessous.
Les mouvements P+ deviennent nécessaires lorsqu'on souhaite changer l'entortillement d'un diagramme. On discutera ce sujet plus tard dans la section « Trouver des Mouvements P0 ».
Trouvez un mouvement P+ remplaçant le brin vert dans le diagramme ci-dessous.
Dans ce diagramme le nouveau brin rouge a plus de passages que le brin vert original. Alors ce diagramme montre un mouvement P+.
Les mouvements P+ deviennent nécessaires lorsqu'on souhaite changer l'entortillement d'un diagramme. On discutera ce sujet plus tard dans la section « Trouver des Mouvements P0 ».
nombre de dénouement:
C'est une propriété d'un noeud et non de ses diagrammes. Il fait partie donc des « invariants » d'un noeud, car il est indépendant de sa projection.
Le nombre minimum de fois qu'on doit inverser des croisements pour obtenir le noeud trivial (c.-à-d. le cercle, noeud sans aucun croisement). Avant la première inversion et entre toutes les autres, le diagramme peut subir des déformations arbitraires.
Pourquoi le noeud de trèfle a-t-il le nombre de dénouement 1?
Le nombre minimum de fois qu'on doit inverser des croisements pour obtenir le noeud trivial (c.-à-d. le cercle, noeud sans aucun croisement). Avant la première inversion et entre toutes les autres, le diagramme peut subir des déformations arbitraires.
Pourquoi le noeud de trèfle a-t-il le nombre de dénouement 1?
Le noeud de trèfle ne peut pas avoir le nombre de dénouement 0 parce qu'il est impossible de le faire transformer en le noeud trivial qui a, quant à lui, le nombre de dénouement 0. Alors, le noeud de trèfle a forcément le nombre de dénouement ≥1. D'autre part, c'est facile de voir qu'il suffit d'inverser n'importe quel croisement sur la projection du noeud de trèfle vue ci-haut pour obtenir le noeud trivial, alors son nombre de dénouement est forcément ≤1. S'il est à la fois ≥1 et ≤1 alors il doit être =1.
Comment Simplifier les Diagrammes de Noeud
Trouver des Mouvements R1
Des cas simples de mouvements R1, comme celui-ci :
où il suffit de retourner une boucle 4 fois pour obtenir le noeud trivial
sont faciles à répérer. Pour ce faire, on doit suivre la ligne du noeud pour chercher un arc dont les deux bouts terminent au même croisement. L'ordre des mouvements R1 n'importe pas.
Cependant, il existe des cas plus généraux de l'application des mouvements de type Reidemeister 1. Si un brin par-dessus commence comme passage par-dessus à un croisement, mais reste par-dessus tous les autres arcs (d'où «brin par-dessus»), alors cette boucle peut certainement être raccourcie et donc supprimée. Par exemple, ici on peut supprimer la boucle au centre de ce diagramme pour commencer, puis toutes les autres boucles là-dessous, une par une, ainsi :
C'est pareil si la boucle se trouve sous toutes les autres :
où il suffit de retourner une boucle 4 fois pour obtenir le noeud trivial
sont faciles à répérer. Pour ce faire, on doit suivre la ligne du noeud pour chercher un arc dont les deux bouts terminent au même croisement. L'ordre des mouvements R1 n'importe pas. Cependant, il existe des cas plus généraux de l'application des mouvements de type Reidemeister 1. Si un brin par-dessus commence comme passage par-dessus à un croisement, mais reste par-dessus tous les autres arcs (d'où «brin par-dessus»), alors cette boucle peut certainement être raccourcie et donc supprimée. Par exemple, ici on peut supprimer la boucle au centre de ce diagramme pour commencer, puis toutes les autres boucles là-dessous, une par une, ainsi :
C'est pareil si la boucle se trouve sous toutes les autres :
Trouver des Mouvements R2
Tout comme pour les mouvements R1, il est assez facile à trouver des possibilités pour des mouvements R2 comme ci-dessous où il faut effectuer deux mouvements R2 de suite avant qu'un mouvement R1 puisse donner le noeud trivial :
Pour le prochain exemple, il faut effectuer deux mouvement R2 de suite sur le même brin, d'abord le passant par dessous un brin, puis par dessus un autre :
La dernière étape n'est pas un mouvement R2. On l'inclut uniquement pour démontrer que ce noeud est la somme de deux noeuds de trèfle.
Pour le prochain exemple, il faut effectuer deux mouvement R2 de suite sur le même brin, d'abord le passant par dessous un brin, puis par dessus un autre :
La dernière étape n'est pas un mouvement R2. On l'inclut uniquement pour démontrer que ce noeud est la somme de deux noeuds de trèfle.
À Propos de l'Interface (1)
L'exemple ci-haut convient bien pour démontrer l'usage optimal de notre interface. Si on fait un clic double pour «couper» la ligne du noeud, on obtient :
Ensuite, il faut effectuer chaque mouvement R2 un par un. On ne peut pas, par exemple, faire reculer un brin sous le premier brin et sous l'autre avant de faire reculer l'autre. Il faut d'abord faire déplacer le premier brin par-dessous pour supprimer un passage par-dessous et mettre les deux extrémités en l'état «par-dessous» ; dans cet état il est impossible de supprimer un passage par-dessus. Il faut changer de bout et déplacer l'autre extrémité vers le même trou, tout en supprimant le deuxième passage par-dessous pour se mettre en l'état «neutre». Ceci fait, on peut faire reculer les deux bouts pour supprimer les deux passages par-dessus et les raboutir. Pour résumer, on doit changer de bout pour enlever d'abord tous les passages par-dessous, puis tous les passages par-dessus, et ainsi de suite.
L'implémentation de ces «états» des extrémités n'est pas une défaillance du logiciel, c'est fait exprès pour qu'on puisse modifier de manière interactive un diagramme sans changer le noeud mathématique représenté.
Ensuite, il faut effectuer chaque mouvement R2 un par un. On ne peut pas, par exemple, faire reculer un brin sous le premier brin et sous l'autre avant de faire reculer l'autre. Il faut d'abord faire déplacer le premier brin par-dessous pour supprimer un passage par-dessous et mettre les deux extrémités en l'état «par-dessous» ; dans cet état il est impossible de supprimer un passage par-dessus. Il faut changer de bout et déplacer l'autre extrémité vers le même trou, tout en supprimant le deuxième passage par-dessous pour se mettre en l'état «neutre». Ceci fait, on peut faire reculer les deux bouts pour supprimer les deux passages par-dessus et les raboutir. Pour résumer, on doit changer de bout pour enlever d'abord tous les passages par-dessous, puis tous les passages par-dessus, et ainsi de suite.
L'implémentation de ces «états» des extrémités n'est pas une défaillance du logiciel, c'est fait exprès pour qu'on puisse modifier de manière interactive un diagramme sans changer le noeud mathématique représenté.
Trouver des Mouvements P-
Un mouvement P- est un mouvement qui remplace un brin par-dessus par un autre brin par-dessus ayant moins de passages par-dessus ou bien un mouvement qui remplace un brin par-dessous par un autre brin par-dessous ayant moins de passages par-dessous. Dans les deux cas, le nombre de croisements est réduit.
Pour trouver de tels mouvements, on marche sur la ligne du noeud en cherchant le plus grand nombre possible de passages par-dessus consécutifs ou de passages par-dessous consécutifs (au moins deux). Si on retrouve un tel brin, disons par exemple un brin par-dessus alors on essaie de trouver une route alternative avec moins de passages par-dessus reliant ces mêmes bouts de passage par-dessous.
Dans l'exemple suivant, on remplace un brin à 4 passages par-dessous par un brin sans aucun passage par-dessous, pour ensuite remplacer ce brin à 3 passages par-dessus consécutifs par un brin à 1 seul passage par-dessus. On effectue encore 2 passages P- pour supprimer 2 croisements de plus. Le diagramme résultant se simplifie encore plus à l'aide de deux mouvements R1, comme montré ci-dessous.
Plus il y a de passages d'un même type consécutifs sur un brin, plus c'est probable qu'on retrouve une déviation alternative ayant moins de passages.
Pour trouver de tels mouvements, on marche sur la ligne du noeud en cherchant le plus grand nombre possible de passages par-dessus consécutifs ou de passages par-dessous consécutifs (au moins deux). Si on retrouve un tel brin, disons par exemple un brin par-dessus alors on essaie de trouver une route alternative avec moins de passages par-dessus reliant ces mêmes bouts de passage par-dessous.
Dans l'exemple suivant, on remplace un brin à 4 passages par-dessous par un brin sans aucun passage par-dessous, pour ensuite remplacer ce brin à 3 passages par-dessus consécutifs par un brin à 1 seul passage par-dessus. On effectue encore 2 passages P- pour supprimer 2 croisements de plus. Le diagramme résultant se simplifie encore plus à l'aide de deux mouvements R1, comme montré ci-dessous.
Trouver des Mouvements R3
Les mouvements de type R1, R2 et P- changent le nombre de croisements. Les mouvements R3, ayant aucun effet sur le nombre de croisements, sont donc décrits ici, en dessous de la description des mouvements de type P-. Si les mouvements R3 ne font pas baisser le nombre de croisements, à quoi servent-ils? Ces mouvements sont quand même utiles puisqu'ils peuvent permettre de débloquer des mouvements P-.
Comme vous pouvez lire plus haut dans les définitions, un trou R3 est un trou délimité par trois arcs, ici un arc supérieur dont les bouts sont deux passages par-dessus (A,B), un arc au milieu dont les bouts sont un passage par-dessus (C) et un passage par-dessous (B), ainsi qu'un arc inférieur dont les bouts sont deux passages par-dessous (A,C).
(ce noeud trivial a été pris sur la page Wikipédia sur le Noeud Trivial)
On a découvert dans la définition des mouvements R3 que ce type de mouvement inverse l'ordre des passages par-dessus et par-dessous pour le brin du milieu. Un mouvement R3 est avantageux lorsque cette inversion fait augmenter le nombre de passages du même type consécutifs sur l'arc du milieu, ce qui dont peut permettre d'effectuer un mouvement P-. Ceci est le cas si, dans la continuation de l'arc du milieu après le passage par-dessous B, on retrouve un passage par-dessus (D et E sont des passages par-dessus) et/ou si après le passage par-dessus C on retrouve un passage par-dessous (F, G, et H sont des passages par-dessous). Le mouvement R3 a pour effet de faire glisser l'arc du milieu BC entre l'arc supérieur et l'arc inférieur au croisement A.
Avant il n'y avait que 2 passages par-dessus consécutifs aux croisements D et E, maintenant il y en a 3 aux croisements C, D, et E. Ce brin par-dessus plus long peut donc se rediriger à l'aide d'un mouvement P- :
ce qui réduit le nombre de croisements de 2. Il reste donc 13 &minusl 2 = 11 croisements.
L'autre côté du brin peut aussi être reconduit par un mouvement P-. Avant, il y avait 3 passages par-dessous aux croisements F, G, et H, maintenant il y en a 4 aux croisements B, F, G, et H. On obtient donc après le mouvement P- :
ce qui réduit donc aussi le nombre de croisements de 2. Ces deux diagrammes peuvent se simplifier encore plus à travers des mouvements P- et R1 pour donner enfin le noeud trivial. Essayez de le faire vous-même à l'aide des indices ci-haut pour reconnaître des mouvements P-.
Pratiquons à l'aide d'un exemple.
Combien y a-t-il de mouvements R3 possibles dans le diagramme suivant?
On a découvert dans la définition des mouvements R3 que ce type de mouvement inverse l'ordre des passages par-dessus et par-dessous pour le brin du milieu. Un mouvement R3 est avantageux lorsque cette inversion fait augmenter le nombre de passages du même type consécutifs sur l'arc du milieu, ce qui dont peut permettre d'effectuer un mouvement P-. Ceci est le cas si, dans la continuation de l'arc du milieu après le passage par-dessous B, on retrouve un passage par-dessus (D et E sont des passages par-dessus) et/ou si après le passage par-dessus C on retrouve un passage par-dessous (F, G, et H sont des passages par-dessous). Le mouvement R3 a pour effet de faire glisser l'arc du milieu BC entre l'arc supérieur et l'arc inférieur au croisement A.
L'autre côté du brin peut aussi être reconduit par un mouvement P-. Avant, il y avait 3 passages par-dessous aux croisements F, G, et H, maintenant il y en a 4 aux croisements B, F, G, et H. On obtient donc après le mouvement P- :
Pratiquons à l'aide d'un exemple.
Combien y a-t-il de mouvements R3 possibles dans le diagramme suivant?
Trois mouvements R3 sont possibles. Pour chaque possibilité, on colorie en vert ci-bas les trois arcs concernés par le mouvement R3. La troisième possibilité est celle qui est la plus difficile à répérer, car le trou est l'espace entier «entouré» par seulement 3 arcs.
Effectuez le 1er mouvement R3 et déterminez s'il est avantageux :
Effectuez le 2ème mouvement R3 et déterminez s'il est avantageux :
Effectuez le 3ème mouvement R3 et déterminez s'il est avantageux :
1er mouvement R3
2ème mouvement R3
3ème mouvement R3
Effectuez le 1er mouvement R3 et déterminez s'il est avantageux :
This R3 move is beneficial. It allows afterwards a P- moves as shown in a sequence of moves further
below.
Our definition for an R3 move to be beneficial it is not
neccessarily to allow a P- move but to increase the number of consecutive
over- or under-passes and that is easy to see even without
performing all these moves. In the following diagram the middle arc
of the R3 hole has an
over-pass at A, an under-pass at B followed by two over-passes at C and D.
In an R3 move the order of over- and under-pass is reversed for the middle arc
as shown in the Diagram 3 with now 3 consecutive over-passes. This is enough
to find a P- move needing less than 3 passes in Diagram 5.
About the sequence of diagrams below: In Dia 1 we make space to prepare the R3 move in Dia 2 (here by
moving
the top arc) with the result in Dia 3. In Dia 4 we make space to prepare
the P- move in Dia 5 where the green strand with 3 over-passes is replaced
by the red strand with only 1 over-pass in Dia 6. In Dia 7 we shift a strand
to make space for the next P- move in Dia 9 with result in Dia 10 and Dia 11
after shrinking which is easily identified as knot 51.
1. Élargissement
2. Mouvement R3
3. Après R3
4. Élargissement
5. Mouvement P-
6. Après P-
7. Pour le prochain mouvement P-
8. Avant le mouvement P-
9. La 2ème Mouvement P-
10. Après P-
11. condensé
Effectuez le 2ème mouvement R3 et déterminez s'il est avantageux :
La séquence suivante montre que le mouvement R3 est avantageux.
1. Élargissement
2. Mouvement R3
3. Après R3
4. Mouvement P-
5. Après P-
6. Encore un Mouvement P-
7. Après P-
8. Condensé
Effectuez le 3ème mouvement R3 et déterminez s'il est avantageux :
Le 3ème mouvement R3 est avantageux aussi. Pour effectuer ce mouvement, on suit le même principe : le brin du milieu coupe les deux autres brins, qui «entourent» donc le trou extérieur dans l'ordre inverse.
Le résultat, c'est que lorsqu'on marche sur le brin vert vers le haut, on rencontre d'abord un passage par-dessus puis un passage par-dessous. Si, après le mouvement R3, on marche sur le brin rouge, on passe par les 2 autres brins dans l'ordre inverse et alors on rencontre un passage par-dessous puis un passage par-dessous. Comme l'on peut voir dans le Diagramme n°5, la séquence de 3 passages par-dessus consécutifs permet un mouvement P-.
Le résultat, c'est que lorsqu'on marche sur le brin vert vers le haut, on rencontre d'abord un passage par-dessus puis un passage par-dessous. Si, après le mouvement R3, on marche sur le brin rouge, on passe par les 2 autres brins dans l'ordre inverse et alors on rencontre un passage par-dessous puis un passage par-dessous. Comme l'on peut voir dans le Diagramme n°5, la séquence de 3 passages par-dessus consécutifs permet un mouvement P-.
1. Élargissement
2. Mouvement R3
3. Après R3
4. Élargissement
5. Mouvement P-
6. Élargissement
7. Un 2ème Mouvement P-
8. Après P-
9. Raccourcissement
10. Après raccourcissement
11. Raccourcissement
12. Redressement
13. Rotation de ↻90°
À Propos de l'Interface (2)
L'exemple ci-dessus convient à démontrer comment on peut effectuer un mouvement R3 avec l'interface du jeu.
Comme décrit ci-haut où on définit les mouvements Reidemeister de type 3, il y a en effet 3 façons d'effectuer un mouvement R3 : en déplaçant le brin inférieur, celui du milieu, ou le brin supérieur. Comme on a vu, le résultat est le même quelle que soit la façon choisie.
Notre interface permet d'effectuer un mouvement R3 en déplaçant le brin inférieur ou le brin supérieur, mais elle ne permet pas de l'effectuer en déplaçant le brin du milieu. La raison, c'est que dans cette interface les deux extrémités «coupées» ne peuvent ajouter ou supprimer que des passages du même type (soit par-dessus, soit par-dessous) en même temps. Mais, ceci ne vous empêchera pas d'effectuer des mouvements de type R3 car le résultat est le même lorsqu'on déplace les brins supérieur et inférieur.
Comme décrit ci-haut où on définit les mouvements Reidemeister de type 3, il y a en effet 3 façons d'effectuer un mouvement R3 : en déplaçant le brin inférieur, celui du milieu, ou le brin supérieur. Comme on a vu, le résultat est le même quelle que soit la façon choisie.
Notre interface permet d'effectuer un mouvement R3 en déplaçant le brin inférieur ou le brin supérieur, mais elle ne permet pas de l'effectuer en déplaçant le brin du milieu. La raison, c'est que dans cette interface les deux extrémités «coupées» ne peuvent ajouter ou supprimer que des passages du même type (soit par-dessus, soit par-dessous) en même temps. Mais, ceci ne vous empêchera pas d'effectuer des mouvements de type R3 car le résultat est le même lorsqu'on déplace les brins supérieur et inférieur.
Trouver des Mouvements P0
Les mouvements P0 sont des mouvements à passage qui ne modifient pas le nombre de croisements, tout comme les mouvements R3 qui sont effectivement une version spéciale de mouvements P0. Comme les mouvements R3, les mouvements P0 peuvent être quand même utiles en déploquant un mouvement P-. Puisque les mouvements P0 sont moins utiles en moyenne, ils apparaissent plus souvent, mais il n'est pas toujours facile de vérifier s'ils permettent un mouvement P-.
Pour identifier un mouvement P0 possible, on doit chercher un brin par-dessus ou par-dessous, comme lorsqu'on cherche un mouvement P-. Pour vérifier si un mouvement P0 pourrait être avantageux et permettre un mouvement P-, on suit le même principe que pour un mouvement R3. On vérifie donc si la suppression d'un brin augmente le nombre de passages de même type consécutifs et si le brin résultant en aura plus. Dans tous les cas, on doit aussi vérifier s'il est possible de rediriger les brins avec plus de passages de même type consécutifs pour obtenir des brins avec moins de croisements.
Prenez cet exemple :
On étiquète les croisements :
et on cherche un mouvement P0 avantageux pas à pas.
Combien voyez-vous de brins par-dessus avec au moins 2 passages par-dessus et combien voyez-vous de brins par-dessous avec au moins 2 passages par-dessous?
Il n'est pas difficile de voir que le brin IE a un mouvement P0 qui le redirigera pour qu'il traverse par-dessus les brins GC et BH :
mais il reste à savoir si ce mouvement P0 est avantageux.
Est-ce que le fait de déplacer le brin IE a fait augmenter le nombre de passages par-dessus/dessous consécutifs sur les brins DF ou HJ déjà croisés?
Est-ce le fait de déplacer le brin sur les brins GC et BH a créé plus de passages par-dessus/dessous consécutifs?
Est-ce qu'on peut rediriger ce brin BH par un mouvement P- de manière à réduire le nombre de croisements?
À partir du diagramme original, il est possible de compléter ce mouvement P- en effectuant d'abord un mouvement P0 suivi par le mouvement P0 mentionné ci-haut comme mouvement P- pour le même résultat : 2 croisements de moins.
Face à des problèmes difficiles il peut être nécessaire d'effectuer plusieurs mouvements P0 avant qu'un mouvement P- devienne possible.
Un diagramme est simplifié de manière maximale lorsque le nombre de croisements du diagramme est égal au nombre minimal de croisements (voir dans les définitions). Dans ce cas, les mouvements P0 ne permettront jamais un mouvement P-.
Pour identifier un mouvement P0 possible, on doit chercher un brin par-dessus ou par-dessous, comme lorsqu'on cherche un mouvement P-. Pour vérifier si un mouvement P0 pourrait être avantageux et permettre un mouvement P-, on suit le même principe que pour un mouvement R3. On vérifie donc si la suppression d'un brin augmente le nombre de passages de même type consécutifs et si le brin résultant en aura plus. Dans tous les cas, on doit aussi vérifier s'il est possible de rediriger les brins avec plus de passages de même type consécutifs pour obtenir des brins avec moins de croisements.
Prenez cet exemple :
Combien voyez-vous de brins par-dessus avec au moins 2 passages par-dessus et combien voyez-vous de brins par-dessous avec au moins 2 passages par-dessous?
On obtient trois brins par-dessus avec au moins 2 passages par-dessus : AB, GC, et IE. Pareillement, il y a 3 brins par-dessous avec au moins 2 passages par-dessous : EF, BH, et DJ.
Il n'est pas difficile de voir que le brin IE a un mouvement P0 qui le redirigera pour qu'il traverse par-dessus les brins GC et BH :
Est-ce que le fait de déplacer le brin IE a fait augmenter le nombre de passages par-dessus/dessous consécutifs sur les brins DF ou HJ déjà croisés?
Est-ce que le fait de déplacer le brin IE a fait augmenter le nombre de passages par-dessus/dessous consécutifs sur les brins DF ou HJ déjà croisés?
Est-ce le fait de déplacer le brin sur les brins GC et BH a créé plus de passages par-dessus/dessous consécutifs?
Oui, le brin par-dessous BH avait 2 passages par-dessous, et il en a 3 maintenant.
Est-ce qu'on peut rediriger ce brin BH par un mouvement P- de manière à réduire le nombre de croisements?
Est-ce qu'on peut rediriger ce brin BH par un mouvement P- de manière à réduire le nombre de croisements?
Le fait que le nouveau brin soit plus long (c.-à-d. a plus d'étapes) dans ce diagramme que le brin remplacé n'est pas important. Tout ce qui compte, c'est la réduction du nombre de croisements de 10 à 8 qui permet d'identifier ce noeud comme le noeud 817. À partir du diagramme original, il est possible de compléter ce mouvement P- en effectuant d'abord un mouvement P0 suivi par le mouvement P0 mentionné ci-haut comme mouvement P- pour le même résultat : 2 croisements de moins.
Face à des problèmes difficiles il peut être nécessaire d'effectuer plusieurs mouvements P0 avant qu'un mouvement P- devienne possible.
Un diagramme est simplifié de manière maximale lorsque le nombre de croisements du diagramme est égal au nombre minimal de croisements (voir dans les définitions). Dans ce cas, les mouvements P0 ne permettront jamais un mouvement P-.
Trouver des Mouvements U1
Un mouvement U1 inverse un croisement, ce qui permet de simplifier le diagramme pour supprimer tous les croisements et démontrer que l'inversion produit le noeud trivial.
En général, des casse-têtes intéressants devraient avoir des solutions uniques alors nos casse-têtes de type U1 et U2 ne donnent que des diagrammes où l'inversion d'un seul croisement produit le noeud trivial. Cet indice permet de chercher la solution plus efficacement.
Si le diagramme inclut une torsion de la ligne de noeud comme celle-ci :
est-ce qu'il est important d'inverser un des croisements plutôt que l'autre?
S'il reste toujours plusieurs candidats pour le croisement recherché, il est utile d'imaginer le nombre de mouvements R1 et R2 qui seront débloqués par l'inversion pour tenter l'inversion qui a l'air de permettre le plus de simplifications par la suite.
Encore une astuce pour minimiser le nombre d'inversions qu'il faut essayer pour trouver la bonne : demandez-vous si, après l'inversion, il y aura un noeud qui reste, par exemple le noeud de trèfle. Si la réponse est oui, alors ce n'est pas la bonne inversion pour les casse-têtes de type U1.
Dans l'interface, le nombre d'inversions disponibles est limité au minimum nécessaire pour obtenir le noeud trivial. Malheureusement, une fois qu'on a inversé un croisement on ne peut pas l'annuler pour revenir en arrière car le diagramme a peut-être changé, alors il faut recommencer.
En général, des casse-têtes intéressants devraient avoir des solutions uniques alors nos casse-têtes de type U1 et U2 ne donnent que des diagrammes où l'inversion d'un seul croisement produit le noeud trivial. Cet indice permet de chercher la solution plus efficacement.
Si le diagramme inclut une torsion de la ligne de noeud comme celle-ci :
Non, ce n'est pas important car les deux résultats sont équivalents :
Alors, soit les deux croisements sont des inversions permettant d'obtenir le noeud trivial, soit ni l'un ni l'autre ne permet le dénouement. Donc, vu que nos casse-têtes ne peuvent avoir qu'une seule « inversion à dénouement », on peut ignorer ces deux croisements.
=
=
S'il reste toujours plusieurs candidats pour le croisement recherché, il est utile d'imaginer le nombre de mouvements R1 et R2 qui seront débloqués par l'inversion pour tenter l'inversion qui a l'air de permettre le plus de simplifications par la suite.
Encore une astuce pour minimiser le nombre d'inversions qu'il faut essayer pour trouver la bonne : demandez-vous si, après l'inversion, il y aura un noeud qui reste, par exemple le noeud de trèfle. Si la réponse est oui, alors ce n'est pas la bonne inversion pour les casse-têtes de type U1.
Dans l'interface, le nombre d'inversions disponibles est limité au minimum nécessaire pour obtenir le noeud trivial. Malheureusement, une fois qu'on a inversé un croisement on ne peut pas l'annuler pour revenir en arrière car le diagramme a peut-être changé, alors il faut recommencer.
Trouver des Mouvements U2
Pour des casse-têtes de type U2, on peut se servir du même indice que pour les casse-têtes de type U1 sur les inversions équivalentes puisqu'il y a un seul croisement qui réduira le nombre de dénouement c.-à-d. qui fera avancer vers le noeud trivial. Après avoir inversé ce croisement unique et simplifié le diagramme résultant, il peut y avoir plus d'une inversion possible pour obtenir le noeud trivial.
Trouver des Mouvements P0U
Des recherches effectuées par les Concours Caribou sur les nombres de dénouement montrent qu'il existe des diagrammes de noeud simplifiés de façon maximale (c.-à-d. avec le nombre minimum de croisements) qui n'ont pas d'inversion simplifiante. Autrement dit, il y a des diagrammes où le fait d'inverser un croisement ne peut pas faire approcher le diagramme du noeud trivial. Dans ce cas il faut d'abord effectuer un ou plusieurs mouvements P0 qui changent le casse-tête en type U2. Heureusement, de tels diagrammes exigeant des mouvements P0 du premier coup sont rares et donc il est très probable que tout mouvement P0 le fera changer en casse-tête de type U2.
Des Références Supplémentaires sur les Noeuds Mathématiques
D'une part, la Théorie des Noeuds est un vieux sujet de recherche, alors il existe une bonne quantité de litérature matématique à son égard. D'autre part, il est relativement jeune car certaines découvertes importantes n'ont été faites que récemment au cours de ces dernières décennies. Par exemple, il existe une revue scientifique mensuelle intitulée « Journal of Knot Theory and Its Ramifications » (Revue sur la Théorie des Noeuds et ses Ramifications) dédiée aux noeuds.
Voici un excellent livre qu'on recommande : Adams, Colin (2004), The Knot Book: An Elementary Introduction to the Mathematical Theory of Knots, American Mathematical Society, ISBN 978-0-8218-3678-1
Il existe aussi plusieurs sites web sur les noeuds. Pour explorer ce sujet, >la page Wikipédia sur la Théorie des Noeuds est un bon point de départ.
Pour des vidéos à ce sujet, nous suggérons la playlist de vidéos sur les noeuds créées par la chaîne YouTube Numberphile. Cette chaîne offre aussi une excellente explication du Coloriage des Noeuds, qui est une autre façon d'identifier deux diagrammes d'un même noeud.
Caribou a créé deux affiches sur le dénouement et le coloriage des noeuds.
Voici un excellent livre qu'on recommande : Adams, Colin (2004), The Knot Book: An Elementary Introduction to the Mathematical Theory of Knots, American Mathematical Society, ISBN 978-0-8218-3678-1
Il existe aussi plusieurs sites web sur les noeuds. Pour explorer ce sujet, >la page Wikipédia sur la Théorie des Noeuds est un bon point de départ.
Pour des vidéos à ce sujet, nous suggérons la playlist de vidéos sur les noeuds créées par la chaîne YouTube Numberphile. Cette chaîne offre aussi une excellente explication du Coloriage des Noeuds, qui est une autre façon d'identifier deux diagrammes d'un même noeud.
Caribou a créé deux affiches sur le dénouement et le coloriage des noeuds.
EN PERSAN! :)
This guide will introduce you to the subject of Knot Theory. The first section gives an introduction to important math theory concepts and terminology. If you are only looking for quick tips to solve the unknotting puzzles, skip ahead to the second section. You can always consult the first section if the meaning of some words is unclear.
Introduction with Definitions
Introduction
First things first : what are knots? Knots in mathematics are different from knots in your everyday life.
What kinds of knots do you use every day?
How are these different from mathematical knots?
Are there any mathematical knots in everyday life?
How does the Unknotting game work?
Definitions
From here on, we only talk about mathematical knots. To avoid misunderstandings and to be able to decide whether any statement is true or false we should start with defining the meaning of several words for the rest of this page.
knot line:
knot diagram:
mathematical knot (or simply, knot):
ambient isotopy:
step:
crossing:
pass:
switch:
orientation:
crossing handedness:
writhe number:
knot invariant:
crossing number:
arc:
hole:
over-strand:
under-strand:
Reidemeister moves:
Reidemeister 1 move:
Reidemeister 2 move:
Reidemeister 3 move:
pass move:
P- move:
P0 move:
P+ move:
unknotting number:
How to Simplify Diagrams
Finding R1 Moves
Finding R2 Moves
About the Interface (1)
Finding P- Moves
Finding R3 Moves
About the Interface (2)
Finding P0 Moves
Finding U1 Moves
Finding U2 Moves
Finding P0U Moves
More References about Knots
This guide will introduce you to the subject of Knot Theory. The first section gives an introduction to important math theory concepts and terminology. If you are only looking for quick tips to solve the unknotting puzzles, skip ahead to the second section. You can always consult the first section if the meaning of some words is unclear.
Introduction with Definitions
Introduction
First things first : what are knots? Knots in mathematics are different from knots in your everyday life.
What kinds of knots do you use every day?
Most of us use knots to tie our shoelaces, put on a necktie or scarf, to close up a bag, and so on... You might
know many more knots if you go sailing, camping, fishing, or if you sew, knit, or style hair.
However, none of these are mathematical knots!
How are these different from mathematical knots?
Have a look at the following two drawings of knots. Confusingly, these are both known as 'figure-eight' knots,
because of the figure 8 they contain.
Everyday Figure-eight Knot Mathematical Figure-eight Knot
What big difference can you see? We're sure you can figure it out!
Everyday Figure-eight Knot Mathematical Figure-eight Knot
What big difference can you see? We're sure you can figure it out!
The biggest difference is that the mathematical knot is a closed curve − that is, there are no loose
ends, it's a closed loop. What we call 'knots' in everyday life are known as 'braids' in
mathematics.
Also, while everyday knots might include more than one strand of material, in mathematics knots are a single, closed, continuous strand. Objects that involve more than one knot woven together are known as 'links'.
Also, while everyday knots might include more than one strand of material, in mathematics knots are a single, closed, continuous strand. Objects that involve more than one knot woven together are known as 'links'.
Are there any mathematical knots in everyday life?
Of course! You now know that in mathematics, a knot is a single, closed, continuous strand.
With this definition in mind, what is the simplest mathematical knot?
How can you make a mathematical knot from a piece of string?
With this definition in mind, what is the simplest mathematical knot?
The simplest mathematical knot is just a single loop or circle, like this:
We will talk more about this knot later, but there are plenty of examples of simple loops like this in real life.
We will talk more about this knot later, but there are plenty of examples of simple loops like this in real life.
How can you make a mathematical knot from a piece of string?
The simplest mathematical knot is a circle. To make it, simply glue the ends of your string together.
What happens if you twist your circle once?
Is this a different knot?
Can all knots be deformed to make a circle?
How different can a knot diagram look from the simplest form?
What happens if you twist your circle once?
If you take your loop of string, twist it, and lay it flat, you might get something like this:
Is this a different knot?
Of course knot! All you did was twist it.
This might seem easy, but finding ways to tell whether two diagrams (pictures) show the same knot is a very important and difficult question for mathematicians who study Knot Theory.
One way to tell that two pictures represent the same knot is whether you can deform one of them to look like the other. For example, here you just need to twist it back to get the loop.
This might seem easy, but finding ways to tell whether two diagrams (pictures) show the same knot is a very important and difficult question for mathematicians who study Knot Theory.
One way to tell that two pictures represent the same knot is whether you can deform one of them to look like the other. For example, here you just need to twist it back to get the loop.
Can all knots be deformed to make a circle?
To answer this question, try this out:
Can you deform this to get a circle?
- take a strand of string
- twist it to form a loop
- pass one end through the loop
Can you deform this to get a circle?
Try as you might, there is no way to deform this knot into a circle. At least, not without cutting
the string and gluing it back together.
This is actually a different mathematical knot, called the trefoil knot because it looks like a three-leaf clover.
This is actually a different mathematical knot, called the trefoil knot because it looks like a three-leaf clover.
How different can a knot diagram look from the simplest form?
Another important property of mathematical knots is that they can be arbirtrarily stretched and bent.
For example, our diagram of the simplest knot looks more like a square than a circle − we could
draw it as a perfect circle, and it would be the same knot. You could take the simplest knot, a circle,
and stretch it out into a long thin ellipse, then use it as a string to tie it into the 'everyday
figure-8 knot'. Mathematically, it is still a circle.
As you might expect, then, the different diagrams of the same underlying knot can look drastically different! For example, the Gordian Knot and this diagram can both be deformed into a circle, with enough patience. They are the diagrams of the same knot as the circle.
The circle is just the easiest way of drawing this knot, but the knot isn't really a circle, it's an abstract mathematical object that we can represent in many different ways. In the same way that "1 car" and "1 apple" is not the number 1, a circle is just one way to represent this knot.
As you might expect, then, the different diagrams of the same underlying knot can look drastically different! For example, the Gordian Knot and this diagram can both be deformed into a circle, with enough patience. They are the diagrams of the same knot as the circle.
The circle is just the easiest way of drawing this knot, but the knot isn't really a circle, it's an abstract mathematical object that we can represent in many different ways. In the same way that "1 car" and "1 apple" is not the number 1, a circle is just one way to represent this knot.
How does the Unknotting game work?
In this game, you deform mathematical knot diagrams to reduce the number of
crossings as much as possible.
While you are allowed to 'cut' the strands by clicking on them, you are only modifying the diagram, not the underlying knot. This is why there is a restriction on how you can reattach the ends you have 'cut'. This restriction guarantees that the mathematical knot is unchanged in any re-routing of strands even if the knot changes its appearance.
When do you undo everyday knots in real life?
Try some other mathematical knot games!
While you are allowed to 'cut' the strands by clicking on them, you are only modifying the diagram, not the underlying knot. This is why there is a restriction on how you can reattach the ends you have 'cut'. This restriction guarantees that the mathematical knot is unchanged in any re-routing of strands even if the knot changes its appearance.
When do you undo everyday knots in real life?
You might be familiar with the struggle of disentangling electronics cables like for earbuds. If you tie
your shoes, you have to untie the laces.
The same property of real knots is what makes them useful, but also harder to undo.
What makes knots in real life so hard to undo?
The same property of real knots is what makes them useful, but also harder to undo.
What makes knots in real life so hard to undo?
The answer is friction! However, mathematical knots have no friction. You can think of them as
'infinitely slippery'.
Try some other mathematical knot games!
Our unknotting game is one way to have fun with knots on a screen, but here are some other games for you to
try:
- 1 player : Eiffel Tower and other string tricks
- 2 players : Cat's cradle
- Group : Human Knot
Definitions
From here on, we only talk about mathematical knots. To avoid misunderstandings and to be able to decide whether any statement is true or false we should start with defining the meaning of several words for the rest of this page.
knot line:
a closed curve in 3-dimensional space which does not
intersect itself and which has a finite thickness (to avoid
infinitely many smaller and smaller knots along the line), Example:
the figure-8 knot.
knot diagram:
a projection of a knot line into 2 dimensions where
different parts of the knot line can cross each other (on this website
lines cross under an angle of 90°), but do not lie on top of each other.
Examples for knot diagrams:
Unknot Trefoil Figure-eight Cinquefoil Three-twist
The 'unknot' is also called the 'trivial knot'.
Which knot does each of these 3 diagrams represent?
Which knot does this diagram represent?
Do you see how to deform this into the earlier trefoil diagram?
Examples for knot diagrams:
Unknot Trefoil Figure-eight Cinquefoil Three-twist
The 'unknot' is also called the 'trivial knot'.
Which knot does each of these 3 diagrams represent?
The unknot. Do you see how they can be deformed to a rectangle?
Which knot does this diagram represent?
The trefoil.
Do you see how to deform this into the earlier trefoil diagram?
You can deform this diagram into the other by flipping down the top arc.
→
→
mathematical knot (or simply, knot):
the abstract object behind
a set of (infinitely many) knot diagrams that can all be deformed,
stretched, and shifted into each other without being cut. Example:
the knot 31 also called 'trefoil' which is the simplest
non-trivial knot.
ambient isotopy:
the mathematical term
when one knot line can be continuously distorted to another one.
step:
On this website knot diagrams are drawn using only 6 tiles which we call steps:
crossing:
the place in a diagram where two steps cross, one on top of
the other:
pass:
a step that is part of a crossing, there are over-passes (fully
visible) and under-passes (partially covered).
switch:
swapping over- and under-pass of a crossing, i.e. switching between these two crossings:
If a crossing is switched, the old and new diagram in general
represent different knots. Switching all crossings is equivalent to
changing a knot to its mirror image.
Some knots are identical to their mirrored version, that means there is an
ambient isotopy between them. These are called 'achiral'. For example, the figure-8 knot is achiral.
Others can not be deformed to their mirrored version, like the
trefoil. They are called 'chiral'.
Can this knot be deformed into its mirror image?
Can this knot be deformed into its mirror image?
Yes! This diagram represents the figure-eight knot which is achiral and can be deformed into its mirror
image as follows:
Notice that the only difference between the first diagram and the last one is that all of the crossings are switched!
Initial position
A 180° rotation
Moving a strand
The mirror image
Notice that the only difference between the first diagram and the last one is that all of the crossings are switched!
orientation:
This is not a property of the knot line, nor of the knot.
It is a question of how to move along the knot line. One can move in 2
directions, also called 2 orientations.
A knot that can be deformed via an ambient isotopy into itself but with the
orientation reversed is called 'invertible' otherwise it is called
noninvertible. The smallest noninvertible knot is 817 which is achiral but
if an orientation is added it becomes chiral (find more on
the Invertible Knot Wikipedia page). Adding more
structure (here
an orientation) causes it to lose symmetry (not identical anymore to its mirror
image).
A chiral knot (a knot that can not be deformed to its mirror image) can still be invertible (symmetric against change of orientation). Such knots are called 'reversible'.
A chiral knot (a knot that can not be deformed to its mirror image) can still be invertible (symmetric against change of orientation). Such knots are called 'reversible'.
crossing handedness:
For a given knot diagram, crossings are either right- or left-handed.
In the following we will explore two types of crossings and determine which is right- or left-handed.
How many different crossings are there if we consider
which pass is an over-/under-pass and consider both orientations?
If one keeps the diagram unchanged and only switches one crossing,
does the handedness of that crossing change?
By using one's hands, how can one remember whether a crossing is
right- or left-handed?
In the following we will explore two types of crossings and determine which is right- or left-handed.
How many different crossings are there if we consider
which pass is an over-/under-pass and consider both orientations?
In total there are 8 cases:
If the horizontal pass is the over-pass then there are 4 options:
Similarly if the vertical pass is the over-pass then there are 4 more options:
The crossings in group 1, 7, 4, 6 are called right-handed crossings and
the crossings in group 2, 5, 3, 8 are called left-handed crossings.
If the horizontal pass is the over-pass then there are 4 options:
1
2
3
4
5
6
7
8
- Principle: Handedness should not depend on the orientation (direction of stepping through the knot line), so reversing both arrows we identify 4 pairs of crossings: 1 = 4, 2 = 3, 5 = 8, 6 = 7. Therefore, whichever groups we end up with, crossings 1 and 4 should be in the same group and so on.
- Principle: The group that a crossing belongs to should not change if we rotate the whole knot. We therefore identify crossings 1 = 7 = 4 = 6 and 2 = 5 = 3 = 8.
The crossings in group 1, 7, 4, 6 are called right-handed crossings and
the crossings in group 2, 5, 3, 8 are called left-handed crossings.
If one keeps the diagram unchanged and only switches one crossing,
does the handedness of that crossing change?
Yes. Try switching any of the eight crossings,
then check which one it has become and check whether it is
still in the same handedness group. For example, switching crossing 1
gives crossing 5, both are in different handedness groups.
Statement: The question whether a crossing is right-handed or left-handed does not only depend on the crossing itself but on the diagram around it.
Proof: Whether the horizontal pass is above or below does not alone decide the handedness. Both cases can be right- and left-handed (see the 8 crossings above). If one rotates the knot so that the over-pass is horizontal and if one then leaves the crossing through the overhead pass to the right (to the East) then it depends on the rest of the knot whether one returns to the crossing from the South (then the crossing is right-handed) or from the North (then the crossing is left-handed).
The fact that the two groups of crossings are called left- and right-handed gives a hint that one can distinguish crossings with the left and right hands.
Statement: The question whether a crossing is right-handed or left-handed does not only depend on the crossing itself but on the diagram around it.
Proof: Whether the horizontal pass is above or below does not alone decide the handedness. Both cases can be right- and left-handed (see the 8 crossings above). If one rotates the knot so that the over-pass is horizontal and if one then leaves the crossing through the overhead pass to the right (to the East) then it depends on the rest of the knot whether one returns to the crossing from the South (then the crossing is right-handed) or from the North (then the crossing is left-handed).
The fact that the two groups of crossings are called left- and right-handed gives a hint that one can distinguish crossings with the left and right hands.
By using one's hands, how can one remember whether a crossing is
right- or left-handed?
Stretch out your fingers so that all are in one plane, and your
thumb is at a right angle to all others which are parallel to
each other. Rotate your hand so that you can see your palm and
your thumb points towards the outgoing direction of the
over-pass and your fingers point towards the outgoing direction of
the under-pass. The hand that can do that decides the
handedness.
For example, for the crossing below, you would stretch your hand out like this:
Since you can only do this with your left hand, this is a left-handed crossing.
Right- and left-handed crossings are also referred to as positive or negative crossings.
Since you can only do this with your left hand, this is a left-handed crossing.
Right- and left-handed crossings are also referred to as positive or negative crossings.
writhe number:
the difference between the number of left- and right-handed
crossings in one diagram. The writhe number characterizes a diagram, not a knot as there can be 2
different diagrams of the same knot with different writhe numbers.
What is the writhe number of this diagram?
What is the writhe number of this diagram?
The right-handed crossings in the diagram are highlighted in red and the left-handed crossings in green.
To get the writhe number, we can count the number of left- and right-handed crossings, then subtract the
number of right-handed crossings
from the number of left-handed crossings.
This diagram has 2 left-handed crossings and 4 right-handed crossings, so its writhe number is 2 − 4 =
−2.
knot invariant:
a number or a polynomial or a feasibility statement that is characteristic for all (infinitely many)
diagrams of a knot. The properties of a knot being chiral/achiral, invertible/noninvertible,
reversible are knot invariants.
crossing number:
the minimal number of crossings that any diagram of
this knot can have after deformation, this is a characteristic of each knot and therefore a knot
invariant.
How many crossings does this diagram have?
What is the crossing number of the knot represented through the above diagram?
What are the two lowest crossing numbers that a knot can have?
How many crossings does this diagram have?
This diagram has 5 crossings.
What is the crossing number of the knot represented through the above diagram?
Zero! The crossing number is a property of the abstract
mathematical knot, it is not the property of a diagram. The
diagram above can be deformed to get the unknot
which has zero crossings.
Do you see how?
Because the crossing number is the minimal number of crossings of any diagram of a knot and because one cannot have less than zero crossings, the crossing number of the knot represented by the above diagram is zero.
which has zero crossings. Do you see how?
Because the crossing number is the minimal number of crossings of any diagram of a knot and because one cannot have less than zero crossings, the crossing number of the knot represented by the above diagram is zero.
What are the two lowest crossing numbers that a knot can have?
The lowest crossing number belongs to the unknot which is 0.
A knot diagram with 1 crossing would look like:
and could be deformed to the unknot. A knot diagram with 2 crossings would look like
and could also be deformed to the unknot.
The trefoil diagram shown further above has 3 crossings and can not be deformed to the unknot, so the lowest crossing numbers are 0 and 3.
and could be deformed to the unknot. A knot diagram with 2 crossings would look like
and could also be deformed to the unknot.The trefoil diagram shown further above has 3 crossings and can not be deformed to the unknot, so the lowest crossing numbers are 0 and 3.
arc:
The part of the knot line in a diagram from one crossing to the next.
How many arcs does a diagram with N crossings have?
How many arcs does a diagram with N crossings have?
Each crossing has 4 ends of arcs. Each arc has 2 ends, so there are 4/2 = 2
times as many arcs as crossings, so 2N arcs.
hole:
empty space in a diagram that is surrounded by arcs. The whole
empty space outside the diagram is also one hole.
How many holes does a diagram with N crossings have?
How many holes does a diagram with N crossings have?
One could draw several knots and guess a formula but one can derive
it too. Euler's formula says that for any drawing in the plane where
m lines (here m=2N arcs) each connect 2 out of n points (here n=N
crossings)
then the number f of faces (here holes) is f = 2 + m − n. That gives
for the number of holes of a knot: 2 + 2N − N = N + 2.
over-strand:
A sequence of consecutive arcs in a diagram
(i.e. arcs following one another) which starts and ends at an
under-pass and otherwise involves 0, 1 or more over-passes.
under-strand:
A sequence of consecutive arcs in a diagram
(i.e. arcs following one another) which starts and ends at an
over-pass and otherwise involves 0, 1 or more under-passes.
(In literature, a 'strand' is often used for what we call an over-strand. For us the number of over-passes of an over-strand matters as well as the number of under-passes of an under-strand. We therefore consider under-strands as well as over-strands.)
What kind of strand is the shown horizontal line consisting of five arcs?
How many over-strands does a diagram with N crossings have?
(In literature, a 'strand' is often used for what we call an over-strand. For us the number of over-passes of an over-strand matters as well as the number of under-passes of an under-strand. We therefore consider under-strands as well as over-strands.)
What kind of strand is the shown horizontal line consisting of five arcs?
This is an under-strand with 4 under-passes.
How many over-strands does a diagram with N crossings have?
At each crossing there are 2 ends of strands (either one end of
2 different strands or both ends from one strand). On the other hand,
each strand has 2 ends which are at a crossing. Therefore the number
of crossings is equal to the number of over-strands and because of
symmetry also equal to the number of under-strands, so there are
N of each.
Reidemeister moves:
In 1927 the German mathematician Kurt
Reidemeister and, independently, James Waddell Alexander and Garland
Baird Briggs (1926), proved that any two diagrams that represent the
same knot can be deformed into each other through a sequence of only 3
different types of moves. The problem is that during the deformation
the number of crossings may temporarily rise and a sharp upper bound for
this increase is unknown as well as the number of needed moves.
Reidemeister 1 move:
removes or adds a hole surrounded by one arc:
Which diagram shows a left-handed crossing and which
shows a right-handed crossing?
Which diagram shows a left-handed crossing and which
shows a right-handed crossing?
The left diagram shows a right-handed crossing and the right diagram shows a
left-handed crossing. A Reidemeister 1 move therefore changes the number
of right- or left-handed crossings by 1 and thus changes the
writhe number of the diagram.
Reidemeister 2 move:
removes or adds a hole surrounded by 2 arcs:
What can one say about the handedness of the two crossings that are
added or removed in a Reidemeister 2 move?
What can one say about the handedness of the two crossings that are
added or removed in a Reidemeister 2 move?
One of the two crossings is right-handed and one is left-handed.
A Reidemeister 2 move therefore does not change the writhe
number of a diagram.
Reidemeister 3 move:
removes and adds a hole surrounded by 3 arcs.
Which 2 types of holes surrounded by 3 arcs can you think of?
Which 2 types of holes surrounded by 3 arcs can you think of?
Either:
Verify that the result of the 3 moves is always the same.
Does the handedness of the 3 crossings change in a Reidemeister 3 move?
What have we learned?
- 1) each arc has 1 over-pass and 1 under-pass:
then none of the 3 arcs can be moved over/under/through the other crossing
- 2) one arc has 2 over-passes, one has 1 over- and 1
under-pass, and one has 2 under-passes:
then there are 3 moves. One can move the over-over-strand which stays an over-over-strand:
or move the under-over-strand which becomes an over-under-strand:
or move the under-under-strand which stays an under-under-strand:
Verify that the result of the 3 moves is always the same.
When comparing the right-hand sides of the above moves it is easy to see that
all 3 moves produce identical results. Therefore, if there is a Reidemeister 3 move
then there is only one. All that changes is that for all 3 arcs the other two arcs are
now crossed in the reverse order. This means that for the middle arc the order of over-pass
and under-pass is reversed.
Does the handedness of the 3 crossings change in a Reidemeister 3 move?
No. To see that, pick any orientation for each strand and use the
hand rule above.
What have we learned?
We learned:
- how to spot holes with 3 arcs that allow a Reidemeister 3 move,
- that for such a hole it does not matter which arc is moved,
- that the handedness of the 3 crossings does not change,
- that the order of over- and under-pass is reversed for the middle arc.
pass move:
This has nothing to do with a 'pass' defined above.
A pass move replaces an over-(under-)strand with another
over-(under-)strand where both strands have the same ends. For examples, please see P-, P0 and P+ moves below.
P- move:
a pass move where the new strand has less passes than the
old strand.
Find a P- move replacing the green strand in this diagram:
Find a P- move replacing the green strand in this diagram:
In this diagram the new red strand has fewer passes than the old green strand. Therefore this diagram shows
a P- move.
P0 move:
a pass move where the new strand has the same number of passes
as the old strand.
Find a P0 move replacing the green strand in this diagram:
Find a P0 move replacing the green strand in this diagram:
In this diagram the new red strand has the same number of passes as the old green strand. Therefore this
diagram shows a P0 move.
P+ move:
a pass move where the new strand has more passes
than the old strand.
Find a P+ move replacing the green strand in this diagram:
P+ moves become necessary if one wants to change the writhe number of a diagram. More about that is described further below under 'Finding P0 Moves'.
Find a P+ move replacing the green strand in this diagram:
In this diagram the new red strand has one more pass than the old green strand, therefore this is a P+ move.
P+ moves become necessary if one wants to change the writhe number of a diagram. More about that is described further below under 'Finding P0 Moves'.
unknotting number:
The unknotting number is the property of a knot, not the property of a diagram and is therefore
a knot invariant.
Starting with a knot diagram it is the minimal number of times that one or more crossings need to be switched in order to obtain the unknot. Before the first switch and inbetween switches the diagram can be arbitrarily deformed. The unknotting number is therefore not an easy number to determine because any deformation is allowed.
Why does the trefoil have unknotting number 1?
Starting with a knot diagram it is the minimal number of times that one or more crossings need to be switched in order to obtain the unknot. Before the first switch and inbetween switches the diagram can be arbitrarily deformed. The unknotting number is therefore not an easy number to determine because any deformation is allowed.
Why does the trefoil have unknotting number 1?
The trefoil cannot have the unknotting number 0 because it cannot
be deformed into the unknot (this needs to be and can be proven). The unknot has unknotting number 0. So the
trefoil has unknotting number ≥1. On the other hand one can easily
see that switching any one crossing of the trefoil diagram
shown further above produces the unknot, so the unknotting
number of the trefoil is ≤1. If it is ≥1 and ≤1 then it must be =1.
How to Simplify Diagrams
Finding R1 Moves
Simple cases of R1 moves, like here:
where one can flip a loop 4 times and instantly get the unknot
are easy to spot by following the knot line and looking for an arc
with both ends at the same crossing. The order of performing R1 moves
does not matter.
But there are more general cases of applying Reidemeister 1 moves. If an over-strand starts at a crossing of which it is an overpass but otherwise lies completely on top of other arcs (therefore being called 'over-strand') then this loop can surely be shortcut and thus removed. For example, at first the loop on top in the center can be removed and then the other ones, one by one:
This loop can also be removed when lying entirely underneath:
where one can flip a loop 4 times and instantly get the unknot
are easy to spot by following the knot line and looking for an arc
with both ends at the same crossing. The order of performing R1 moves
does not matter. But there are more general cases of applying Reidemeister 1 moves. If an over-strand starts at a crossing of which it is an overpass but otherwise lies completely on top of other arcs (therefore being called 'over-strand') then this loop can surely be shortcut and thus removed. For example, at first the loop on top in the center can be removed and then the other ones, one by one:
This loop can also be removed when lying entirely underneath:
Finding R2 Moves
Similarly to R1 moves it is easy to spot prototype R2 moves like
here where two R2 moves need to be done before an R1 move yields
the unknot:
In the following example one has to perform an R2 move with the same strand twice, once pulling this strand from underneath and once from above:
The last step is not an R2 move. It is only added to show that the knot is the sum of two trefoil knots.
In the following example one has to perform an R2 move with the same strand twice, once pulling this strand from underneath and once from above:
The last step is not an R2 move. It is only added to show that the knot is the sum of two trefoil knots.
About the Interface (1)
The example above is suitable to demonstrate the optimal use of the
interface. After intercepting the knot line:
One can NOT retreat one end all the way; and then retreat the other end all the way because one would first remove an under-pass and have both ends in the under-pass status and in this status it is not possible to remove an over-pass. Instead, one removes one under-pass, jumps to the other end, removes the other under-pass which brings both ends into the same hole, which changes the ends into neutral status and then allows to remove over-passes and then to re-unite the ends. In short, one jumps between ends to remove all under-passes then all over-passes and so on.
This implementation of a status of ends is not a weakness of the program but it guarantees that interactive modification of the diagram does not change the mathematical knot.
One can NOT retreat one end all the way; and then retreat the other end all the way because one would first remove an under-pass and have both ends in the under-pass status and in this status it is not possible to remove an over-pass. Instead, one removes one under-pass, jumps to the other end, removes the other under-pass which brings both ends into the same hole, which changes the ends into neutral status and then allows to remove over-passes and then to re-unite the ends. In short, one jumps between ends to remove all under-passes then all over-passes and so on.
This implementation of a status of ends is not a weakness of the program but it guarantees that interactive modification of the diagram does not change the mathematical knot.
Finding P- Moves
P- moves replace an over-strand with one having less over-passes
or an under-strand with one having less under-passes. In both
cases the number of crossings is reduced.
To find such moves one steps through the knot line and looks for as many consecutive over-passes as possible or for as many consecutive under-passes as possible, at least two. If one found such a strand, for example, an over-strand then one tries to find an alternative route with less over-passes that links the same two under-pass ends.
In the following example a strand with 4 consecutive under-passes is replaced by a strand without underpasses and then this strand with 3 consecutive over-passes is replaced by a strand with one over-pass. Two more P- moves remove each 2 more crossings. The resulting diagram can be simplified further by two more R1 moves, as seen below.
The more consecutive passes of one sort one finds, the higher the
chance to find a different route that needs less passes.
To find such moves one steps through the knot line and looks for as many consecutive over-passes as possible or for as many consecutive under-passes as possible, at least two. If one found such a strand, for example, an over-strand then one tries to find an alternative route with less over-passes that links the same two under-pass ends.
In the following example a strand with 4 consecutive under-passes is replaced by a strand without underpasses and then this strand with 3 consecutive over-passes is replaced by a strand with one over-pass. Two more P- moves remove each 2 more crossings. The resulting diagram can be simplified further by two more R1 moves, as seen below.
Finding R3 Moves
R1, R2 and P- moves change the number of crossings. An R3
move does not change the number of crossings, therefore we place
its description after the P- move. The following example shows how
R3 moves can still be useful by making P- moves possible.
As described in the first section, an R3 hole is surrounded by a top arc
with 2 over-pass ends (here A,B), a middle arc with 1 over-pass end (C) and 1
under-pass end (B) and a bottom arc with 2 under-pass ends (here
A,C).
(unknot taken from the Unknot Wikipedia page)
We found in the above section that an R3 move reverses for the middle strand (here through B,C) the order of under- and over-pass. An R3 move is beneficial if the R3 move leads for the continuation of the middle arc to an increase of consecutive over-passes and/or consecutive under-passes and thus increases the chance to find a P- move. This is the case if in the continuation of the middle arc after the under-pass B comes an over-pass (D,E are over-passes) and/or after the over-pass (C) follows an under-pass (F,G,H are under-passes). The R3 move slides the middle arc BC between the top arc and bottom arc at A:
Before, there were only 2 consecutive over-passes at D and E,
now there are 3 at C, D and E. This longer over-strand can now be
re-routed in a P- move:
reducing the number of crossings by 2 from 13 to 11.
Also, the other side of the strand can be re-routed in a P- move. Before, there were 3 consecutive under-passes at F,G and H, now there are 4 at B, F, G, and H. This P- move leads to:
and is also reducing the number of crossings by 2. Both diagrams can be
simplified further through P- and R1 moves resulting finally in the unknot.
Can you see how? Just follow the hints on how to spot P- moves given above.
Let us practise that with an example.
How many R3 moves are possible in this diagram:
We found in the above section that an R3 move reverses for the middle strand (here through B,C) the order of under- and over-pass. An R3 move is beneficial if the R3 move leads for the continuation of the middle arc to an increase of consecutive over-passes and/or consecutive under-passes and thus increases the chance to find a P- move. This is the case if in the continuation of the middle arc after the under-pass B comes an over-pass (D,E are over-passes) and/or after the over-pass (C) follows an under-pass (F,G,H are under-passes). The R3 move slides the middle arc BC between the top arc and bottom arc at A:
Also, the other side of the strand can be re-routed in a P- move. Before, there were 3 consecutive under-passes at F,G and H, now there are 4 at B, F, G, and H. This P- move leads to:
Let us practise that with an example.
How many R3 moves are possible in this diagram:
Three R3 moves are possible. For each one we show in light blue the three arcs that are
involved. What is easily overlooked is the third one where the hole
is the whole outside space which is 'surrounded' by only 3 arcs.
Perform the 1. R3 move and find out whether it is beneficial:
Perform the 2. R3 move and find out whether it is beneficial:
Perform the 3. R3 move and find out whether it is beneficial:
1. R3 move
2. R3 move
3. R3 move
Perform the 1. R3 move and find out whether it is beneficial:
This R3 move is beneficial. It allows afterwards a P- moves as shown in a sequence of moves further
below.
Our definition for an R3 move to be beneficial it is not
neccessarily to allow a P- move but to increase the number of consecutive
over- or under-passes and that is easy to see even without
performing all these moves. In the following diagram the middle arc
of the R3 hole has an
over-pass at A, an under-pass at B followed by two over-passes at C and D.
In an R3 move the order of over- and under-pass is reversed for the middle arc
as shown in the Diagram 3 with now 3 consecutive over-passes. This is enough
to find a P- move needing less than 3 passes in Diagram 5.
About the sequence of diagrams below: In Dia 1 we make space to prepare the R3 move in Dia 2 (here by
moving
the top arc) with the result in Dia 3. In Dia 4 we make space to prepare
the P- move in Dia 5 where the green strand with 3 over-passes is replaced
by the red strand with only 1 over-pass in Dia 6. In Dia 7 we shift a strand
to make space for the next P- move in Dia 9 with result in Dia 10 and Dia 11
after shrinking which is easily identified as knot 51.
1. Widening
2. The R3 move
3. After the R3 move
4. Widening
5. A P- move
6. After the P- move
7. For the next P- move
8. Before the P- move
9. The 2nd P- move
10. Afterwards
11. Contracted
Perform the 2. R3 move and find out whether it is beneficial:
The sequence shows that the R3 move is beneficial.
1. Widening
2. The R3 move
3. After the R3 move
4. A P- move
5. After the P- move
6. Another P- move
7. After the P- move
8. Contracted
Perform the 3. R3 move and find out whether it is beneficial:
The 3rd R3 move is also beneficial. To execute this move, one follows the same principle: the middle
strand cuts the two other strands, which this time
'surround' the outside hole, in reverse order.
The result is that travelling upwards in the Dia 2 along the light blue strand one first gets to an overpass and then an underpass. If after the R3 move one travels along the red strand then one passes the other 2 strands in reverse order and therefore first gets to an underpass and then an overpass. As one can see in Dia 5, the sequence of now 3 consecutive underpasses of the light blue strand there enables a P- move.
The result is that travelling upwards in the Dia 2 along the light blue strand one first gets to an overpass and then an underpass. If after the R3 move one travels along the red strand then one passes the other 2 strands in reverse order and therefore first gets to an underpass and then an overpass. As one can see in Dia 5, the sequence of now 3 consecutive underpasses of the light blue strand there enables a P- move.
1. Widening
2. An R3 move
3. After the R3 move
4. Widening
5. A P- move
6. Widening
7. A 2nd P- move
8. After the P- move
9. Shortening
10. After shortening
11. Shortening
12. Straightening
13. ↻90° rotation
About the Interface (2)
The example above is suitable to demonstrate how to perform an R3
move with our interface.
As described in the 'Introduction with Definitions' > 'Reidemeister 3 move' there are 3 ways to perform a Reidemeister 3 move: moving the bottom strand, moving the middle strand, or, moving the top strand. As shown there, all 3 ways have the same result, they perform the same Reidemeister 3 move.
Our interface only allows one to complete an R3 move by moving the bottom strand or the top strand but not the middle strand. The reason is the feature of our interface that BOTH ends can only add/remove over-passes or BOTH ends can only add/remove under-passes at a time. But that does not prevent us from performing R3 moves, as moving any one of the 3 strands gives the same result.
As described in the 'Introduction with Definitions' > 'Reidemeister 3 move' there are 3 ways to perform a Reidemeister 3 move: moving the bottom strand, moving the middle strand, or, moving the top strand. As shown there, all 3 ways have the same result, they perform the same Reidemeister 3 move.
Our interface only allows one to complete an R3 move by moving the bottom strand or the top strand but not the middle strand. The reason is the feature of our interface that BOTH ends can only add/remove over-passes or BOTH ends can only add/remove under-passes at a time. But that does not prevent us from performing R3 moves, as moving any one of the 3 strands gives the same result.
Finding P0 Moves
P0 moves are pass moves that do not change the number of
crossings, just like R3 moves which are special versions of
P0 moves. Like R3 moves, a P0 move may be beneficial and
enable a P- move. Because P0 moves are less useful on
average, they occur more frequently but it is more
challenging to see whether they enable a P- move.
To find a P0 move one looks for an over-strand or an under-strand just like for a P- move. To check whether a P0 move may be beneficial and enable a P- move one proceeds like in the case of the R3 move. One looks whether the removal of the strand increases the number of consecutive over- or under-passes of strands that were crossed and one checks whether after re-routing the new strand increases the number of consecutive over- or under-passes of strands that are crossed now. In either one of these cases one checks whether the strands with an increased number of consecutive over- or under-passes can be re-routed with less crossings.
Let us look at this example:
We label the crossings:
and find a beneficial P0 move step by step.
How many over-strands with at least 2 over-passes
and how many under-strands with at least 2
under-passes do you see?
It is not difficult to see that the IE strand has a P0 move that re-locates it to cross over the GC and BH strand:
but the question is whether this P0 move is beneficial.
Did moving the IE strand increase the number of consecutive over-/under-passes
of the previously crossed DF or HJ strands?
Did more consecutive over-/under-passes get created when placing the strand on top of the
2 strands GC and BH?
Can this BH strand be re-routed in a P- move to reduce the number of crossings?
In the original diagram this P- move could have been done first as a P0 move and the P0 move mentioned above afterwards as P- move with the same total saving of 2 crossings.
In difficult problems it may be necessary to perform several P0 moves before a P- move becomes possible.
A diagram is maximally simplified if the number of crossings is the crossing number (see first section). In that case P0 moves will never enable a P- move.
To find a P0 move one looks for an over-strand or an under-strand just like for a P- move. To check whether a P0 move may be beneficial and enable a P- move one proceeds like in the case of the R3 move. One looks whether the removal of the strand increases the number of consecutive over- or under-passes of strands that were crossed and one checks whether after re-routing the new strand increases the number of consecutive over- or under-passes of strands that are crossed now. In either one of these cases one checks whether the strands with an increased number of consecutive over- or under-passes can be re-routed with less crossings.
Let us look at this example:
How many over-strands with at least 2 over-passes
and how many under-strands with at least 2
under-passes do you see?
We get three over-strands with at least 2 over-passes:
AB, GC, IE and three under-strands with at least 2
under-passes: EF, BH, DJ.
It is not difficult to see that the IE strand has a P0 move that re-locates it to cross over the GC and BH strand:
Did moving the IE strand increase the number of consecutive over-/under-passes
of the previously crossed DF or HJ strands?
Yes, the HJ strand now has 2 over-passes but this strand can not be re-routed
to link the same two holes with less over-passes.
Did more consecutive over-/under-passes get created when placing the strand on top of the
2 strands GC and BH?
Yes, the BH under-strand had 2 under-passes and now has 3 under-passes.
Can this BH strand be re-routed in a P- move to reduce the number of crossings?
Yes: The new route of the BH strand links the same holes but with only 1 under-pass instead of
3 under-passes.
The fact that the new strand is longer (involves more steps) in this
diagram than the replaced strand does not matter. All that matters
is the reduction of the number of crossings from 10 to 8 which now
allows to identify this knot as knot 817. In the original diagram this P- move could have been done first as a P0 move and the P0 move mentioned above afterwards as P- move with the same total saving of 2 crossings.
In difficult problems it may be necessary to perform several P0 moves before a P- move becomes possible.
A diagram is maximally simplified if the number of crossings is the crossing number (see first section). In that case P0 moves will never enable a P- move.
Finding U1 Moves
A U1 move switches a crossing which afterwards allows to simplify
the diagram to remove all crossings and show that the switch
produced the unknot.
In general nice puzzles should have unique solutions so our U1 and U2 puzzles show diagrams where the switch of only one crossing results in the unknot. This hint allows you to cut down the search.
If the diagram includes a twist of the knot line like this:
would it matter which one of the crossings is switched?
If there are still several crossing candidates, one should try to imagine how many R1, R2 moves will become available through the switch and try that switch first which seems to allow the most simplifications afterwards.
Another hint to minimize trying switches is to imagine whether after the switch there will definitely be a knot remaining, like a trefoil. If so then that switch is not the correct one in U1 puzzles.
On the worksheet the number of available switches is limited to the minimum that is needed to get the unknot. If you did a switch one can not switch it back because the diagram may have been changed, so one would have to reset the diagram.
In general nice puzzles should have unique solutions so our U1 and U2 puzzles show diagrams where the switch of only one crossing results in the unknot. This hint allows you to cut down the search.
If the diagram includes a twist of the knot line like this:
It will not matter which crossing is switched. Both results are equivalent:
Therefore either both of these crossings are unknotting switches or
none of them. Because our puzzles have only one unknotting switch,
these two crossings can be ignored.
=
=
If there are still several crossing candidates, one should try to imagine how many R1, R2 moves will become available through the switch and try that switch first which seems to allow the most simplifications afterwards.
Another hint to minimize trying switches is to imagine whether after the switch there will definitely be a knot remaining, like a trefoil. If so then that switch is not the correct one in U1 puzzles.
On the worksheet the number of available switches is limited to the minimum that is needed to get the unknot. If you did a switch one can not switch it back because the diagram may have been changed, so one would have to reset the diagram.
Finding U2 Moves
For U2 puzzles the same hint about equivalent switches applies as
for U1 puzzles. Also for U2 puzzles there is just one crossing that
reduces the unknotting number, i.e. makes progress towards the
unknot. Once that unique first crossing is switched and the
resulting diagram is simplified, there may be more than one switch
possible that creates the unknot.
Finding P0U Moves
Research performed by Caribou Contests on unknotting numbers showed
that there exists maximally simplified knot diagrams (with the
minimal number of crossings), which do not have a simplifying
switch. In other words, there are diagrams where switching any
crossings will not make progress to reach the unknot. In that case
one first has to perform one or more P0 moves that change the puzzle
into a U2 puzzle. The good news is that diagrams requiring P0 moves
first are rare and therefore it will be likely that any P0 moves
will change it into a U2 puzzle.
More References about Knots
Mathematical Knot Theory is an old research subject so there
exists a vast amount of literature for it. However, it also is a young
subject as several milestones have only been reached in recent
decades. For example, there is a scientific "Journal of Knot
Theory and Its Ramifications" dedicated to knots which has
a new issue every month.
A great book that we recommend is: Adams, Colin (2004), The Knot Book: An Elementary Introduction to the Mathematical Theory of Knots, American Mathematical Society, ISBN 978-0-8218-3678-1
There are also many websites about knots. A good place to start is the Knot Theory page on Wikipedia.
For videos, check out Numberphile's Playlist of Knot videos on YouTube. They also have a great explanation of Colouring Knots, which is another way to help identify diagrams of the same knot.
Caribou produced two posters on Unknotting and colouring knots.
A great book that we recommend is: Adams, Colin (2004), The Knot Book: An Elementary Introduction to the Mathematical Theory of Knots, American Mathematical Society, ISBN 978-0-8218-3678-1
There are also many websites about knots. A good place to start is the Knot Theory page on Wikipedia.
For videos, check out Numberphile's Playlist of Knot videos on YouTube. They also have a great explanation of Colouring Knots, which is another way to help identify diagrams of the same knot.
Caribou produced two posters on Unknotting and colouring knots.
这篇指南将向你介绍绳结理论的主要内容。第一部分介绍重要的数学理论概念和术语。
如果想获得解题思路，可直接跳过第一部分，如果对个别词的理解有困难，可参阅第一部分。
介绍与定义
介绍
首先：什么是绳结？数学概念中的绳结与日常生活中的绳结不同。
日常生活中都有什么绳结?
日常绳结与数学绳结有什么不同?
日常生活中存在数学绳结吗?
解绳结游戏是如何进行的？
定义
从现在开始，我们只讨论数学绳结。为了避免误解，也为了能够更好地判断一个陈述的真假，我们从下述的几个定义开始。
绳结线:
绳结图:
数学结(或简单地说，绳结):
环绕同痕:
阶:
交叉:
通道:
转换:
方向:
交叉偏向性:
扭数:
纽结不变量:
相交数:
弧::
孔:
上方连续弧线:
下方连续弧线:
赖德迈斯特移动:
赖德迈斯特移动 1:
赖德迈斯特移动 2:
赖德迈斯特移动 3:
通道移动:
P-移动:
P0 移动:
P+移动:
可解绳结数量:
如何简化绳结图
解锁 R1 移动
解锁 R2 移动
关于界面(1)
解锁 P-移动
解锁 R3 移动
关于界面 (2)
解锁 P0 移动
解锁 U1移动
解锁 U2 移动
解锁 P0U 移动
绳结参考书
介绍与定义
介绍
首先：什么是绳结？数学概念中的绳结与日常生活中的绳结不同。
日常生活中都有什么绳结?
大多数人利用绳结来系鞋带，戴领带或围巾，以及合上包包，等等……在航海、野营、钓鱼、缝纫、编织和梳头发的过程中，你能见到更多的绳结。
然而，它们并不属于数学绳结！
日常绳结与数学绳结有什么不同?
观察下图所示的两个绳结。令人困惑的是，它们都被称为“八字绳结”，因为它们包含数字8。
日常八字绳结 数学八字绳结
你能发现它们最大的不同吗? 我们知道你可以 指出 它们的不同!
日常八字绳结 数学八字绳结
你能发现它们最大的不同吗? 我们知道你可以 指出 它们的不同!
它们最大的不同在于数学绳结是一个闭合曲线 − 也就是说，它是一个闭环，没有解开的末端。而我们日常生活中所说的 “绳结” 在数学中被称为“辫子” 。
此外，日常的绳结可能有多种材料，在数学概念里，绳结是一个单一的，封闭的，连续的线。多个绳结交织在一起被称为‘链接’。
此外，日常的绳结可能有多种材料，在数学概念里，绳结是一个单一的，封闭的，连续的线。多个绳结交织在一起被称为‘链接’。
日常生活中存在数学绳结吗?
当然！你现在了解到，在数学概念里，绳结是一个单一的、封闭的、连续的线。
在这个概念下, 最简单的数学绳结是什么?
如何用一根绳子打一个数学绳结？
在这个概念下, 最简单的数学绳结是什么?
最简单的数学绳结就是一个环或圆。例如:
我们将在后面对这种绳结进行讨论，但是在现实生活中有很多类似这样的简单环形的例子。
我们将在后面对这种绳结进行讨论，但是在现实生活中有很多类似这样的简单环形的例子。
如何用一根绳子打一个数学绳结？
最简单的数学绳结是圆。所以，做一个数学绳结，只要把绳子的两端粘在一起就可以了。
如果你扭转一次这个圆会发生什么？
这是一个与之前不同的绳结吗？
所有的绳结都能变形成一个圆吗？
一个绳结图可以与最简绳结形式有多大区别？
如果你扭转一次这个圆会发生什么？
如果你拿着一个绳环，扭转，放平，你可能会得到这样的结果:
这是一个与之前不同的绳结吗？
当然不是！你只是扭转了它一下。
这看起来很简单，但是对于研究绳结理论的数学家来说，找到方法来分辨两幅图(图片)是否表示同一个绳结是一个非常重要且困难的问题。
这看起来很简单，但是对于研究绳结理论的数学家来说，找到方法来分辨两幅图(图片)是否表示同一个绳结是一个非常重要且困难的问题。
所有的绳结都能变形成一个圆吗？
想要回答这个问题，可以这样做：
你能把它变形成一个圆吗?
- 拿一根绳子
- 把它拧成一个闭环
- 将绳子的一端通过闭环
你能把它变形成一个圆吗?
不管你怎么试，都没有办法把这个绳结变成一个圆。变成圆至少要剪断绳子，再把它粘在一起。
这实际上是一个不同的数学结，叫做三叶结，因为它看起来像三叶草。
这实际上是一个不同的数学结，叫做三叶结，因为它看起来像三叶草。
一个绳结图可以与最简绳结形式有多大区别？
数学绳结的另一个重要性质是它们可以被任意地拉伸和弯曲。例如，我们图中的最简绳结看起来更像一个正方形而不是圆形 − 我们可以把它画成一个完美的圆，而它仍然是同一个绳结。你可以取一个最简单的绳结，一个圆，把它拉成一个又长又细的椭圆，然后把它当成一根绳子，把它打一个“八字绳结”。在数学概念上，它仍然是一个圆。
如你所想，拥有相同基础绳结的不同绳结图可以看起来完全不同！例如，只要有足够的耐心， 戈尔迪之结 和 这个绳结图 都可以变形成一个圆。它们是拥有相同环形绳结的绳结图。
圆是画这个绳结最简单的方法，但这个绳结不是一个真正的圆，它是一个抽象的数学对象，我们可以用很多不同的方式来表示。同样，“一辆车”和“一个苹果”不是数字1，一个圆只是表示这个绳结的一种方式。
如你所想，拥有相同基础绳结的不同绳结图可以看起来完全不同！例如，只要有足够的耐心， 戈尔迪之结 和 这个绳结图 都可以变形成一个圆。它们是拥有相同环形绳结的绳结图。
圆是画这个绳结最简单的方法，但这个绳结不是一个真正的圆，它是一个抽象的数学对象，我们可以用很多不同的方式来表示。同样，“一辆车”和“一个苹果”不是数字1，一个圆只是表示这个绳结的一种方式。
解绳结游戏是如何进行的？
在这个游戏中，通过变形绳结图来尽可能的减少交叉的数量。
当点击鼠标“切断”这些线时，你只是改变了图，而不是基础的绳结。这就是设置重新连接被“切断”末端限制的原因。 这一限制保证了数学绳结在重新连接后不变，即使绳结的外观有改变。
在现实生活中，我们都在什么情况下解开日常绳结？
尝试一些其他的数学绳结游戏!
当点击鼠标“切断”这些线时，你只是改变了图，而不是基础的绳结。这就是设置重新连接被“切断”末端限制的原因。 这一限制保证了数学绳结在重新连接后不变，即使绳结的外观有改变。
在现实生活中，我们都在什么情况下解开日常绳结？
你可能对像耳机那样的导线很熟悉。如果你系 鞋带，首先必须解开它。
现实中的绳结也有同样的性质，这使得它们很有用，但也更难解开。
是什么让现实生活中的绳结如此难解？
现实中的绳结也有同样的性质，这使得它们很有用，但也更难解开。
是什么让现实生活中的绳结如此难解？
答案是摩擦! 然而，数学绳结不存在摩擦。你可以把它们的表面想象成‘非常光滑’。
尝试一些其他的数学绳结游戏!定义
从现在开始，我们只讨论数学绳结。为了避免误解，也为了能够更好地判断一个陈述的真假，我们从下述的几个定义开始。
绳结线:
在三维空间中的一条闭合曲线，该曲线本身并不 相交，并且有一定的厚度(以避免绳结线上出现无限多个越来越小的绳结)，比如: 八字绳结。
绳结图:
一条绳结线在两个维度上的投影，其中绳结线的不同部分可以互相交叉(在此网站上，线交叉的角度为 90°)，但不相互重叠。
绳结图示例：
可解绳结 三叶草绳结 八字绳结 五叶形绳结 三维绳结
‘可解绳结’又称为‘平凡绳结’。
下列三个绳结图分别表示的是哪一种绳结？
这个绳结展示的是哪一个绳结图？
你能看出是如何将它变形成基础三叶草绳结的吗？
绳结图示例：
可解绳结 三叶草绳结 八字绳结 五叶形绳结 三维绳结
‘可解绳结’又称为‘平凡绳结’。
下列三个绳结图分别表示的是哪一种绳结？
可解绳结。你能看出它是如何变形成矩形的吗?
这个绳结展示的是哪一个绳结图？
三叶草绳结。
你能看出是如何将它变形成基础三叶草绳结的吗？
你可以通过向下翻转顶部的弧线将这个绳结图变形成一个新的绳结图。
→
→
数学结(或简单地说，绳结):
一组(无限多个)绳结图背后的抽象对象，这些结图都可以变形、拉伸和移动，且不需要切割。例如: 31绳结也被称为“三叶草绳结”，这是最简单的特殊绳结。
环绕同痕:
数学术语，指一条绳结线可以不被切断而变形成另一条绳结线。
阶:
本网站仅使用 6 个图块绘制了绳结图：我们称为阶。
交叉:
图中阶和阶相互交叉的地方。
通道:
是交叉的一部分，有上方通道(完全可见)和下方通道(部分覆盖)，
转换:
可转换绳结图中交叉的上方通道和下方通道。
如果一个交叉被转换，原绳结图和新绳结图表示完全不同的绳结。转换所有的交叉相当于把一个绳结变换成它的镜像。一些绳结与它们的镜像完全相同，比如八字绳结，其他则并不相同，比如三叶草绳结。
这个绳结可以变形成它的镜像吗?
这个绳结可以变形成它的镜像吗?
可以！此图表示的为非手性八字绳结，可以变形成其镜像，如下所示:
注意，第一个图和最后一个图之间唯一的区别是所有的交叉都被转换了!
初始位置
一个180° 度旋转
移动一个绳
镜像
注意，第一个图和最后一个图之间唯一的区别是所有的交叉都被转换了!
方向:
方向不是绳结线或绳结的属性，而是一个如何沿着绳结线移动的问题。绳结线可向 2 个方向移动。
如果一个绳结可以通过环境同痕转换成与自己方向相反的绳结，则可以称其为‘可逆绳结’，否则，则为不可逆绳结。最小的不可逆绳结为817 ，它们是非手性的，但如果加入一个方向，就变成了手性的（详见可逆绳结维基百科）。添加更多的结构(这里指添加一个方向)会导致它失去对称性(不再与它的镜像相同)。
手性结(不能变形成其镜像绳结的绳结)仍然可以是可逆的(对方向变化对称)。这样的绳结称为“可逆绳结”。
手性结(不能变形成其镜像绳结的绳结)仍然可以是可逆的(对方向变化对称)。这样的绳结称为“可逆绳结”。
交叉偏向性:
对于给定的绳结图，交叉要么是左交叉要么是右交叉。
下文中我们将从基本原理出发，推导出两种交叉类型，并分别命名为右交叉和左交叉。
如果我们将上通道/下通道以及两个方向都考虑在内的话，那么有多少个不同的交叉？
如果在绳结图不变的前提下仅转换一个交叉，交叉的偏向性是否会发生变化？
如何我们的手来判断一个交叉是左交叉还是右交叉？
下文中我们将从基本原理出发，推导出两种交叉类型，并分别命名为右交叉和左交叉。
如果我们将上通道/下通道以及两个方向都考虑在内的话，那么有多少个不同的交叉？
共有 8 种情况:
如果水平通道在上方，则有 4 种可能：
如果水平通道在上方，则有 4 种可能：
交叉1、7、4、6 可称作右交叉，而
交叉2、5、3、8可称作左交叉。
如果水平通道在上方，则有 4 种可能：
1
2
3
4
5
6
7
8
- 原理1：交叉偏向不应取决于方向(穿过绳结线的方向)，所以翻转两个箭头，我们确定了 4 对交叉:1 = 4, 2 = 3, 5 = 8, 6 = 7。所以，不管我们得到的是哪一组，交叉 1 和 4 都应该在同一组中，以此类推。
- 原理2: 如果我们旋转整个绳结，交叉所属的组不应该改变。因此我们确定交叉 1 = 7 = 4 = 6 和交叉 2 = 5 = 3 = 8。
交叉1、7、4、6 可称作右交叉，而
交叉2、5、3、8可称作左交叉。
如果在绳结图不变的前提下仅转换一个交叉，交叉的偏向性是否会发生变化？
会。试着转换八个交叉中的任意一个，看看它变成了哪一个，并检查它是否仍然在原交叉组。例如，将交叉1 转换得到交叉 5，两者在不同的交叉组。
在此说明：交叉为左交叉还是右交叉的问题不仅仅取决于交叉本身，还与其周围的绳结图有关。
证明：水平通道在上方还是下方并不决定交叉的偏向性。两种情况都可能是左交叉和右交叉（见上述 8 个交叉）。如果旋转绳结，则上方通道是水平方向。如果使交叉穿过上方通道向右转换（朝东），交叉的方向则取决于剩下的绳结是从南面返回交叉（右交叉），还是从北面返回交叉（左交叉）。
这两组交叉被称为左交叉和右交叉，提示我们可以以此来区分交叉的方向。
在此说明：交叉为左交叉还是右交叉的问题不仅仅取决于交叉本身，还与其周围的绳结图有关。
证明：水平通道在上方还是下方并不决定交叉的偏向性。两种情况都可能是左交叉和右交叉（见上述 8 个交叉）。如果旋转绳结，则上方通道是水平方向。如果使交叉穿过上方通道向右转换（朝东），交叉的方向则取决于剩下的绳结是从南面返回交叉（右交叉），还是从北面返回交叉（左交叉）。
这两组交叉被称为左交叉和右交叉，提示我们可以以此来区分交叉的方向。
如何我们的手来判断一个交叉是左交叉还是右交叉？
把手张开，让所有手指处于同一平面，大拇指与相互平行的其他手指成直角，旋转手部，可以看到手掌和大拇指指向上方通道向外延伸的方向，而其他手指指向下方通道的延伸方向。可以用手判断出交叉的偏向。
例如，交叉如下所示，像这样伸出手：
因为这个动作只能用左手来完成，所以这是一个左交叉。
右交叉和左交叉也可表示为正交叉和负交叉。
因为这个动作只能用左手来完成，所以这是一个左交叉。
右交叉和左交叉也可表示为正交叉和负交叉。
扭数:
在一个绳结图中左交叉和右交叉的数量之差，以绳结图为例，这个数字可以用来分辨具有同样绳结的 2 个不同绳结图。
这个绳结图的扭数是多少?
这个绳结图的扭数是多少?
绳结图中的右交叉显示为红色，左交叉显示为绿色。想要知道扭数，需要先计算左交叉和右交叉的数量，然后用左交叉数减去右交叉数。
这个绳结图有二个左交叉和4个右交叉，所以，它的扭数为2 − 4 = −2。
纽结不变量:
表示一个绳结所有(无限多）特征的一种数字或多项式或一种可行性陈述。一个绳结的手性/非手性，可逆/不可逆，可反转的性质都是扭结不变量。
相交数:
一个绳结在任意图形的最小相交数，这是每个绳结自身的特征，因此是一个不变量。
这个绳结图有多少个交叉:
图中所表示的绳结的相交数是多少?
一个绳结可拥有的最少相交数是哪两个?
这个绳结图有多少个交叉:
有5个交叉
图中所表示的绳结的相交数是多少?
0！相交数是抽象数学绳结的一种属性，而不是绳结图的属性。上面的图可以变形为具
有零交叉的图。
你知道为什么可以这样吗？
因为相交数是任意绳结图的最小相交数，而且一个绳结的相交数不能小于零，所以上图中所表示的绳结的相交数为零。
有零交叉的图。 你知道为什么可以这样吗？
因为相交数是任意绳结图的最小相交数，而且一个绳结的相交数不能小于零，所以上图中所表示的绳结的相交数为零。
一个绳结可拥有的最少相交数是哪两个?
可解绳结的最小相交数为0。
一个有1个交叉的绳结图如下：
可以将其变形为可解绳结。 2个有1个交叉的绳结图如下：
可以将其变形为可解绳结。
上面的三叶草绳结图有 3 个交叉，不能变形为可解绳结，所以最小相交数为 0 和 3。
可以将其变形为可解绳结。 2个有1个交叉的绳结图如下：
可以将其变形为可解绳结。上面的三叶草绳结图有 3 个交叉，不能变形为可解绳结，所以最小相交数为 0 和 3。
弧::
图中从一个交叉到下一个交叉的绳结线部分。
带有 N 个交叉的绳结图有多少个弧？
带有 N 个交叉的绳结图有多少个弧？
每个交叉有 4 条弧线的末端。每个弧有两个端点，所以弧的数量是交叉的 4/2 =2 倍，因此有 2N 个弧。
孔:
绳结图中被弧线包围的空白区域。绳结图外部的整个空白区域也是一个孔。
一个有 N 个交叉的绳结图有多少个孔？
一个有 N 个交叉的绳结图有多少个孔？
可以画多个绳结然后猜出一个公式，但是也可以推导出来。欧拉公式表示，对于平面上的任何图形，其中 m 条线（此处有 m=2N 个弧）各连接 n个点 (此处 n=N 个交叉)中的 2 个，那么面的数量（此处为孔）为 f = 2 + m − n。因此一个绳结的孔的数量为：2 + 2N − N = N + 2。
上方连续弧线:
纽结图中的连续弧线（即彼此相继的弧），起始端和终端都在下方通道。因此有0，1 或者多条上方通道。
下方连续弧线:
A纽结图中的连续弧线（即彼此相继的弧），起始端和终端都在上方通道。因此有0，1或者多条下方通道。
（在文献中，我们通常称一条上方连续弧线为“线段”。对于我们来说，一条上方弧线的上方通道数量与一条下方弧线的下方通道数量同样重要，因此我们需要同时考虑两种情况。）
哪种线段是由 5 条弧线组成的平行线？
带有 N个交叉的纽结图有多少个上方连续弧线？
（在文献中，我们通常称一条上方连续弧线为“线段”。对于我们来说，一条上方弧线的上方通道数量与一条下方弧线的下方通道数量同样重要，因此我们需要同时考虑两种情况。）
哪种线段是由 5 条弧线组成的平行线？
这是一个带有 4 个下方通道的下方连续弧线。
带有 N个交叉的纽结图有多少个上方连续弧线？
每一个交叉有两个线段的终端（两条不同线段的同一个终端或者一条线段的两个终端）。另一方面，处于交叉位置的每条线段有两个终端。因此交叉数量与上方连续弧线数量相同。由于对称数量与下方弧线的数量相同，所以每条下方弧线有 N 个交叉。
赖德迈斯特移动:
1927年，德国数学家库尔特·赖德迈斯特、詹姆斯·亚历山大和加兰·贝尔德·布里格斯(1926年) 分别证明了任何两个表示同一绳结的图形都可以通过 3 种不同的步法彼此变形。但问题是，在变形过程中，交叉的数量可能会暂时增加，而这种增加的上限和需要移动的数量都是未知的。
赖德迈斯特移动 1:
删除或添加由一个弧围成的孔:
哪个图形表示左交叉，哪个图形表示右交叉？
哪个图形表示左交叉，哪个图形表示右交叉？
左图是左交叉，右边的图是右交叉。因此，赖德迈斯特移动 1 会将右交叉或左交叉的次数减 1，从而绳结图的扭数发生改变。
赖德迈斯特移动 2:
移除或添加由两个弧围成的孔:
在赖德迈斯特移动 2中，增加或移除的两个交叉偏向应该怎样描述？
在赖德迈斯特移动 2中，增加或移除的两个交叉偏向应该怎样描述？
在这两个交叉中，一个是右交叉，一个是左交叉。因此，赖德迈斯特移动 2 不会改变绳结图的扭数。
赖德迈斯特移动 3:
删除或添加一个由 3 个弧围成的孔。
你能想出哪两种被 3 个弧线围成的孔?
你能想出哪两种被 3 个弧线围成的孔?
任何一个：
证明 3 次移动的结果是一致的。
在这 3 次移动中，3 个交叉偏向有没有变化？
我们学到了什么?
- 1)每一个弧线都有一个上方通道和一个下方通道：
那么这三条弧线中任意一条都无法移动到其他交叉的上方/下方或穿过交叉
- 一个弧有 2 个上方通道，一个弧有一个上方通道和一个下方通道，还有一个弧有 2 个下方通道：
之后进行 3 次移动。可以移动处于上方通道的上方的连续弧线：
或者移动变成下方通道弧线上方的弧线:
或者移动处于下方通道弧线下方的弧线:
证明 3 次移动的结果是一致的。
在 3 次移动的每一次移动中，发生的变化为：3 条弧线中的其他两条是以相反的顺序进行交叉的。即对于中间的弧线来说，上方通道和下方通道的顺序是相反的。
在这 3 次移动中，3 个交叉偏向有没有变化？
没有。每一条线段选择任意一个方向，利用手势法进行分析。
我们学到了什么?
我们了解到：
- 如何利用3条弧线去识别孔，以实现赖德迈斯特移动 3,
- 对于这样的孔，需要考虑进行赖德迈斯特移动 3,
- 这三个交叉偏向没有改变,
- 中间弧线的上方通道和下方通道的顺序是相反的。
通道移动:
这与上面定义的“通道”无关。通道移动用另一个上方-（下方-）弧线来代替先前的上方-（下方-）弧线，这两个弧线的终端相同。例如：参考 P-, P0 and P+的移动如下。
P-移动:
新弧线的通道移动要少于原有弧线的通道移动。
在该绳结图中找到可以替换青色线段的P-移动：
在该绳结图中找到可以替换青色线段的P-移动：
在该绳结图中新的红色线段通道数量少于原有的青色线段的通道数量。因此该绳结图属于P-步法。
P0 移动:
新线段的通道数量与原线段的通道数量相等。
在该绳结图中找出可以替换青色弧线的 P0 移动:
在该绳结图中找出可以替换青色弧线的 P0 移动:
在该绳结图中新的红色线段的通道数量与原有的青色线段的通道数量相等。因此该绳结图表示的是 P0 移动。
P+移动:
新线段通道数量多于原有线段的通道数量。这些步法包括所有其他可能的通道移动。除非 P+移动有其他一些有用的属性，否则并没有什么意义，因此不在该网站显示。
在这个图中找到一个P+移动取代青色线段:
如想要改变绳结图的扭数，P+移动是必要的。关于这一点的更多信息将在下方“寻找P0移动”中进一步描述。
在这个图中找到一个P+移动取代青色线段:
在这个图中，新的红色线段的通道数量比原有的青色线段的通道数量多一个，因此这是一个P+移动。
如想要改变绳结图的扭数，P+移动是必要的。关于这一点的更多信息将在下方“寻找P0移动”中进一步描述。
可解绳结数量:
这是绳结的属性而不是绳结图的属性。因此我们叫做“不变量”，它与绳结图的“形态”无关。
可解绳结的数量是为得到可解绳结进行交叉转换的最小数量。在交叉第一次转换之前以及转换中间，绳结图可以进行任意变形。因此可解绳结的数量很难确定。
为什么三叶草绳结的可解绳结数量为 1？
可解绳结的数量是为得到可解绳结进行交叉转换的最小数量。在交叉第一次转换之前以及转换中间，绳结图可以进行任意变形。因此可解绳结的数量很难确定。
为什么三叶草绳结的可解绳结数量为 1？
由于三叶草绳结无法通过变形使得可解绳结数量为 0，所以它的可解绳结数量无法为 0. 所以三叶草纽结的可解绳结大于等于1。我们很容易发现，按上面进一步显示的三叶草图形所示，切换到图形中的任意一个交叉都会产生可解绳结，所以三叶草纽结的可解绳结的数量小于等于1。如果它既大于等于1，又小于等于1，那么它只能等于1.
如何简化绳结图
解锁 R1 移动
关于R1 移动的简单案例如下：
我们可以把线圈翻转 4 次，绳结马上就会解开，沿着结绳找到有两个有相同终端的弧线。执行 R1 移动的顺序并不重要。
关于赖德迈斯特移动 1 的应用还有更多普遍的案例。如果一个上方连续弧线起点和终点相同，但完全位于与其他弧线的顶部（因此可以称为“上方连续弧线”），那么可以切断或移除此线圈。例如，首先可以把顶端的青色线圈移除，然后再依次移除其他线圈：
当此线圈完全处于最底部时也可以移除：
我们可以把线圈翻转 4 次，绳结马上就会解开，沿着结绳找到有两个有相同终端的弧线。执行 R1 移动的顺序并不重要。
关于赖德迈斯特移动 1 的应用还有更多普遍的案例。如果一个上方连续弧线起点和终点相同，但完全位于与其他弧线的顶部（因此可以称为“上方连续弧线”），那么可以切断或移除此线圈。例如，首先可以把顶端的青色线圈移除，然后再依次移除其他线圈：
当此线圈完全处于最底部时也可以移除：
解锁 R2 移动
与 R1 移动相似，我们很容易就可以找到 R2 移动的原型，如下：当 R2 移动在 R1 之前完成时会产生可解绳结：
在以下示例中，需要用相同的弧线执行两次 R2 移动，一次是从底部拉动线段，一次是从上面拉动线段：
最后一步不是R2移动。仅用于说明最后的绳结是两个三叶草绳结共同呈现的结果。
在以下示例中，需要用相同的弧线执行两次 R2 移动，一次是从底部拉动线段，一次是从上面拉动线段：
最后一步不是R2移动。仅用于说明最后的绳结是两个三叶草绳结共同呈现的结果。
关于界面(1)
以上示例可以很好地证明游戏界面的最佳使用情况。在截断绳结之后：
我们不能一直撤回某一端后继续撤回另一端，因为我们首先移除了一个下方通道，使两端都处于下方通道的状态；在这一状态下，无法移除一个上方通道。相反，我们移除一个下方通道，跳到另一端，移除另一个下方通道，让两端处于同一个孔中，两端处于一个中立状态。然后我们可以移除上方通道，重新连接两端。简言之，在端点之间跳动以移除两端之间所有的下方通道和上方通道。
两端达到的状态并不是该程序的一个缺陷，反而保证了绳结图交互式的修改不会改变数学绳结。
我们不能一直撤回某一端后继续撤回另一端，因为我们首先移除了一个下方通道，使两端都处于下方通道的状态；在这一状态下，无法移除一个上方通道。相反，我们移除一个下方通道，跳到另一端，移除另一个下方通道，让两端处于同一个孔中，两端处于一个中立状态。然后我们可以移除上方通道，重新连接两端。简言之，在端点之间跳动以移除两端之间所有的下方通道和上方通道。
两端达到的状态并不是该程序的一个缺陷，反而保证了绳结图交互式的修改不会改变数学绳结。
解锁 P-移动
P-移动通过上方通道较少的上方连续弧线替代上方通道较多的上方连续弧线，或者通过下方通道较少的弧线替代下方通道较多的下方连续弧线。在这两种情况中，交叉点的数量减少了。
为解锁这种移动，需要穿过结绳，尽可能多的找到连续的上方通道或者下方通道，至少要找到两个。如果找到这样的线段，比如一个上方连续弧线，尝试找到一条替代路径，这条路径的上方通道较少，可以连接相同的下方路径的两端。
在该例子中，有 4 条上方通道的原青色上方连续弧线被红色上方连续弧线所取代，红色上方连续弧线较长，但上方通道较少。
解锁的连续通道越多，找到不同路径（需要较少通道）的几率就越高。
为解锁这种移动，需要穿过结绳，尽可能多的找到连续的上方通道或者下方通道，至少要找到两个。如果找到这样的线段，比如一个上方连续弧线，尝试找到一条替代路径，这条路径的上方通道较少，可以连接相同的下方路径的两端。
在该例子中，有 4 条上方通道的原青色上方连续弧线被红色上方连续弧线所取代，红色上方连续弧线较长，但上方通道较少。
解锁 R3 移动
R1, R2和 P- 移动改变了交叉的数量。R3 移动不会改变交叉的数量，因此我们把对 R3 移动的描述放在了P-移动之后。以下例子表明通过执行P-移动，R3 移动为有效移动。如第一部分所述，一个R3 孔由三部分组成：一个顶部弧线（2个上方通道终端）（此处为A,B），一个中部弧线（1个上方通道（C）和1个下方通道（B）），一个底部弧线（两个下方通道终端（此处为A,C）。
(可解绳结来源)
在上一部分中，我们发现 R3 移动翻转了中间线段(这里是通过 B,C)下方和上方通道的顺序。如果 R3 移动导致中间弧线延续，连续上方通道或连续下方通道增加，进而增加找到 P-移动的几率，那么 R3 移动就是有效的。如果在中间弧线的延长线上，下方通道B之后有一个上方通道（D、E 都是上方通道），或者在上方通道（C）之后紧跟着一个下方通道（F、G、H都是下方通道），就会出现上述情况。R3 移动使中间弧线 BC 在点 A 的上方弧线和底部弧线之间滑动。
在这之前，D 和 E 仅有 2 个上方通道，现在 C, D 和 E 有 3 个上方通道。这个较长的上方连续弧线可以以P-移动的方式改变路径。
把交叉数量从 13 减到 11。
同样，线段的另一侧也可以以P-移动的方式改变路径。在改变路径之前，F, G 和 H 有3个连续的下方通道，现在B, F 和 G 有4个连续下方通道。这种P-移动会导致：
交叉数量减少2个。通过P-和R1 移动，两个绳结图都可以进一步简化，最终得到可解绳结。你知道怎么做吗？请参考上文——如何识别P-移动。
我们来通过一个例子来练习一下。
在这个绳结图中有多少个可能的R3移动:
在上一部分中，我们发现 R3 移动翻转了中间线段(这里是通过 B,C)下方和上方通道的顺序。如果 R3 移动导致中间弧线延续，连续上方通道或连续下方通道增加，进而增加找到 P-移动的几率，那么 R3 移动就是有效的。如果在中间弧线的延长线上，下方通道B之后有一个上方通道（D、E 都是上方通道），或者在上方通道（C）之后紧跟着一个下方通道（F、G、H都是下方通道），就会出现上述情况。R3 移动使中间弧线 BC 在点 A 的上方弧线和底部弧线之间滑动。
同样，线段的另一侧也可以以P-移动的方式改变路径。在改变路径之前，F, G 和 H 有3个连续的下方通道，现在B, F 和 G 有4个连续下方通道。这种P-移动会导致：
我们来通过一个例子来练习一下。
在这个绳结图中有多少个可能的R3移动:
有三种可能的R3 移动。对于每一个，我们用浅蓝色表示所涉及的三个弧。很容易被忽略的是第三个洞，洞是整个外部空间，只有3个弧线“包围”。
执行 1. R3 移动，看看是否有效:
执行 2. R3 移动，看看是否有效:
执行 3. R3 移动，看看是否有效:
1. R3 移动
2. R3 移动
3. R3 移动
执行 1. R3 移动，看看是否有效:
这个R3 移动时候有效移动。它允许之后的 P- 移动，如以下系列移动所示。定义R3 移动有效，并不一定需要P- 移动，而是即使不执行所有这些移动，也可以很容易地看到连续上方或下方通道数量的增加。如以下绳结图所示，R3 孔的中间弧线在A处有一个上方通道，在B处有一个下放通道，其后在C和D处有两个上方通道。在R3 移动中，中间弧线的上方通道和下放通道调换，如图3所示， 其现在有3个连续的上方通道。这足以在图5 中解锁一个至少需要3个通道的P- 移动。
关于下面图表的顺序:在 Dia 1 中，我们留出空间准备在 Dia 2 中完成 R3 移动(这里通过移动顶部的弧)，结果在 Dia 3 中。在 Dia 4 中，我们为 Dia 5 中的P-移动做了准备，在 Dia 6 中，有3个青色线段的上方通道被只有1个上方通道的红色线段取代。在 Dia 7 中，我们移动一条线段，为 Dia 9 的下一个P-移动腾出空间，结果在 Dia 10 和 Dia 11中，很容易发现knot 51。
1. 扩大
2. R3 移动
3. R3 移动之后
4. 扩大
5. P- 移动
6. P- 移动之后
7. 准备下一个P- 移动
8. P- 移动之前
第二个P- 移动
10. 后续
11. 缩小
执行 2. R3 移动，看看是否有效:
结果显示，R3 移动有效。
1. 扩大
2. R3 移动
3. R3 移动之后
4. P- 移动
5. P- 移动之后
6. P- 移动之后
7. P- 移动之后
8. 缩小
执行 3. R3 移动，看看是否有效:
第三个 R3 移动也是有效的。完成这一移动需遵循同样的原则：中间的一股线切断另外两股以相反顺序“围绕”外面的洞的线。
结果就是，沿着淡蓝色的 Dia 2 向上延伸，首先是一个上方通道，然后是一个下方通道。如果进行R3移动后，其中一条沿着红色线段移动，另一条就会以相反的顺序通过另外两条线，因此首先遇到一个下方通道，然后是一个上方通道。正如我们在 Dia 5 中看到的，这里的浅蓝色线段的3个连续下方通道的序列使P-移动成为可能。
结果就是，沿着淡蓝色的 Dia 2 向上延伸，首先是一个上方通道，然后是一个下方通道。如果进行R3移动后，其中一条沿着红色线段移动，另一条就会以相反的顺序通过另外两条线，因此首先遇到一个下方通道，然后是一个上方通道。正如我们在 Dia 5 中看到的，这里的浅蓝色线段的3个连续下方通道的序列使P-移动成为可能。
1. 扩大
2. R3 移动
3. R3 移动之后
4. 扩大
5. P- 移动
6. 扩大
7. 第二个P- 移动
8. P- 移动之后
9. 缩短
10. 缩短之后
11. 缩短
12. 校正
13. ↻旋转90°
关于界面 (2)
上面的例子很好地展示了如何用我们的界面执行R3移动。
正如‘定义介绍’>‘赖德迈斯特移动 3’所示，有三种完成赖德迈斯特移动 3的方法：移动底部线段，移动中间线段，或者移动顶部线段。如图所示，这3种方法结果相同，均执行了Reidemeister 3移动。
我们的界面只允许通过移动底部线段或顶部线段来完成R3移动，但不允许移动中间线段。因为在我们的界面里，两端一次只能添加/移除一个上方通道，或者添加/移除一个下方通道。但这并不妨碍我们执行R3移动，因为移动三条线段中的任何一条都会得到相同的结果。
正如‘定义介绍’>‘赖德迈斯特移动 3’所示，有三种完成赖德迈斯特移动 3的方法：移动底部线段，移动中间线段，或者移动顶部线段。如图所示，这3种方法结果相同，均执行了Reidemeister 3移动。
我们的界面只允许通过移动底部线段或顶部线段来完成R3移动，但不允许移动中间线段。因为在我们的界面里，两端一次只能添加/移除一个上方通道，或者添加/移除一个下方通道。但这并不妨碍我们执行R3移动，因为移动三条线段中的任何一条都会得到相同的结果。
解锁 P0 移动
P0 移动是通道移动，未改变交叉数量，就像 R3 移动是 P0 移动的特殊形式。正如 R3 移动，P0 移动可以是有效的，并且可以进行P-移动。由于P0 移动通常情况下用处较小，它们出现的次数较为频繁，但是P0 移动是否能够进行P-移动的操作，挑战性较大。
为找到 P0 移动，我们需要找到一个类似于 P0 移动的上方弧和一个下方弧。为检验 P0 移动是否有效以及是否能够进行P-移动，我们需要执行与 R3 移动相似的步骤。一个人看一下移除这条线段是否会增加交叉线段连续上方通道或者连续下方通道的数量，另一个人检查在重新修改新的线段的路径之后，是否会增加交叉线段的连续上方通道或者连续下方通道的数量。选择两种情况中的任意一个，都要检查是否可以以较少交叉重新设置具有更多连续上方或下方通道的线段。
让我们看一下这个例子：
我们标出交叉：
逐步解锁有效的 P0 移动。
你能看到多少个至少有2个上方通道的上方连续弧线？多少个至少有2个下方通道的下方连续弧线？
不难看出 IE 弧线进行了一次 P0 移动，IE 弧线的新位置穿过了 GC 和 BH 弧线：
但问题是这个P0 移动是否有效。
移动 IE 线段会增加先前与 DF 或 HJ 交叉线段的连续上方通道或者连续下方通道的数量吗？
当该线段与 GC 和 BH 线段重合时会产生更多连续上方或下方通道吗？
BH 线段可以通过P-移动改变其路径减少交叉数量吗？
在最初的绳结图中，P-移动可以首先按照 P0 移动完成，然后再将上述 P0 移动按照 P- 移动完成，共减少两个交叉。
处理较为困难的问题时，可能需先执行几次P0 移动，然后才能进行P- 移动。
如果交叉数量是像这样的交叉数量（参见第一部分），那就需要最大程度地简化绳结图。在这种情况下，执行P0 移动之后无法进行P- 移动。
为找到 P0 移动，我们需要找到一个类似于 P0 移动的上方弧和一个下方弧。为检验 P0 移动是否有效以及是否能够进行P-移动，我们需要执行与 R3 移动相似的步骤。一个人看一下移除这条线段是否会增加交叉线段连续上方通道或者连续下方通道的数量，另一个人检查在重新修改新的线段的路径之后，是否会增加交叉线段的连续上方通道或者连续下方通道的数量。选择两种情况中的任意一个，都要检查是否可以以较少交叉重新设置具有更多连续上方或下方通道的线段。
让我们看一下这个例子：
你能看到多少个至少有2个上方通道的上方连续弧线？多少个至少有2个下方通道的下方连续弧线？
我们找到 4 个至少有 2 个上方通道的上方连续弧线：AB, GC, IE，以及 4 个至少有两个下方通道的下方连续弧线：EF, BH, DJ.
不难看出 IE 弧线进行了一次 P0 移动，IE 弧线的新位置穿过了 GC 和 BH 弧线：
移动 IE 线段会增加先前与 DF 或 HJ 交叉线段的连续上方通道或者连续下方通道的数量吗？
是的，HJ 弧线现在有 2 个上方通道，但此弧线没有办法改变路径来以较少的上方通道去连接两个相同的孔。
当该线段与 GC 和 BH 线段重合时会产生更多连续上方或下方通道吗？
是的，BH下方连续弧线原本有2个下方通道，现在有3个下方通道。
BH 线段可以通过P-移动改变其路径减少交叉数量吗？
是的：BH线段新的路径连接了相同的孔，但是只有一个下方通道而不是3个下方通道。
在该绳结图中，新的线段比（涉及到更多阶）被替代的线段更长。但这不是重点。重点是把交叉点数量从10 减到8，这样可以确定该绳结为绳结17。 在最初的绳结图中，P-移动可以首先按照 P0 移动完成，然后再将上述 P0 移动按照 P- 移动完成，共减少两个交叉。
处理较为困难的问题时，可能需先执行几次P0 移动，然后才能进行P- 移动。
如果交叉数量是像这样的交叉数量（参见第一部分），那就需要最大程度地简化绳结图。在这种情况下，执行P0 移动之后无法进行P- 移动。
解锁 U1移动
U1移动转换交叉，这样可以简化绳结图，删除所有交叉。并显示该转换产生了可解绳结。
通常完美的绳结应该有独特的解决方案。因此我们的 U1,U2 所展示的绳结图为仅转换一个交叉就可以产生可解绳结。该结果提示您可以减少搜索次数。
如果此图形包括类似的结绳:
对哪一个交叉进行转换很重要吗？
如果还有几个剩余的交叉，应试着去想象一下通过转换可以实现多少次 R1,R2移动。尝试首先进行转换，这样之后便可以生成最简图。
尽量减少尝试转换的另一个提示是，想象一下转换之后是否一定会像三叶草纽一样保留一个绳结。如果是这样的话，那么在 U1 中进行这样的转换就是错误的。
在工作表上，限制可用转换的数量为获得可解绳结所需的最少数量。如果进行了转换，则由于该图可能已更改而无法将其切换回原来的状态，因此必须重置该图。
通常完美的绳结应该有独特的解决方案。因此我们的 U1,U2 所展示的绳结图为仅转换一个交叉就可以产生可解绳结。该结果提示您可以减少搜索次数。
如果此图形包括类似的结绳:
转换哪一个交叉并不重要。两个转换的结果是相同的。
因此，这两个交叉既可以进行可解绳结转换也可以都不进行可解绳结转换。所以这两个交叉可以忽略。
=
=
如果还有几个剩余的交叉，应试着去想象一下通过转换可以实现多少次 R1,R2移动。尝试首先进行转换，这样之后便可以生成最简图。
尽量减少尝试转换的另一个提示是，想象一下转换之后是否一定会像三叶草纽一样保留一个绳结。如果是这样的话，那么在 U1 中进行这样的转换就是错误的。
在工作表上，限制可用转换的数量为获得可解绳结所需的最少数量。如果进行了转换，则由于该图可能已更改而无法将其切换回原来的状态，因此必须重置该图。
解锁 U2 移动
对于 U2 图来说，关于等效转换的提示也同样适用于U1图。同样，对于 U2 图形来说，只有一个交叉可以减少可解绳结的数量，即找到可解绳结。一旦转换了唯一的第一个交叉，简化了绳结图，那么就有可能出现多个可转换的交叉，产生更多的可解绳结。
解锁 P0U 移动
驯鹿竞赛关于可解绳结所进行的研究表明，最大程度简化的绳结图形是存在的（这种图形交叉的数量最少），但没有办法进行简化转换。换句话说，有些图形即使进行交叉点转换也无法达到可解绳结的状态。在这种情况下，首先应该进行 1 次或多次 P0移动，这样可以把图形变成U2图形。好消息是，首先进行 P0 移动的绳结图是不常见的。因此，任何P0 移动都有可能将其改变为 U2 图形。
绳结参考书
数学的绳结理论是一个比较旧的研究课题，有大量关于此课题的文献。但是它又是一个比较新的课题，因为在近几十年才实现了一些里程碑式的成果。例如有一本专门研究绳结科学的《结理论及其分支期刊》，每月出版一期。
好书推荐： Adams, Colin (2004), The Knot Book: An Elementary Introduction to the Mathematical Theory of Knots, American Mathematical Society, ISBN 978-0-8218-3678-1
Adams，Colin（2004），《绳结之书：绳结的数学理论基础》，美国数学学会，ISBN 978-0-8218-3678-1
关于绳结的网站有很多 可从维基百科查起。
关于视频，请查看 数字狂绳结视频 YouTube 播放列表. 他们对 彩色绳结也有出色的讲解，这是另一种方式来帮助识别拥有相同绳结的绳结图。
驯鹿制作了两张关于 可解绳结 和 着色绳结 的海报。
好书推荐： Adams, Colin (2004), The Knot Book: An Elementary Introduction to the Mathematical Theory of Knots, American Mathematical Society, ISBN 978-0-8218-3678-1
Adams，Colin（2004），《绳结之书：绳结的数学理论基础》，美国数学学会，ISBN 978-0-8218-3678-1
关于绳结的网站有很多 可从维基百科查起。
关于视频，请查看 数字狂绳结视频 YouTube 播放列表. 他们对 彩色绳结也有出色的讲解，这是另一种方式来帮助识别拥有相同绳结的绳结图。
驯鹿制作了两张关于 可解绳结 和 着色绳结 的海报。
EN UKRANIEN! :)
This guide will introduce you to the subject of Knot Theory. The first section gives an introduction to important math theory concepts and terminology. If you are only looking for quick tips to solve the unknotting puzzles, skip ahead to the second section. You can always consult the first section if the meaning of some words is unclear.
Introduction with Definitions
Introduction
First things first : what are knots? Knots in mathematics are different from knots in your everyday life.
What kinds of knots do you use every day?
How are these different from mathematical knots?
Are there any mathematical knots in everyday life?
How does the Unknotting game work?
Definitions
From here on, we only talk about mathematical knots. To avoid misunderstandings and to be able to decide whether any statement is true or false we should start with defining the meaning of several words for the rest of this page.
knot line:
knot diagram:
mathematical knot (or simply, knot):
ambient isotopy:
step:
crossing:
pass:
switch:
orientation:
crossing handedness:
writhe number:
knot invariant:
crossing number:
arc:
hole:
over-strand:
under-strand:
Reidemeister moves:
Reidemeister 1 move:
Reidemeister 2 move:
Reidemeister 3 move:
pass move:
P- move:
P0 move:
P+ move:
unknotting number:
How to Simplify Diagrams
Finding R1 Moves
Finding R2 Moves
About the Interface (1)
Finding P- Moves
Finding R3 Moves
About the Interface (2)
Finding P0 Moves
Finding U1 Moves
Finding U2 Moves
Finding P0U Moves
More References about Knots
This guide will introduce you to the subject of Knot Theory. The first section gives an introduction to important math theory concepts and terminology. If you are only looking for quick tips to solve the unknotting puzzles, skip ahead to the second section. You can always consult the first section if the meaning of some words is unclear.
Introduction with Definitions
Introduction
First things first : what are knots? Knots in mathematics are different from knots in your everyday life.
What kinds of knots do you use every day?
Most of us use knots to tie our shoelaces, put on a necktie or scarf, to close up a bag, and so on... You might
know many more knots if you go sailing, camping, fishing, or if you sew, knit, or style hair.
However, none of these are mathematical knots!
How are these different from mathematical knots?
Have a look at the following two drawings of knots. Confusingly, these are both known as 'figure-eight' knots,
because of the figure 8 they contain.
Everyday Figure-eight Knot Mathematical Figure-eight Knot
What big difference can you see? We're sure you can figure it out!
Everyday Figure-eight Knot Mathematical Figure-eight Knot
What big difference can you see? We're sure you can figure it out!
The biggest difference is that the mathematical knot is a closed curve − that is, there are no loose
ends, it's a closed loop. What we call 'knots' in everyday life are known as 'braids' in
mathematics.
Also, while everyday knots might include more than one strand of material, in mathematics knots are a single, closed, continuous strand. Objects that involve more than one knot woven together are known as 'links'.
Also, while everyday knots might include more than one strand of material, in mathematics knots are a single, closed, continuous strand. Objects that involve more than one knot woven together are known as 'links'.
Are there any mathematical knots in everyday life?
Of course! You now know that in mathematics, a knot is a single, closed, continuous strand.
With this definition in mind, what is the simplest mathematical knot?
How can you make a mathematical knot from a piece of string?
With this definition in mind, what is the simplest mathematical knot?
The simplest mathematical knot is just a single loop or circle, like this:
We will talk more about this knot later, but there are plenty of examples of simple loops like this in real life.
We will talk more about this knot later, but there are plenty of examples of simple loops like this in real life.
How can you make a mathematical knot from a piece of string?
The simplest mathematical knot is a circle. To make it, simply glue the ends of your string together.
What happens if you twist your circle once?
Is this a different knot?
Can all knots be deformed to make a circle?
How different can a knot diagram look from the simplest form?
What happens if you twist your circle once?
If you take your loop of string, twist it, and lay it flat, you might get something like this:
Is this a different knot?
Of course knot! All you did was twist it.
This might seem easy, but finding ways to tell whether two diagrams (pictures) show the same knot is a very important and difficult question for mathematicians who study Knot Theory.
One way to tell that two pictures represent the same knot is whether you can deform one of them to look like the other. For example, here you just need to twist it back to get the loop.
This might seem easy, but finding ways to tell whether two diagrams (pictures) show the same knot is a very important and difficult question for mathematicians who study Knot Theory.
One way to tell that two pictures represent the same knot is whether you can deform one of them to look like the other. For example, here you just need to twist it back to get the loop.
Can all knots be deformed to make a circle?
To answer this question, try this out:
Can you deform this to get a circle?
- take a strand of string
- twist it to form a loop
- pass one end through the loop
Can you deform this to get a circle?
Try as you might, there is no way to deform this knot into a circle. At least, not without cutting
the string and gluing it back together.
This is actually a different mathematical knot, called the trefoil knot because it looks like a three-leaf clover.
This is actually a different mathematical knot, called the trefoil knot because it looks like a three-leaf clover.
How different can a knot diagram look from the simplest form?
Another important property of mathematical knots is that they can be arbirtrarily stretched and bent.
For example, our diagram of the simplest knot looks more like a square than a circle − we could
draw it as a perfect circle, and it would be the same knot. You could take the simplest knot, a circle,
and stretch it out into a long thin ellipse, then use it as a string to tie it into the 'everyday
figure-8 knot'. Mathematically, it is still a circle.
As you might expect, then, the different diagrams of the same underlying knot can look drastically different! For example, the Gordian Knot and this diagram can both be deformed into a circle, with enough patience. They are the diagrams of the same knot as the circle.
The circle is just the easiest way of drawing this knot, but the knot isn't really a circle, it's an abstract mathematical object that we can represent in many different ways. In the same way that "1 car" and "1 apple" is not the number 1, a circle is just one way to represent this knot.
As you might expect, then, the different diagrams of the same underlying knot can look drastically different! For example, the Gordian Knot and this diagram can both be deformed into a circle, with enough patience. They are the diagrams of the same knot as the circle.
The circle is just the easiest way of drawing this knot, but the knot isn't really a circle, it's an abstract mathematical object that we can represent in many different ways. In the same way that "1 car" and "1 apple" is not the number 1, a circle is just one way to represent this knot.
How does the Unknotting game work?
In this game, you deform mathematical knot diagrams to reduce the number of
crossings as much as possible.
While you are allowed to 'cut' the strands by clicking on them, you are only modifying the diagram, not the underlying knot. This is why there is a restriction on how you can reattach the ends you have 'cut'. This restriction guarantees that the mathematical knot is unchanged in any re-routing of strands even if the knot changes its appearance.
When do you undo everyday knots in real life?
Try some other mathematical knot games!
While you are allowed to 'cut' the strands by clicking on them, you are only modifying the diagram, not the underlying knot. This is why there is a restriction on how you can reattach the ends you have 'cut'. This restriction guarantees that the mathematical knot is unchanged in any re-routing of strands even if the knot changes its appearance.
When do you undo everyday knots in real life?
You might be familiar with the struggle of disentangling electronics cables like for earbuds. If you tie
your shoes, you have to untie the laces.
The same property of real knots is what makes them useful, but also harder to undo.
What makes knots in real life so hard to undo?
The same property of real knots is what makes them useful, but also harder to undo.
What makes knots in real life so hard to undo?
The answer is friction! However, mathematical knots have no friction. You can think of them as
'infinitely slippery'.
Try some other mathematical knot games!
Our unknotting game is one way to have fun with knots on a screen, but here are some other games for you to
try:
- 1 player : Eiffel Tower and other string tricks
- 2 players : Cat's cradle
- Group : Human Knot
Definitions
From here on, we only talk about mathematical knots. To avoid misunderstandings and to be able to decide whether any statement is true or false we should start with defining the meaning of several words for the rest of this page.
knot line:
a closed curve in 3-dimensional space which does not
intersect itself and which has a finite thickness (to avoid
infinitely many smaller and smaller knots along the line), Example:
the figure-8 knot.
knot diagram:
a projection of a knot line into 2 dimensions where
different parts of the knot line can cross each other (on this website
lines cross under an angle of 90°), but do not lie on top of each other.
Examples for knot diagrams:
Unknot Trefoil Figure-eight Cinquefoil Three-twist
The 'unknot' is also called the 'trivial knot'.
Which knot does each of these 3 diagrams represent?
Which knot does this diagram represent?
Do you see how to deform this into the earlier trefoil diagram?
Examples for knot diagrams:
Unknot Trefoil Figure-eight Cinquefoil Three-twist
The 'unknot' is also called the 'trivial knot'.
Which knot does each of these 3 diagrams represent?
The unknot. Do you see how they can be deformed to a rectangle?
Which knot does this diagram represent?
The trefoil.
Do you see how to deform this into the earlier trefoil diagram?
You can deform this diagram into the other by flipping down the top arc.
→
→
mathematical knot (or simply, knot):
the abstract object behind
a set of (infinitely many) knot diagrams that can all be deformed,
stretched, and shifted into each other without being cut. Example:
the knot 31 also called 'trefoil' which is the simplest
non-trivial knot.
ambient isotopy:
the mathematical term
when one knot line can be continuously distorted to another one.
step:
On this website knot diagrams are drawn using only 6 tiles which we call steps:
crossing:
the place in a diagram where two steps cross, one on top of
the other:
pass:
a step that is part of a crossing, there are over-passes (fully
visible) and under-passes (partially covered).
switch:
swapping over- and under-pass of a crossing, i.e. switching between these two crossings:
If a crossing is switched, the old and new diagram in general
represent different knots. Switching all crossings is equivalent to
changing a knot to its mirror image.
Some knots are identical to their mirrored version, that means there is an
ambient isotopy between them. These are called 'achiral'. For example, the figure-8 knot is achiral.
Others can not be deformed to their mirrored version, like the
trefoil. They are called 'chiral'.
Can this knot be deformed into its mirror image?
Can this knot be deformed into its mirror image?
Yes! This diagram represents the figure-eight knot which is achiral and can be deformed into its mirror
image as follows:
Notice that the only difference between the first diagram and the last one is that all of the crossings are switched!
Initial position
A 180° rotation
Moving a strand
The mirror image
Notice that the only difference between the first diagram and the last one is that all of the crossings are switched!
orientation:
This is not a property of the knot line, nor of the knot.
It is a question of how to move along the knot line. One can move in 2
directions, also called 2 orientations.
A knot that can be deformed via an ambient isotopy into itself but with the
orientation reversed is called 'invertible' otherwise it is called
noninvertible. The smallest noninvertible knot is 817 which is achiral but
if an orientation is added it becomes chiral (find more on
the Invertible Knot Wikipedia page). Adding more
structure (here
an orientation) causes it to lose symmetry (not identical anymore to its mirror
image).
A chiral knot (a knot that can not be deformed to its mirror image) can still be invertible (symmetric against change of orientation). Such knots are called 'reversible'.
A chiral knot (a knot that can not be deformed to its mirror image) can still be invertible (symmetric against change of orientation). Such knots are called 'reversible'.
crossing handedness:
For a given knot diagram, crossings are either right- or left-handed.
In the following we will explore two types of crossings and determine which is right- or left-handed.
How many different crossings are there if we consider
which pass is an over-/under-pass and consider both orientations?
If one keeps the diagram unchanged and only switches one crossing,
does the handedness of that crossing change?
By using one's hands, how can one remember whether a crossing is
right- or left-handed?
In the following we will explore two types of crossings and determine which is right- or left-handed.
How many different crossings are there if we consider
which pass is an over-/under-pass and consider both orientations?
In total there are 8 cases:
If the horizontal pass is the over-pass then there are 4 options:
Similarly if the vertical pass is the over-pass then there are 4 more options:
The crossings in group 1, 7, 4, 6 are called right-handed crossings and
the crossings in group 2, 5, 3, 8 are called left-handed crossings.
If the horizontal pass is the over-pass then there are 4 options:
1
2
3
4
5
6
7
8
- Principle: Handedness should not depend on the orientation (direction of stepping through the knot line), so reversing both arrows we identify 4 pairs of crossings: 1 = 4, 2 = 3, 5 = 8, 6 = 7. Therefore, whichever groups we end up with, crossings 1 and 4 should be in the same group and so on.
- Principle: The group that a crossing belongs to should not change if we rotate the whole knot. We therefore identify crossings 1 = 7 = 4 = 6 and 2 = 5 = 3 = 8.
The crossings in group 1, 7, 4, 6 are called right-handed crossings and
the crossings in group 2, 5, 3, 8 are called left-handed crossings.
If one keeps the diagram unchanged and only switches one crossing,
does the handedness of that crossing change?
Yes. Try switching any of the eight crossings,
then check which one it has become and check whether it is
still in the same handedness group. For example, switching crossing 1
gives crossing 5, both are in different handedness groups.
Statement: The question whether a crossing is right-handed or left-handed does not only depend on the crossing itself but on the diagram around it.
Proof: Whether the horizontal pass is above or below does not alone decide the handedness. Both cases can be right- and left-handed (see the 8 crossings above). If one rotates the knot so that the over-pass is horizontal and if one then leaves the crossing through the overhead pass to the right (to the East) then it depends on the rest of the knot whether one returns to the crossing from the South (then the crossing is right-handed) or from the North (then the crossing is left-handed).
The fact that the two groups of crossings are called left- and right-handed gives a hint that one can distinguish crossings with the left and right hands.
Statement: The question whether a crossing is right-handed or left-handed does not only depend on the crossing itself but on the diagram around it.
Proof: Whether the horizontal pass is above or below does not alone decide the handedness. Both cases can be right- and left-handed (see the 8 crossings above). If one rotates the knot so that the over-pass is horizontal and if one then leaves the crossing through the overhead pass to the right (to the East) then it depends on the rest of the knot whether one returns to the crossing from the South (then the crossing is right-handed) or from the North (then the crossing is left-handed).
The fact that the two groups of crossings are called left- and right-handed gives a hint that one can distinguish crossings with the left and right hands.
By using one's hands, how can one remember whether a crossing is
right- or left-handed?
Stretch out your fingers so that all are in one plane, and your
thumb is at a right angle to all others which are parallel to
each other. Rotate your hand so that you can see your palm and
your thumb points towards the outgoing direction of the
over-pass and your fingers point towards the outgoing direction of
the under-pass. The hand that can do that decides the
handedness.
For example, for the crossing below, you would stretch your hand out like this:
Since you can only do this with your left hand, this is a left-handed crossing.
Right- and left-handed crossings are also referred to as positive or negative crossings.
Since you can only do this with your left hand, this is a left-handed crossing.
Right- and left-handed crossings are also referred to as positive or negative crossings.
writhe number:
the difference between the number of left- and right-handed
crossings in one diagram. The writhe number characterizes a diagram, not a knot as there can be 2
different diagrams of the same knot with different writhe numbers.
What is the writhe number of this diagram?
What is the writhe number of this diagram?
The right-handed crossings in the diagram are highlighted in red and the left-handed crossings in green.
To get the writhe number, we can count the number of left- and right-handed crossings, then subtract the
number of right-handed crossings
from the number of left-handed crossings.
This diagram has 2 left-handed crossings and 4 right-handed crossings, so its writhe number is 2 − 4 =
−2.
knot invariant:
a number or a polynomial or a feasibility statement that is characteristic for all (infinitely many)
diagrams of a knot. The properties of a knot being chiral/achiral, invertible/noninvertible,
reversible are knot invariants.
crossing number:
the minimal number of crossings that any diagram of
this knot can have after deformation, this is a characteristic of each knot and therefore a knot
invariant.
How many crossings does this diagram have?
What is the crossing number of the knot represented through the above diagram?
What are the two lowest crossing numbers that a knot can have?
How many crossings does this diagram have?
This diagram has 5 crossings.
What is the crossing number of the knot represented through the above diagram?
Zero! The crossing number is a property of the abstract
mathematical knot, it is not the property of a diagram. The
diagram above can be deformed to get the unknot
which has zero crossings.
Do you see how?
Because the crossing number is the minimal number of crossings of any diagram of a knot and because one cannot have less than zero crossings, the crossing number of the knot represented by the above diagram is zero.
which has zero crossings. Do you see how?
Because the crossing number is the minimal number of crossings of any diagram of a knot and because one cannot have less than zero crossings, the crossing number of the knot represented by the above diagram is zero.
What are the two lowest crossing numbers that a knot can have?
The lowest crossing number belongs to the unknot which is 0.
A knot diagram with 1 crossing would look like:
and could be deformed to the unknot. A knot diagram with 2 crossings would look like
and could also be deformed to the unknot.
The trefoil diagram shown further above has 3 crossings and can not be deformed to the unknot, so the lowest crossing numbers are 0 and 3.
and could be deformed to the unknot. A knot diagram with 2 crossings would look like
and could also be deformed to the unknot.The trefoil diagram shown further above has 3 crossings and can not be deformed to the unknot, so the lowest crossing numbers are 0 and 3.
arc:
The part of the knot line in a diagram from one crossing to the next.
How many arcs does a diagram with N crossings have?
How many arcs does a diagram with N crossings have?
Each crossing has 4 ends of arcs. Each arc has 2 ends, so there are 4/2 = 2
times as many arcs as crossings, so 2N arcs.
hole:
empty space in a diagram that is surrounded by arcs. The whole
empty space outside the diagram is also one hole.
How many holes does a diagram with N crossings have?
How many holes does a diagram with N crossings have?
One could draw several knots and guess a formula but one can derive
it too. Euler's formula says that for any drawing in the plane where
m lines (here m=2N arcs) each connect 2 out of n points (here n=N
crossings)
then the number f of faces (here holes) is f = 2 + m − n. That gives
for the number of holes of a knot: 2 + 2N − N = N + 2.
over-strand:
A sequence of consecutive arcs in a diagram
(i.e. arcs following one another) which starts and ends at an
under-pass and otherwise involves 0, 1 or more over-passes.
under-strand:
A sequence of consecutive arcs in a diagram
(i.e. arcs following one another) which starts and ends at an
over-pass and otherwise involves 0, 1 or more under-passes.
(In literature, a 'strand' is often used for what we call an over-strand. For us the number of over-passes of an over-strand matters as well as the number of under-passes of an under-strand. We therefore consider under-strands as well as over-strands.)
What kind of strand is the shown horizontal line consisting of five arcs?
How many over-strands does a diagram with N crossings have?
(In literature, a 'strand' is often used for what we call an over-strand. For us the number of over-passes of an over-strand matters as well as the number of under-passes of an under-strand. We therefore consider under-strands as well as over-strands.)
What kind of strand is the shown horizontal line consisting of five arcs?
This is an under-strand with 4 under-passes.
How many over-strands does a diagram with N crossings have?
At each crossing there are 2 ends of strands (either one end of
2 different strands or both ends from one strand). On the other hand,
each strand has 2 ends which are at a crossing. Therefore the number
of crossings is equal to the number of over-strands and because of
symmetry also equal to the number of under-strands, so there are
N of each.
Reidemeister moves:
In 1927 the German mathematician Kurt
Reidemeister and, independently, James Waddell Alexander and Garland
Baird Briggs (1926), proved that any two diagrams that represent the
same knot can be deformed into each other through a sequence of only 3
different types of moves. The problem is that during the deformation
the number of crossings may temporarily rise and a sharp upper bound for
this increase is unknown as well as the number of needed moves.
Reidemeister 1 move:
removes or adds a hole surrounded by one arc:
Which diagram shows a left-handed crossing and which
shows a right-handed crossing?
Which diagram shows a left-handed crossing and which
shows a right-handed crossing?
The left diagram shows a right-handed crossing and the right diagram shows a
left-handed crossing. A Reidemeister 1 move therefore changes the number
of right- or left-handed crossings by 1 and thus changes the
writhe number of the diagram.
Reidemeister 2 move:
removes or adds a hole surrounded by 2 arcs:
What can one say about the handedness of the two crossings that are
added or removed in a Reidemeister 2 move?
What can one say about the handedness of the two crossings that are
added or removed in a Reidemeister 2 move?
One of the two crossings is right-handed and one is left-handed.
A Reidemeister 2 move therefore does not change the writhe
number of a diagram.
Reidemeister 3 move:
removes and adds a hole surrounded by 3 arcs.
Which 2 types of holes surrounded by 3 arcs can you think of?
Which 2 types of holes surrounded by 3 arcs can you think of?
Either:
Verify that the result of the 3 moves is always the same.
Does the handedness of the 3 crossings change in a Reidemeister 3 move?
What have we learned?
- 1) each arc has 1 over-pass and 1 under-pass:
then none of the 3 arcs can be moved over/under/through the other crossing
- 2) one arc has 2 over-passes, one has 1 over- and 1
under-pass, and one has 2 under-passes:
then there are 3 moves. One can move the over-over-strand which stays an over-over-strand:
or move the under-over-strand which becomes an over-under-strand:
or move the under-under-strand which stays an under-under-strand:
Verify that the result of the 3 moves is always the same.
When comparing the right-hand sides of the above moves it is easy to see that
all 3 moves produce identical results. Therefore, if there is a Reidemeister 3 move
then there is only one. All that changes is that for all 3 arcs the other two arcs are
now crossed in the reverse order. This means that for the middle arc the order of over-pass
and under-pass is reversed.
Does the handedness of the 3 crossings change in a Reidemeister 3 move?
No. To see that, pick any orientation for each strand and use the
hand rule above.
What have we learned?
We learned:
- how to spot holes with 3 arcs that allow a Reidemeister 3 move,
- that for such a hole it does not matter which arc is moved,
- that the handedness of the 3 crossings does not change,
- that the order of over- and under-pass is reversed for the middle arc.
pass move:
This has nothing to do with a 'pass' defined above.
A pass move replaces an over-(under-)strand with another
over-(under-)strand where both strands have the same ends. For examples, please see P-, P0 and P+ moves below.
P- move:
a pass move where the new strand has less passes than the
old strand.
Find a P- move replacing the green strand in this diagram:
Find a P- move replacing the green strand in this diagram:
In this diagram the new red strand has fewer passes than the old green strand. Therefore this diagram shows
a P- move.
P0 move:
a pass move where the new strand has the same number of passes
as the old strand.
Find a P0 move replacing the green strand in this diagram:
Find a P0 move replacing the green strand in this diagram:
In this diagram the new red strand has the same number of passes as the old green strand. Therefore this
diagram shows a P0 move.
P+ move:
a pass move where the new strand has more passes
than the old strand.
Find a P+ move replacing the green strand in this diagram:
P+ moves become necessary if one wants to change the writhe number of a diagram. More about that is described further below under 'Finding P0 Moves'.
Find a P+ move replacing the green strand in this diagram:
In this diagram the new red strand has one more pass than the old green strand, therefore this is a P+ move.
P+ moves become necessary if one wants to change the writhe number of a diagram. More about that is described further below under 'Finding P0 Moves'.
unknotting number:
The unknotting number is the property of a knot, not the property of a diagram and is therefore
a knot invariant.
Starting with a knot diagram it is the minimal number of times that one or more crossings need to be switched in order to obtain the unknot. Before the first switch and inbetween switches the diagram can be arbitrarily deformed. The unknotting number is therefore not an easy number to determine because any deformation is allowed.
Why does the trefoil have unknotting number 1?
Starting with a knot diagram it is the minimal number of times that one or more crossings need to be switched in order to obtain the unknot. Before the first switch and inbetween switches the diagram can be arbitrarily deformed. The unknotting number is therefore not an easy number to determine because any deformation is allowed.
Why does the trefoil have unknotting number 1?
The trefoil cannot have the unknotting number 0 because it cannot
be deformed into the unknot (this needs to be and can be proven). The unknot has unknotting number 0. So the
trefoil has unknotting number ≥1. On the other hand one can easily
see that switching any one crossing of the trefoil diagram
shown further above produces the unknot, so the unknotting
number of the trefoil is ≤1. If it is ≥1 and ≤1 then it must be =1.
How to Simplify Diagrams
Finding R1 Moves
Simple cases of R1 moves, like here:
where one can flip a loop 4 times and instantly get the unknot
are easy to spot by following the knot line and looking for an arc
with both ends at the same crossing. The order of performing R1 moves
does not matter.
But there are more general cases of applying Reidemeister 1 moves. If an over-strand starts at a crossing of which it is an overpass but otherwise lies completely on top of other arcs (therefore being called 'over-strand') then this loop can surely be shortcut and thus removed. For example, at first the loop on top in the center can be removed and then the other ones, one by one:
This loop can also be removed when lying entirely underneath:
where one can flip a loop 4 times and instantly get the unknot
are easy to spot by following the knot line and looking for an arc
with both ends at the same crossing. The order of performing R1 moves
does not matter. But there are more general cases of applying Reidemeister 1 moves. If an over-strand starts at a crossing of which it is an overpass but otherwise lies completely on top of other arcs (therefore being called 'over-strand') then this loop can surely be shortcut and thus removed. For example, at first the loop on top in the center can be removed and then the other ones, one by one:
This loop can also be removed when lying entirely underneath:
Finding R2 Moves
Similarly to R1 moves it is easy to spot prototype R2 moves like
here where two R2 moves need to be done before an R1 move yields
the unknot:
In the following example one has to perform an R2 move with the same strand twice, once pulling this strand from underneath and once from above:
The last step is not an R2 move. It is only added to show that the knot is the sum of two trefoil knots.
In the following example one has to perform an R2 move with the same strand twice, once pulling this strand from underneath and once from above:
The last step is not an R2 move. It is only added to show that the knot is the sum of two trefoil knots.
About the Interface (1)
The example above is suitable to demonstrate the optimal use of the
interface. After intercepting the knot line:
One can NOT retreat one end all the way; and then retreat the other end all the way because one would first remove an under-pass and have both ends in the under-pass status and in this status it is not possible to remove an over-pass. Instead, one removes one under-pass, jumps to the other end, removes the other under-pass which brings both ends into the same hole, which changes the ends into neutral status and then allows to remove over-passes and then to re-unite the ends. In short, one jumps between ends to remove all under-passes then all over-passes and so on.
This implementation of a status of ends is not a weakness of the program but it guarantees that interactive modification of the diagram does not change the mathematical knot.
One can NOT retreat one end all the way; and then retreat the other end all the way because one would first remove an under-pass and have both ends in the under-pass status and in this status it is not possible to remove an over-pass. Instead, one removes one under-pass, jumps to the other end, removes the other under-pass which brings both ends into the same hole, which changes the ends into neutral status and then allows to remove over-passes and then to re-unite the ends. In short, one jumps between ends to remove all under-passes then all over-passes and so on.
This implementation of a status of ends is not a weakness of the program but it guarantees that interactive modification of the diagram does not change the mathematical knot.
Finding P- Moves
P- moves replace an over-strand with one having less over-passes
or an under-strand with one having less under-passes. In both
cases the number of crossings is reduced.
To find such moves one steps through the knot line and looks for as many consecutive over-passes as possible or for as many consecutive under-passes as possible, at least two. If one found such a strand, for example, an over-strand then one tries to find an alternative route with less over-passes that links the same two under-pass ends.
In the following example a strand with 4 consecutive under-passes is replaced by a strand without underpasses and then this strand with 3 consecutive over-passes is replaced by a strand with one over-pass. Two more P- moves remove each 2 more crossings. The resulting diagram can be simplified further by two more R1 moves, as seen below.
The more consecutive passes of one sort one finds, the higher the
chance to find a different route that needs less passes.
To find such moves one steps through the knot line and looks for as many consecutive over-passes as possible or for as many consecutive under-passes as possible, at least two. If one found such a strand, for example, an over-strand then one tries to find an alternative route with less over-passes that links the same two under-pass ends.
In the following example a strand with 4 consecutive under-passes is replaced by a strand without underpasses and then this strand with 3 consecutive over-passes is replaced by a strand with one over-pass. Two more P- moves remove each 2 more crossings. The resulting diagram can be simplified further by two more R1 moves, as seen below.
Finding R3 Moves
R1, R2 and P- moves change the number of crossings. An R3
move does not change the number of crossings, therefore we place
its description after the P- move. The following example shows how
R3 moves can still be useful by making P- moves possible.
As described in the first section, an R3 hole is surrounded by a top arc
with 2 over-pass ends (here A,B), a middle arc with 1 over-pass end (C) and 1
under-pass end (B) and a bottom arc with 2 under-pass ends (here
A,C).
(unknot taken from the Unknot Wikipedia page)
We found in the above section that an R3 move reverses for the middle strand (here through B,C) the order of under- and over-pass. An R3 move is beneficial if the R3 move leads for the continuation of the middle arc to an increase of consecutive over-passes and/or consecutive under-passes and thus increases the chance to find a P- move. This is the case if in the continuation of the middle arc after the under-pass B comes an over-pass (D,E are over-passes) and/or after the over-pass (C) follows an under-pass (F,G,H are under-passes). The R3 move slides the middle arc BC between the top arc and bottom arc at A:
Before, there were only 2 consecutive over-passes at D and E,
now there are 3 at C, D and E. This longer over-strand can now be
re-routed in a P- move:
reducing the number of crossings by 2 from 13 to 11.
Also, the other side of the strand can be re-routed in a P- move. Before, there were 3 consecutive under-passes at F,G and H, now there are 4 at B, F, G, and H. This P- move leads to:
and is also reducing the number of crossings by 2. Both diagrams can be
simplified further through P- and R1 moves resulting finally in the unknot.
Can you see how? Just follow the hints on how to spot P- moves given above.
Let us practise that with an example.
How many R3 moves are possible in this diagram:
We found in the above section that an R3 move reverses for the middle strand (here through B,C) the order of under- and over-pass. An R3 move is beneficial if the R3 move leads for the continuation of the middle arc to an increase of consecutive over-passes and/or consecutive under-passes and thus increases the chance to find a P- move. This is the case if in the continuation of the middle arc after the under-pass B comes an over-pass (D,E are over-passes) and/or after the over-pass (C) follows an under-pass (F,G,H are under-passes). The R3 move slides the middle arc BC between the top arc and bottom arc at A:
Also, the other side of the strand can be re-routed in a P- move. Before, there were 3 consecutive under-passes at F,G and H, now there are 4 at B, F, G, and H. This P- move leads to:
Let us practise that with an example.
How many R3 moves are possible in this diagram:
Three R3 moves are possible. For each one we show in light blue the three arcs that are
involved. What is easily overlooked is the third one where the hole
is the whole outside space which is 'surrounded' by only 3 arcs.
Perform the 1. R3 move and find out whether it is beneficial:
Perform the 2. R3 move and find out whether it is beneficial:
Perform the 3. R3 move and find out whether it is beneficial:
1. R3 move
2. R3 move
3. R3 move
Perform the 1. R3 move and find out whether it is beneficial:
This R3 move is beneficial. It allows afterwards a P- moves as shown in a sequence of moves further
below.
Our definition for an R3 move to be beneficial it is not
neccessarily to allow a P- move but to increase the number of consecutive
over- or under-passes and that is easy to see even without
performing all these moves. In the following diagram the middle arc
of the R3 hole has an
over-pass at A, an under-pass at B followed by two over-passes at C and D.
In an R3 move the order of over- and under-pass is reversed for the middle arc
as shown in the Diagram 3 with now 3 consecutive over-passes. This is enough
to find a P- move needing less than 3 passes in Diagram 5.
About the sequence of diagrams below: In Dia 1 we make space to prepare the R3 move in Dia 2 (here by
moving
the top arc) with the result in Dia 3. In Dia 4 we make space to prepare
the P- move in Dia 5 where the green strand with 3 over-passes is replaced
by the red strand with only 1 over-pass in Dia 6. In Dia 7 we shift a strand
to make space for the next P- move in Dia 9 with result in Dia 10 and Dia 11
after shrinking which is easily identified as knot 51.
1. Widening
2. The R3 move
3. After the R3 move
4. Widening
5. A P- move
6. After the P- move
7. For the next P- move
8. Before the P- move
9. The 2nd P- move
10. Afterwards
11. Contracted
Perform the 2. R3 move and find out whether it is beneficial:
The sequence shows that the R3 move is beneficial.
1. Widening
2. The R3 move
3. After the R3 move
4. A P- move
5. After the P- move
6. Another P- move
7. After the P- move
8. Contracted
Perform the 3. R3 move and find out whether it is beneficial:
The 3rd R3 move is also beneficial. To execute this move, one follows the same principle: the middle
strand cuts the two other strands, which this time
'surround' the outside hole, in reverse order.
The result is that travelling upwards in the Dia 2 along the light blue strand one first gets to an overpass and then an underpass. If after the R3 move one travels along the red strand then one passes the other 2 strands in reverse order and therefore first gets to an underpass and then an overpass. As one can see in Dia 5, the sequence of now 3 consecutive underpasses of the light blue strand there enables a P- move.
The result is that travelling upwards in the Dia 2 along the light blue strand one first gets to an overpass and then an underpass. If after the R3 move one travels along the red strand then one passes the other 2 strands in reverse order and therefore first gets to an underpass and then an overpass. As one can see in Dia 5, the sequence of now 3 consecutive underpasses of the light blue strand there enables a P- move.
1. Widening
2. An R3 move
3. After the R3 move
4. Widening
5. A P- move
6. Widening
7. A 2nd P- move
8. After the P- move
9. Shortening
10. After shortening
11. Shortening
12. Straightening
13. ↻90° rotation
About the Interface (2)
The example above is suitable to demonstrate how to perform an R3
move with our interface.
As described in the 'Introduction with Definitions' > 'Reidemeister 3 move' there are 3 ways to perform a Reidemeister 3 move: moving the bottom strand, moving the middle strand, or, moving the top strand. As shown there, all 3 ways have the same result, they perform the same Reidemeister 3 move.
Our interface only allows one to complete an R3 move by moving the bottom strand or the top strand but not the middle strand. The reason is the feature of our interface that BOTH ends can only add/remove over-passes or BOTH ends can only add/remove under-passes at a time. But that does not prevent us from performing R3 moves, as moving any one of the 3 strands gives the same result.
As described in the 'Introduction with Definitions' > 'Reidemeister 3 move' there are 3 ways to perform a Reidemeister 3 move: moving the bottom strand, moving the middle strand, or, moving the top strand. As shown there, all 3 ways have the same result, they perform the same Reidemeister 3 move.
Our interface only allows one to complete an R3 move by moving the bottom strand or the top strand but not the middle strand. The reason is the feature of our interface that BOTH ends can only add/remove over-passes or BOTH ends can only add/remove under-passes at a time. But that does not prevent us from performing R3 moves, as moving any one of the 3 strands gives the same result.
Finding P0 Moves
P0 moves are pass moves that do not change the number of
crossings, just like R3 moves which are special versions of
P0 moves. Like R3 moves, a P0 move may be beneficial and
enable a P- move. Because P0 moves are less useful on
average, they occur more frequently but it is more
challenging to see whether they enable a P- move.
To find a P0 move one looks for an over-strand or an under-strand just like for a P- move. To check whether a P0 move may be beneficial and enable a P- move one proceeds like in the case of the R3 move. One looks whether the removal of the strand increases the number of consecutive over- or under-passes of strands that were crossed and one checks whether after re-routing the new strand increases the number of consecutive over- or under-passes of strands that are crossed now. In either one of these cases one checks whether the strands with an increased number of consecutive over- or under-passes can be re-routed with less crossings.
Let us look at this example:
We label the crossings:
and find a beneficial P0 move step by step.
How many over-strands with at least 2 over-passes
and how many under-strands with at least 2
under-passes do you see?
It is not difficult to see that the IE strand has a P0 move that re-locates it to cross over the GC and BH strand:
but the question is whether this P0 move is beneficial.
Did moving the IE strand increase the number of consecutive over-/under-passes
of the previously crossed DF or HJ strands?
Did more consecutive over-/under-passes get created when placing the strand on top of the
2 strands GC and BH?
Can this BH strand be re-routed in a P- move to reduce the number of crossings?
In the original diagram this P- move could have been done first as a P0 move and the P0 move mentioned above afterwards as P- move with the same total saving of 2 crossings.
In difficult problems it may be necessary to perform several P0 moves before a P- move becomes possible.
A diagram is maximally simplified if the number of crossings is the crossing number (see first section). In that case P0 moves will never enable a P- move.
To find a P0 move one looks for an over-strand or an under-strand just like for a P- move. To check whether a P0 move may be beneficial and enable a P- move one proceeds like in the case of the R3 move. One looks whether the removal of the strand increases the number of consecutive over- or under-passes of strands that were crossed and one checks whether after re-routing the new strand increases the number of consecutive over- or under-passes of strands that are crossed now. In either one of these cases one checks whether the strands with an increased number of consecutive over- or under-passes can be re-routed with less crossings.
Let us look at this example:
How many over-strands with at least 2 over-passes
and how many under-strands with at least 2
under-passes do you see?
We get three over-strands with at least 2 over-passes:
AB, GC, IE and three under-strands with at least 2
under-passes: EF, BH, DJ.
It is not difficult to see that the IE strand has a P0 move that re-locates it to cross over the GC and BH strand:
Did moving the IE strand increase the number of consecutive over-/under-passes
of the previously crossed DF or HJ strands?
Yes, the HJ strand now has 2 over-passes but this strand can not be re-routed
to link the same two holes with less over-passes.
Did more consecutive over-/under-passes get created when placing the strand on top of the
2 strands GC and BH?
Yes, the BH under-strand had 2 under-passes and now has 3 under-passes.
Can this BH strand be re-routed in a P- move to reduce the number of crossings?
Yes: The new route of the BH strand links the same holes but with only 1 under-pass instead of
3 under-passes.
The fact that the new strand is longer (involves more steps) in this
diagram than the replaced strand does not matter. All that matters
is the reduction of the number of crossings from 10 to 8 which now
allows to identify this knot as knot 817. In the original diagram this P- move could have been done first as a P0 move and the P0 move mentioned above afterwards as P- move with the same total saving of 2 crossings.
In difficult problems it may be necessary to perform several P0 moves before a P- move becomes possible.
A diagram is maximally simplified if the number of crossings is the crossing number (see first section). In that case P0 moves will never enable a P- move.
Finding U1 Moves
A U1 move switches a crossing which afterwards allows to simplify
the diagram to remove all crossings and show that the switch
produced the unknot.
In general nice puzzles should have unique solutions so our U1 and U2 puzzles show diagrams where the switch of only one crossing results in the unknot. This hint allows you to cut down the search.
If the diagram includes a twist of the knot line like this:
would it matter which one of the crossings is switched?
If there are still several crossing candidates, one should try to imagine how many R1, R2 moves will become available through the switch and try that switch first which seems to allow the most simplifications afterwards.
Another hint to minimize trying switches is to imagine whether after the switch there will definitely be a knot remaining, like a trefoil. If so then that switch is not the correct one in U1 puzzles.
On the worksheet the number of available switches is limited to the minimum that is needed to get the unknot. If you did a switch one can not switch it back because the diagram may have been changed, so one would have to reset the diagram.
In general nice puzzles should have unique solutions so our U1 and U2 puzzles show diagrams where the switch of only one crossing results in the unknot. This hint allows you to cut down the search.
If the diagram includes a twist of the knot line like this:
It will not matter which crossing is switched. Both results are equivalent:
Therefore either both of these crossings are unknotting switches or
none of them. Because our puzzles have only one unknotting switch,
these two crossings can be ignored.
=
=
If there are still several crossing candidates, one should try to imagine how many R1, R2 moves will become available through the switch and try that switch first which seems to allow the most simplifications afterwards.
Another hint to minimize trying switches is to imagine whether after the switch there will definitely be a knot remaining, like a trefoil. If so then that switch is not the correct one in U1 puzzles.
On the worksheet the number of available switches is limited to the minimum that is needed to get the unknot. If you did a switch one can not switch it back because the diagram may have been changed, so one would have to reset the diagram.
Finding U2 Moves
For U2 puzzles the same hint about equivalent switches applies as
for U1 puzzles. Also for U2 puzzles there is just one crossing that
reduces the unknotting number, i.e. makes progress towards the
unknot. Once that unique first crossing is switched and the
resulting diagram is simplified, there may be more than one switch
possible that creates the unknot.
Finding P0U Moves
Research performed by Caribou Contests on unknotting numbers showed
that there exists maximally simplified knot diagrams (with the
minimal number of crossings), which do not have a simplifying
switch. In other words, there are diagrams where switching any
crossings will not make progress to reach the unknot. In that case
one first has to perform one or more P0 moves that change the puzzle
into a U2 puzzle. The good news is that diagrams requiring P0 moves
first are rare and therefore it will be likely that any P0 moves
will change it into a U2 puzzle.
More References about Knots
Mathematical Knot Theory is an old research subject so there
exists a vast amount of literature for it. However, it also is a young
subject as several milestones have only been reached in recent
decades. For example, there is a scientific "Journal of Knot
Theory and Its Ramifications" dedicated to knots which has
a new issue every month.
A great book that we recommend is: Adams, Colin (2004), The Knot Book: An Elementary Introduction to the Mathematical Theory of Knots, American Mathematical Society, ISBN 978-0-8218-3678-1
There are also many websites about knots. A good place to start is the Knot Theory page on Wikipedia.
For videos, check out Numberphile's Playlist of Knot videos on YouTube. They also have a great explanation of Colouring Knots, which is another way to help identify diagrams of the same knot.
Caribou produced two posters on Unknotting and colouring knots.
A great book that we recommend is: Adams, Colin (2004), The Knot Book: An Elementary Introduction to the Mathematical Theory of Knots, American Mathematical Society, ISBN 978-0-8218-3678-1
There are also many websites about knots. A good place to start is the Knot Theory page on Wikipedia.
For videos, check out Numberphile's Playlist of Knot videos on YouTube. They also have a great explanation of Colouring Knots, which is another way to help identify diagrams of the same knot.
Caribou produced two posters on Unknotting and colouring knots.
EN AZÉRI! :)
This guide will introduce you to the subject of Knot Theory. The first section gives an introduction to important math theory concepts and terminology. If you are only looking for quick tips to solve the unknotting puzzles, skip ahead to the second section. You can always consult the first section if the meaning of some words is unclear.
Introduction with Definitions
Introduction
First things first : what are knots? Knots in mathematics are different from knots in your everyday life.
What kinds of knots do you use every day?
How are these different from mathematical knots?
Are there any mathematical knots in everyday life?
How does the Unknotting game work?
Definitions
From here on, we only talk about mathematical knots. To avoid misunderstandings and to be able to decide whether any statement is true or false we should start with defining the meaning of several words for the rest of this page.
knot line:
knot diagram:
mathematical knot (or simply, knot):
ambient isotopy:
step:
crossing:
pass:
switch:
orientation:
crossing handedness:
writhe number:
knot invariant:
crossing number:
arc:
hole:
over-strand:
under-strand:
Reidemeister moves:
Reidemeister 1 move:
Reidemeister 2 move:
Reidemeister 3 move:
pass move:
P- move:
P0 move:
P+ move:
unknotting number:
How to Simplify Diagrams
Finding R1 Moves
Finding R2 Moves
About the Interface (1)
Finding P- Moves
Finding R3 Moves
About the Interface (2)
Finding P0 Moves
Finding U1 Moves
Finding U2 Moves
Finding P0U Moves
More References about Knots
This guide will introduce you to the subject of Knot Theory. The first section gives an introduction to important math theory concepts and terminology. If you are only looking for quick tips to solve the unknotting puzzles, skip ahead to the second section. You can always consult the first section if the meaning of some words is unclear.
Introduction with Definitions
Introduction
First things first : what are knots? Knots in mathematics are different from knots in your everyday life.
What kinds of knots do you use every day?
Most of us use knots to tie our shoelaces, put on a necktie or scarf, to close up a bag, and so on... You might
know many more knots if you go sailing, camping, fishing, or if you sew, knit, or style hair.
However, none of these are mathematical knots!
How are these different from mathematical knots?
Have a look at the following two drawings of knots. Confusingly, these are both known as 'figure-eight' knots,
because of the figure 8 they contain.
Everyday Figure-eight Knot Mathematical Figure-eight Knot
What big difference can you see? We're sure you can figure it out!
Everyday Figure-eight Knot Mathematical Figure-eight Knot
What big difference can you see? We're sure you can figure it out!
The biggest difference is that the mathematical knot is a closed curve − that is, there are no loose
ends, it's a closed loop. What we call 'knots' in everyday life are known as 'braids' in
mathematics.
Also, while everyday knots might include more than one strand of material, in mathematics knots are a single, closed, continuous strand. Objects that involve more than one knot woven together are known as 'links'.
Also, while everyday knots might include more than one strand of material, in mathematics knots are a single, closed, continuous strand. Objects that involve more than one knot woven together are known as 'links'.
Are there any mathematical knots in everyday life?
Of course! You now know that in mathematics, a knot is a single, closed, continuous strand.
With this definition in mind, what is the simplest mathematical knot?
How can you make a mathematical knot from a piece of string?
With this definition in mind, what is the simplest mathematical knot?
The simplest mathematical knot is just a single loop or circle, like this:
We will talk more about this knot later, but there are plenty of examples of simple loops like this in real life.
We will talk more about this knot later, but there are plenty of examples of simple loops like this in real life.
How can you make a mathematical knot from a piece of string?
The simplest mathematical knot is a circle. To make it, simply glue the ends of your string together.
What happens if you twist your circle once?
Is this a different knot?
Can all knots be deformed to make a circle?
How different can a knot diagram look from the simplest form?
What happens if you twist your circle once?
If you take your loop of string, twist it, and lay it flat, you might get something like this:
Is this a different knot?
Of course knot! All you did was twist it.
This might seem easy, but finding ways to tell whether two diagrams (pictures) show the same knot is a very important and difficult question for mathematicians who study Knot Theory.
One way to tell that two pictures represent the same knot is whether you can deform one of them to look like the other. For example, here you just need to twist it back to get the loop.
This might seem easy, but finding ways to tell whether two diagrams (pictures) show the same knot is a very important and difficult question for mathematicians who study Knot Theory.
One way to tell that two pictures represent the same knot is whether you can deform one of them to look like the other. For example, here you just need to twist it back to get the loop.
Can all knots be deformed to make a circle?
To answer this question, try this out:
Can you deform this to get a circle?
- take a strand of string
- twist it to form a loop
- pass one end through the loop
Can you deform this to get a circle?
Try as you might, there is no way to deform this knot into a circle. At least, not without cutting
the string and gluing it back together.
This is actually a different mathematical knot, called the trefoil knot because it looks like a three-leaf clover.
This is actually a different mathematical knot, called the trefoil knot because it looks like a three-leaf clover.
How different can a knot diagram look from the simplest form?
Another important property of mathematical knots is that they can be arbirtrarily stretched and bent.
For example, our diagram of the simplest knot looks more like a square than a circle − we could
draw it as a perfect circle, and it would be the same knot. You could take the simplest knot, a circle,
and stretch it out into a long thin ellipse, then use it as a string to tie it into the 'everyday
figure-8 knot'. Mathematically, it is still a circle.
As you might expect, then, the different diagrams of the same underlying knot can look drastically different! For example, the Gordian Knot and this diagram can both be deformed into a circle, with enough patience. They are the diagrams of the same knot as the circle.
The circle is just the easiest way of drawing this knot, but the knot isn't really a circle, it's an abstract mathematical object that we can represent in many different ways. In the same way that "1 car" and "1 apple" is not the number 1, a circle is just one way to represent this knot.
As you might expect, then, the different diagrams of the same underlying knot can look drastically different! For example, the Gordian Knot and this diagram can both be deformed into a circle, with enough patience. They are the diagrams of the same knot as the circle.
The circle is just the easiest way of drawing this knot, but the knot isn't really a circle, it's an abstract mathematical object that we can represent in many different ways. In the same way that "1 car" and "1 apple" is not the number 1, a circle is just one way to represent this knot.
How does the Unknotting game work?
In this game, you deform mathematical knot diagrams to reduce the number of
crossings as much as possible.
While you are allowed to 'cut' the strands by clicking on them, you are only modifying the diagram, not the underlying knot. This is why there is a restriction on how you can reattach the ends you have 'cut'. This restriction guarantees that the mathematical knot is unchanged in any re-routing of strands even if the knot changes its appearance.
When do you undo everyday knots in real life?
Try some other mathematical knot games!
While you are allowed to 'cut' the strands by clicking on them, you are only modifying the diagram, not the underlying knot. This is why there is a restriction on how you can reattach the ends you have 'cut'. This restriction guarantees that the mathematical knot is unchanged in any re-routing of strands even if the knot changes its appearance.
When do you undo everyday knots in real life?
You might be familiar with the struggle of disentangling electronics cables like for earbuds. If you tie
your shoes, you have to untie the laces.
The same property of real knots is what makes them useful, but also harder to undo.
What makes knots in real life so hard to undo?
The same property of real knots is what makes them useful, but also harder to undo.
What makes knots in real life so hard to undo?
The answer is friction! However, mathematical knots have no friction. You can think of them as
'infinitely slippery'.
Try some other mathematical knot games!
Our unknotting game is one way to have fun with knots on a screen, but here are some other games for you to
try:
- 1 player : Eiffel Tower and other string tricks
- 2 players : Cat's cradle
- Group : Human Knot
Definitions
From here on, we only talk about mathematical knots. To avoid misunderstandings and to be able to decide whether any statement is true or false we should start with defining the meaning of several words for the rest of this page.
knot line:
a closed curve in 3-dimensional space which does not
intersect itself and which has a finite thickness (to avoid
infinitely many smaller and smaller knots along the line), Example:
the figure-8 knot.
knot diagram:
a projection of a knot line into 2 dimensions where
different parts of the knot line can cross each other (on this website
lines cross under an angle of 90°), but do not lie on top of each other.
Examples for knot diagrams:
Unknot Trefoil Figure-eight Cinquefoil Three-twist
The 'unknot' is also called the 'trivial knot'.
Which knot does each of these 3 diagrams represent?
Which knot does this diagram represent?
Do you see how to deform this into the earlier trefoil diagram?
Examples for knot diagrams:
Unknot Trefoil Figure-eight Cinquefoil Three-twist
The 'unknot' is also called the 'trivial knot'.
Which knot does each of these 3 diagrams represent?
The unknot. Do you see how they can be deformed to a rectangle?
Which knot does this diagram represent?
The trefoil.
Do you see how to deform this into the earlier trefoil diagram?
You can deform this diagram into the other by flipping down the top arc.
→
→
mathematical knot (or simply, knot):
the abstract object behind
a set of (infinitely many) knot diagrams that can all be deformed,
stretched, and shifted into each other without being cut. Example:
the knot 31 also called 'trefoil' which is the simplest
non-trivial knot.
ambient isotopy:
the mathematical term
when one knot line can be continuously distorted to another one.
step:
On this website knot diagrams are drawn using only 6 tiles which we call steps:
crossing:
the place in a diagram where two steps cross, one on top of
the other:
pass:
a step that is part of a crossing, there are over-passes (fully
visible) and under-passes (partially covered).
switch:
swapping over- and under-pass of a crossing, i.e. switching between these two crossings:
If a crossing is switched, the old and new diagram in general
represent different knots. Switching all crossings is equivalent to
changing a knot to its mirror image.
Some knots are identical to their mirrored version, that means there is an
ambient isotopy between them. These are called 'achiral'. For example, the figure-8 knot is achiral.
Others can not be deformed to their mirrored version, like the
trefoil. They are called 'chiral'.
Can this knot be deformed into its mirror image?
Can this knot be deformed into its mirror image?
Yes! This diagram represents the figure-eight knot which is achiral and can be deformed into its mirror
image as follows:
Notice that the only difference between the first diagram and the last one is that all of the crossings are switched!
Initial position
A 180° rotation
Moving a strand
The mirror image
Notice that the only difference between the first diagram and the last one is that all of the crossings are switched!
orientation:
This is not a property of the knot line, nor of the knot.
It is a question of how to move along the knot line. One can move in 2
directions, also called 2 orientations.
A knot that can be deformed via an ambient isotopy into itself but with the
orientation reversed is called 'invertible' otherwise it is called
noninvertible. The smallest noninvertible knot is 817 which is achiral but
if an orientation is added it becomes chiral (find more on
the Invertible Knot Wikipedia page). Adding more
structure (here
an orientation) causes it to lose symmetry (not identical anymore to its mirror
image).
A chiral knot (a knot that can not be deformed to its mirror image) can still be invertible (symmetric against change of orientation). Such knots are called 'reversible'.
A chiral knot (a knot that can not be deformed to its mirror image) can still be invertible (symmetric against change of orientation). Such knots are called 'reversible'.
crossing handedness:
For a given knot diagram, crossings are either right- or left-handed.
In the following we will explore two types of crossings and determine which is right- or left-handed.
How many different crossings are there if we consider
which pass is an over-/under-pass and consider both orientations?
If one keeps the diagram unchanged and only switches one crossing,
does the handedness of that crossing change?
By using one's hands, how can one remember whether a crossing is
right- or left-handed?
In the following we will explore two types of crossings and determine which is right- or left-handed.
How many different crossings are there if we consider
which pass is an over-/under-pass and consider both orientations?
In total there are 8 cases:
If the horizontal pass is the over-pass then there are 4 options:
Similarly if the vertical pass is the over-pass then there are 4 more options:
The crossings in group 1, 7, 4, 6 are called right-handed crossings and
the crossings in group 2, 5, 3, 8 are called left-handed crossings.
If the horizontal pass is the over-pass then there are 4 options:
1
2
3
4
5
6
7
8
- Principle: Handedness should not depend on the orientation (direction of stepping through the knot line), so reversing both arrows we identify 4 pairs of crossings: 1 = 4, 2 = 3, 5 = 8, 6 = 7. Therefore, whichever groups we end up with, crossings 1 and 4 should be in the same group and so on.
- Principle: The group that a crossing belongs to should not change if we rotate the whole knot. We therefore identify crossings 1 = 7 = 4 = 6 and 2 = 5 = 3 = 8.
The crossings in group 1, 7, 4, 6 are called right-handed crossings and
the crossings in group 2, 5, 3, 8 are called left-handed crossings.
If one keeps the diagram unchanged and only switches one crossing,
does the handedness of that crossing change?
Yes. Try switching any of the eight crossings,
then check which one it has become and check whether it is
still in the same handedness group. For example, switching crossing 1
gives crossing 5, both are in different handedness groups.
Statement: The question whether a crossing is right-handed or left-handed does not only depend on the crossing itself but on the diagram around it.
Proof: Whether the horizontal pass is above or below does not alone decide the handedness. Both cases can be right- and left-handed (see the 8 crossings above). If one rotates the knot so that the over-pass is horizontal and if one then leaves the crossing through the overhead pass to the right (to the East) then it depends on the rest of the knot whether one returns to the crossing from the South (then the crossing is right-handed) or from the North (then the crossing is left-handed).
The fact that the two groups of crossings are called left- and right-handed gives a hint that one can distinguish crossings with the left and right hands.
Statement: The question whether a crossing is right-handed or left-handed does not only depend on the crossing itself but on the diagram around it.
Proof: Whether the horizontal pass is above or below does not alone decide the handedness. Both cases can be right- and left-handed (see the 8 crossings above). If one rotates the knot so that the over-pass is horizontal and if one then leaves the crossing through the overhead pass to the right (to the East) then it depends on the rest of the knot whether one returns to the crossing from the South (then the crossing is right-handed) or from the North (then the crossing is left-handed).
The fact that the two groups of crossings are called left- and right-handed gives a hint that one can distinguish crossings with the left and right hands.
By using one's hands, how can one remember whether a crossing is
right- or left-handed?
Stretch out your fingers so that all are in one plane, and your
thumb is at a right angle to all others which are parallel to
each other. Rotate your hand so that you can see your palm and
your thumb points towards the outgoing direction of the
over-pass and your fingers point towards the outgoing direction of
the under-pass. The hand that can do that decides the
handedness.
For example, for the crossing below, you would stretch your hand out like this:
Since you can only do this with your left hand, this is a left-handed crossing.
Right- and left-handed crossings are also referred to as positive or negative crossings.
Since you can only do this with your left hand, this is a left-handed crossing.
Right- and left-handed crossings are also referred to as positive or negative crossings.
writhe number:
the difference between the number of left- and right-handed
crossings in one diagram. The writhe number characterizes a diagram, not a knot as there can be 2
different diagrams of the same knot with different writhe numbers.
What is the writhe number of this diagram?
What is the writhe number of this diagram?
The right-handed crossings in the diagram are highlighted in red and the left-handed crossings in green.
To get the writhe number, we can count the number of left- and right-handed crossings, then subtract the
number of right-handed crossings
from the number of left-handed crossings.
This diagram has 2 left-handed crossings and 4 right-handed crossings, so its writhe number is 2 − 4 =
−2.
knot invariant:
a number or a polynomial or a feasibility statement that is characteristic for all (infinitely many)
diagrams of a knot. The properties of a knot being chiral/achiral, invertible/noninvertible,
reversible are knot invariants.
crossing number:
the minimal number of crossings that any diagram of
this knot can have after deformation, this is a characteristic of each knot and therefore a knot
invariant.
How many crossings does this diagram have?
What is the crossing number of the knot represented through the above diagram?
What are the two lowest crossing numbers that a knot can have?
How many crossings does this diagram have?
This diagram has 5 crossings.
What is the crossing number of the knot represented through the above diagram?
Zero! The crossing number is a property of the abstract
mathematical knot, it is not the property of a diagram. The
diagram above can be deformed to get the unknot
which has zero crossings.
Do you see how?
Because the crossing number is the minimal number of crossings of any diagram of a knot and because one cannot have less than zero crossings, the crossing number of the knot represented by the above diagram is zero.
which has zero crossings. Do you see how?
Because the crossing number is the minimal number of crossings of any diagram of a knot and because one cannot have less than zero crossings, the crossing number of the knot represented by the above diagram is zero.
What are the two lowest crossing numbers that a knot can have?
The lowest crossing number belongs to the unknot which is 0.
A knot diagram with 1 crossing would look like:
and could be deformed to the unknot. A knot diagram with 2 crossings would look like
and could also be deformed to the unknot.
The trefoil diagram shown further above has 3 crossings and can not be deformed to the unknot, so the lowest crossing numbers are 0 and 3.
and could be deformed to the unknot. A knot diagram with 2 crossings would look like
and could also be deformed to the unknot.The trefoil diagram shown further above has 3 crossings and can not be deformed to the unknot, so the lowest crossing numbers are 0 and 3.
arc:
The part of the knot line in a diagram from one crossing to the next.
How many arcs does a diagram with N crossings have?
How many arcs does a diagram with N crossings have?
Each crossing has 4 ends of arcs. Each arc has 2 ends, so there are 4/2 = 2
times as many arcs as crossings, so 2N arcs.
hole:
empty space in a diagram that is surrounded by arcs. The whole
empty space outside the diagram is also one hole.
How many holes does a diagram with N crossings have?
How many holes does a diagram with N crossings have?
One could draw several knots and guess a formula but one can derive
it too. Euler's formula says that for any drawing in the plane where
m lines (here m=2N arcs) each connect 2 out of n points (here n=N
crossings)
then the number f of faces (here holes) is f = 2 + m − n. That gives
for the number of holes of a knot: 2 + 2N − N = N + 2.
over-strand:
A sequence of consecutive arcs in a diagram
(i.e. arcs following one another) which starts and ends at an
under-pass and otherwise involves 0, 1 or more over-passes.
under-strand:
A sequence of consecutive arcs in a diagram
(i.e. arcs following one another) which starts and ends at an
over-pass and otherwise involves 0, 1 or more under-passes.
(In literature, a 'strand' is often used for what we call an over-strand. For us the number of over-passes of an over-strand matters as well as the number of under-passes of an under-strand. We therefore consider under-strands as well as over-strands.)
What kind of strand is the shown horizontal line consisting of five arcs?
How many over-strands does a diagram with N crossings have?
(In literature, a 'strand' is often used for what we call an over-strand. For us the number of over-passes of an over-strand matters as well as the number of under-passes of an under-strand. We therefore consider under-strands as well as over-strands.)
What kind of strand is the shown horizontal line consisting of five arcs?
This is an under-strand with 4 under-passes.
How many over-strands does a diagram with N crossings have?
At each crossing there are 2 ends of strands (either one end of
2 different strands or both ends from one strand). On the other hand,
each strand has 2 ends which are at a crossing. Therefore the number
of crossings is equal to the number of over-strands and because of
symmetry also equal to the number of under-strands, so there are
N of each.
Reidemeister moves:
In 1927 the German mathematician Kurt
Reidemeister and, independently, James Waddell Alexander and Garland
Baird Briggs (1926), proved that any two diagrams that represent the
same knot can be deformed into each other through a sequence of only 3
different types of moves. The problem is that during the deformation
the number of crossings may temporarily rise and a sharp upper bound for
this increase is unknown as well as the number of needed moves.
Reidemeister 1 move:
removes or adds a hole surrounded by one arc:
Which diagram shows a left-handed crossing and which
shows a right-handed crossing?
Which diagram shows a left-handed crossing and which
shows a right-handed crossing?
The left diagram shows a right-handed crossing and the right diagram shows a
left-handed crossing. A Reidemeister 1 move therefore changes the number
of right- or left-handed crossings by 1 and thus changes the
writhe number of the diagram.
Reidemeister 2 move:
removes or adds a hole surrounded by 2 arcs:
What can one say about the handedness of the two crossings that are
added or removed in a Reidemeister 2 move?
What can one say about the handedness of the two crossings that are
added or removed in a Reidemeister 2 move?
One of the two crossings is right-handed and one is left-handed.
A Reidemeister 2 move therefore does not change the writhe
number of a diagram.
Reidemeister 3 move:
removes and adds a hole surrounded by 3 arcs.
Which 2 types of holes surrounded by 3 arcs can you think of?
Which 2 types of holes surrounded by 3 arcs can you think of?
Either:
Verify that the result of the 3 moves is always the same.
Does the handedness of the 3 crossings change in a Reidemeister 3 move?
What have we learned?
- 1) each arc has 1 over-pass and 1 under-pass:
then none of the 3 arcs can be moved over/under/through the other crossing
- 2) one arc has 2 over-passes, one has 1 over- and 1
under-pass, and one has 2 under-passes:
then there are 3 moves. One can move the over-over-strand which stays an over-over-strand:
or move the under-over-strand which becomes an over-under-strand:
or move the under-under-strand which stays an under-under-strand:
Verify that the result of the 3 moves is always the same.
When comparing the right-hand sides of the above moves it is easy to see that
all 3 moves produce identical results. Therefore, if there is a Reidemeister 3 move
then there is only one. All that changes is that for all 3 arcs the other two arcs are
now crossed in the reverse order. This means that for the middle arc the order of over-pass
and under-pass is reversed.
Does the handedness of the 3 crossings change in a Reidemeister 3 move?
No. To see that, pick any orientation for each strand and use the
hand rule above.
What have we learned?
We learned:
- how to spot holes with 3 arcs that allow a Reidemeister 3 move,
- that for such a hole it does not matter which arc is moved,
- that the handedness of the 3 crossings does not change,
- that the order of over- and under-pass is reversed for the middle arc.
pass move:
This has nothing to do with a 'pass' defined above.
A pass move replaces an over-(under-)strand with another
over-(under-)strand where both strands have the same ends. For examples, please see P-, P0 and P+ moves below.
P- move:
a pass move where the new strand has less passes than the
old strand.
Find a P- move replacing the green strand in this diagram:
Find a P- move replacing the green strand in this diagram:
In this diagram the new red strand has fewer passes than the old green strand. Therefore this diagram shows
a P- move.
P0 move:
a pass move where the new strand has the same number of passes
as the old strand.
Find a P0 move replacing the green strand in this diagram:
Find a P0 move replacing the green strand in this diagram:
In this diagram the new red strand has the same number of passes as the old green strand. Therefore this
diagram shows a P0 move.
P+ move:
a pass move where the new strand has more passes
than the old strand.
Find a P+ move replacing the green strand in this diagram:
P+ moves become necessary if one wants to change the writhe number of a diagram. More about that is described further below under 'Finding P0 Moves'.
Find a P+ move replacing the green strand in this diagram:
In this diagram the new red strand has one more pass than the old green strand, therefore this is a P+ move.
P+ moves become necessary if one wants to change the writhe number of a diagram. More about that is described further below under 'Finding P0 Moves'.
unknotting number:
The unknotting number is the property of a knot, not the property of a diagram and is therefore
a knot invariant.
Starting with a knot diagram it is the minimal number of times that one or more crossings need to be switched in order to obtain the unknot. Before the first switch and inbetween switches the diagram can be arbitrarily deformed. The unknotting number is therefore not an easy number to determine because any deformation is allowed.
Why does the trefoil have unknotting number 1?
Starting with a knot diagram it is the minimal number of times that one or more crossings need to be switched in order to obtain the unknot. Before the first switch and inbetween switches the diagram can be arbitrarily deformed. The unknotting number is therefore not an easy number to determine because any deformation is allowed.
Why does the trefoil have unknotting number 1?
The trefoil cannot have the unknotting number 0 because it cannot
be deformed into the unknot (this needs to be and can be proven). The unknot has unknotting number 0. So the
trefoil has unknotting number ≥1. On the other hand one can easily
see that switching any one crossing of the trefoil diagram
shown further above produces the unknot, so the unknotting
number of the trefoil is ≤1. If it is ≥1 and ≤1 then it must be =1.
How to Simplify Diagrams
Finding R1 Moves
Simple cases of R1 moves, like here:
where one can flip a loop 4 times and instantly get the unknot
are easy to spot by following the knot line and looking for an arc
with both ends at the same crossing. The order of performing R1 moves
does not matter.
But there are more general cases of applying Reidemeister 1 moves. If an over-strand starts at a crossing of which it is an overpass but otherwise lies completely on top of other arcs (therefore being called 'over-strand') then this loop can surely be shortcut and thus removed. For example, at first the loop on top in the center can be removed and then the other ones, one by one:
This loop can also be removed when lying entirely underneath:
where one can flip a loop 4 times and instantly get the unknot
are easy to spot by following the knot line and looking for an arc
with both ends at the same crossing. The order of performing R1 moves
does not matter. But there are more general cases of applying Reidemeister 1 moves. If an over-strand starts at a crossing of which it is an overpass but otherwise lies completely on top of other arcs (therefore being called 'over-strand') then this loop can surely be shortcut and thus removed. For example, at first the loop on top in the center can be removed and then the other ones, one by one:
This loop can also be removed when lying entirely underneath:
Finding R2 Moves
Similarly to R1 moves it is easy to spot prototype R2 moves like
here where two R2 moves need to be done before an R1 move yields
the unknot:
In the following example one has to perform an R2 move with the same strand twice, once pulling this strand from underneath and once from above:
The last step is not an R2 move. It is only added to show that the knot is the sum of two trefoil knots.
In the following example one has to perform an R2 move with the same strand twice, once pulling this strand from underneath and once from above:
The last step is not an R2 move. It is only added to show that the knot is the sum of two trefoil knots.
About the Interface (1)
The example above is suitable to demonstrate the optimal use of the
interface. After intercepting the knot line:
One can NOT retreat one end all the way; and then retreat the other end all the way because one would first remove an under-pass and have both ends in the under-pass status and in this status it is not possible to remove an over-pass. Instead, one removes one under-pass, jumps to the other end, removes the other under-pass which brings both ends into the same hole, which changes the ends into neutral status and then allows to remove over-passes and then to re-unite the ends. In short, one jumps between ends to remove all under-passes then all over-passes and so on.
This implementation of a status of ends is not a weakness of the program but it guarantees that interactive modification of the diagram does not change the mathematical knot.
One can NOT retreat one end all the way; and then retreat the other end all the way because one would first remove an under-pass and have both ends in the under-pass status and in this status it is not possible to remove an over-pass. Instead, one removes one under-pass, jumps to the other end, removes the other under-pass which brings both ends into the same hole, which changes the ends into neutral status and then allows to remove over-passes and then to re-unite the ends. In short, one jumps between ends to remove all under-passes then all over-passes and so on.
This implementation of a status of ends is not a weakness of the program but it guarantees that interactive modification of the diagram does not change the mathematical knot.
Finding P- Moves
P- moves replace an over-strand with one having less over-passes
or an under-strand with one having less under-passes. In both
cases the number of crossings is reduced.
To find such moves one steps through the knot line and looks for as many consecutive over-passes as possible or for as many consecutive under-passes as possible, at least two. If one found such a strand, for example, an over-strand then one tries to find an alternative route with less over-passes that links the same two under-pass ends.
In the following example a strand with 4 consecutive under-passes is replaced by a strand without underpasses and then this strand with 3 consecutive over-passes is replaced by a strand with one over-pass. Two more P- moves remove each 2 more crossings. The resulting diagram can be simplified further by two more R1 moves, as seen below.
The more consecutive passes of one sort one finds, the higher the
chance to find a different route that needs less passes.
To find such moves one steps through the knot line and looks for as many consecutive over-passes as possible or for as many consecutive under-passes as possible, at least two. If one found such a strand, for example, an over-strand then one tries to find an alternative route with less over-passes that links the same two under-pass ends.
In the following example a strand with 4 consecutive under-passes is replaced by a strand without underpasses and then this strand with 3 consecutive over-passes is replaced by a strand with one over-pass. Two more P- moves remove each 2 more crossings. The resulting diagram can be simplified further by two more R1 moves, as seen below.
Finding R3 Moves
R1, R2 and P- moves change the number of crossings. An R3
move does not change the number of crossings, therefore we place
its description after the P- move. The following example shows how
R3 moves can still be useful by making P- moves possible.
As described in the first section, an R3 hole is surrounded by a top arc
with 2 over-pass ends (here A,B), a middle arc with 1 over-pass end (C) and 1
under-pass end (B) and a bottom arc with 2 under-pass ends (here
A,C).
(unknot taken from the Unknot Wikipedia page)
We found in the above section that an R3 move reverses for the middle strand (here through B,C) the order of under- and over-pass. An R3 move is beneficial if the R3 move leads for the continuation of the middle arc to an increase of consecutive over-passes and/or consecutive under-passes and thus increases the chance to find a P- move. This is the case if in the continuation of the middle arc after the under-pass B comes an over-pass (D,E are over-passes) and/or after the over-pass (C) follows an under-pass (F,G,H are under-passes). The R3 move slides the middle arc BC between the top arc and bottom arc at A:
Before, there were only 2 consecutive over-passes at D and E,
now there are 3 at C, D and E. This longer over-strand can now be
re-routed in a P- move:
reducing the number of crossings by 2 from 13 to 11.
Also, the other side of the strand can be re-routed in a P- move. Before, there were 3 consecutive under-passes at F,G and H, now there are 4 at B, F, G, and H. This P- move leads to:
and is also reducing the number of crossings by 2. Both diagrams can be
simplified further through P- and R1 moves resulting finally in the unknot.
Can you see how? Just follow the hints on how to spot P- moves given above.
Let us practise that with an example.
How many R3 moves are possible in this diagram:
We found in the above section that an R3 move reverses for the middle strand (here through B,C) the order of under- and over-pass. An R3 move is beneficial if the R3 move leads for the continuation of the middle arc to an increase of consecutive over-passes and/or consecutive under-passes and thus increases the chance to find a P- move. This is the case if in the continuation of the middle arc after the under-pass B comes an over-pass (D,E are over-passes) and/or after the over-pass (C) follows an under-pass (F,G,H are under-passes). The R3 move slides the middle arc BC between the top arc and bottom arc at A:
Also, the other side of the strand can be re-routed in a P- move. Before, there were 3 consecutive under-passes at F,G and H, now there are 4 at B, F, G, and H. This P- move leads to:
Let us practise that with an example.
How many R3 moves are possible in this diagram:
Three R3 moves are possible. For each one we show in light blue the three arcs that are
involved. What is easily overlooked is the third one where the hole
is the whole outside space which is 'surrounded' by only 3 arcs.
Perform the 1. R3 move and find out whether it is beneficial:
Perform the 2. R3 move and find out whether it is beneficial:
Perform the 3. R3 move and find out whether it is beneficial:
1. R3 move
2. R3 move
3. R3 move
Perform the 1. R3 move and find out whether it is beneficial:
This R3 move is beneficial. It allows afterwards a P- moves as shown in a sequence of moves further
below.
Our definition for an R3 move to be beneficial it is not
neccessarily to allow a P- move but to increase the number of consecutive
over- or under-passes and that is easy to see even without
performing all these moves. In the following diagram the middle arc
of the R3 hole has an
over-pass at A, an under-pass at B followed by two over-passes at C and D.
In an R3 move the order of over- and under-pass is reversed for the middle arc
as shown in the Diagram 3 with now 3 consecutive over-passes. This is enough
to find a P- move needing less than 3 passes in Diagram 5.
About the sequence of diagrams below: In Dia 1 we make space to prepare the R3 move in Dia 2 (here by
moving
the top arc) with the result in Dia 3. In Dia 4 we make space to prepare
the P- move in Dia 5 where the green strand with 3 over-passes is replaced
by the red strand with only 1 over-pass in Dia 6. In Dia 7 we shift a strand
to make space for the next P- move in Dia 9 with result in Dia 10 and Dia 11
after shrinking which is easily identified as knot 51.
1. Widening
2. The R3 move
3. After the R3 move
4. Widening
5. A P- move
6. After the P- move
7. For the next P- move
8. Before the P- move
9. The 2nd P- move
10. Afterwards
11. Contracted
Perform the 2. R3 move and find out whether it is beneficial:
The sequence shows that the R3 move is beneficial.
1. Widening
2. The R3 move
3. After the R3 move
4. A P- move
5. After the P- move
6. Another P- move
7. After the P- move
8. Contracted
Perform the 3. R3 move and find out whether it is beneficial:
The 3rd R3 move is also beneficial. To execute this move, one follows the same principle: the middle
strand cuts the two other strands, which this time
'surround' the outside hole, in reverse order.
The result is that travelling upwards in the Dia 2 along the light blue strand one first gets to an overpass and then an underpass. If after the R3 move one travels along the red strand then one passes the other 2 strands in reverse order and therefore first gets to an underpass and then an overpass. As one can see in Dia 5, the sequence of now 3 consecutive underpasses of the light blue strand there enables a P- move.
The result is that travelling upwards in the Dia 2 along the light blue strand one first gets to an overpass and then an underpass. If after the R3 move one travels along the red strand then one passes the other 2 strands in reverse order and therefore first gets to an underpass and then an overpass. As one can see in Dia 5, the sequence of now 3 consecutive underpasses of the light blue strand there enables a P- move.
1. Widening
2. An R3 move
3. After the R3 move
4. Widening
5. A P- move
6. Widening
7. A 2nd P- move
8. After the P- move
9. Shortening
10. After shortening
11. Shortening
12. Straightening
13. ↻90° rotation
About the Interface (2)
The example above is suitable to demonstrate how to perform an R3
move with our interface.
As described in the 'Introduction with Definitions' > 'Reidemeister 3 move' there are 3 ways to perform a Reidemeister 3 move: moving the bottom strand, moving the middle strand, or, moving the top strand. As shown there, all 3 ways have the same result, they perform the same Reidemeister 3 move.
Our interface only allows one to complete an R3 move by moving the bottom strand or the top strand but not the middle strand. The reason is the feature of our interface that BOTH ends can only add/remove over-passes or BOTH ends can only add/remove under-passes at a time. But that does not prevent us from performing R3 moves, as moving any one of the 3 strands gives the same result.
As described in the 'Introduction with Definitions' > 'Reidemeister 3 move' there are 3 ways to perform a Reidemeister 3 move: moving the bottom strand, moving the middle strand, or, moving the top strand. As shown there, all 3 ways have the same result, they perform the same Reidemeister 3 move.
Our interface only allows one to complete an R3 move by moving the bottom strand or the top strand but not the middle strand. The reason is the feature of our interface that BOTH ends can only add/remove over-passes or BOTH ends can only add/remove under-passes at a time. But that does not prevent us from performing R3 moves, as moving any one of the 3 strands gives the same result.
Finding P0 Moves
P0 moves are pass moves that do not change the number of
crossings, just like R3 moves which are special versions of
P0 moves. Like R3 moves, a P0 move may be beneficial and
enable a P- move. Because P0 moves are less useful on
average, they occur more frequently but it is more
challenging to see whether they enable a P- move.
To find a P0 move one looks for an over-strand or an under-strand just like for a P- move. To check whether a P0 move may be beneficial and enable a P- move one proceeds like in the case of the R3 move. One looks whether the removal of the strand increases the number of consecutive over- or under-passes of strands that were crossed and one checks whether after re-routing the new strand increases the number of consecutive over- or under-passes of strands that are crossed now. In either one of these cases one checks whether the strands with an increased number of consecutive over- or under-passes can be re-routed with less crossings.
Let us look at this example:
We label the crossings:
and find a beneficial P0 move step by step.
How many over-strands with at least 2 over-passes
and how many under-strands with at least 2
under-passes do you see?
It is not difficult to see that the IE strand has a P0 move that re-locates it to cross over the GC and BH strand:
but the question is whether this P0 move is beneficial.
Did moving the IE strand increase the number of consecutive over-/under-passes
of the previously crossed DF or HJ strands?
Did more consecutive over-/under-passes get created when placing the strand on top of the
2 strands GC and BH?
Can this BH strand be re-routed in a P- move to reduce the number of crossings?
In the original diagram this P- move could have been done first as a P0 move and the P0 move mentioned above afterwards as P- move with the same total saving of 2 crossings.
In difficult problems it may be necessary to perform several P0 moves before a P- move becomes possible.
A diagram is maximally simplified if the number of crossings is the crossing number (see first section). In that case P0 moves will never enable a P- move.
To find a P0 move one looks for an over-strand or an under-strand just like for a P- move. To check whether a P0 move may be beneficial and enable a P- move one proceeds like in the case of the R3 move. One looks whether the removal of the strand increases the number of consecutive over- or under-passes of strands that were crossed and one checks whether after re-routing the new strand increases the number of consecutive over- or under-passes of strands that are crossed now. In either one of these cases one checks whether the strands with an increased number of consecutive over- or under-passes can be re-routed with less crossings.
Let us look at this example:
How many over-strands with at least 2 over-passes
and how many under-strands with at least 2
under-passes do you see?
We get three over-strands with at least 2 over-passes:
AB, GC, IE and three under-strands with at least 2
under-passes: EF, BH, DJ.
It is not difficult to see that the IE strand has a P0 move that re-locates it to cross over the GC and BH strand:
Did moving the IE strand increase the number of consecutive over-/under-passes
of the previously crossed DF or HJ strands?
Yes, the HJ strand now has 2 over-passes but this strand can not be re-routed
to link the same two holes with less over-passes.
Did more consecutive over-/under-passes get created when placing the strand on top of the
2 strands GC and BH?
Yes, the BH under-strand had 2 under-passes and now has 3 under-passes.
Can this BH strand be re-routed in a P- move to reduce the number of crossings?
Yes: The new route of the BH strand links the same holes but with only 1 under-pass instead of
3 under-passes.
The fact that the new strand is longer (involves more steps) in this
diagram than the replaced strand does not matter. All that matters
is the reduction of the number of crossings from 10 to 8 which now
allows to identify this knot as knot 817. In the original diagram this P- move could have been done first as a P0 move and the P0 move mentioned above afterwards as P- move with the same total saving of 2 crossings.
In difficult problems it may be necessary to perform several P0 moves before a P- move becomes possible.
A diagram is maximally simplified if the number of crossings is the crossing number (see first section). In that case P0 moves will never enable a P- move.
Finding U1 Moves
A U1 move switches a crossing which afterwards allows to simplify
the diagram to remove all crossings and show that the switch
produced the unknot.
In general nice puzzles should have unique solutions so our U1 and U2 puzzles show diagrams where the switch of only one crossing results in the unknot. This hint allows you to cut down the search.
If the diagram includes a twist of the knot line like this:
would it matter which one of the crossings is switched?
If there are still several crossing candidates, one should try to imagine how many R1, R2 moves will become available through the switch and try that switch first which seems to allow the most simplifications afterwards.
Another hint to minimize trying switches is to imagine whether after the switch there will definitely be a knot remaining, like a trefoil. If so then that switch is not the correct one in U1 puzzles.
On the worksheet the number of available switches is limited to the minimum that is needed to get the unknot. If you did a switch one can not switch it back because the diagram may have been changed, so one would have to reset the diagram.
In general nice puzzles should have unique solutions so our U1 and U2 puzzles show diagrams where the switch of only one crossing results in the unknot. This hint allows you to cut down the search.
If the diagram includes a twist of the knot line like this:
It will not matter which crossing is switched. Both results are equivalent:
Therefore either both of these crossings are unknotting switches or
none of them. Because our puzzles have only one unknotting switch,
these two crossings can be ignored.
=
=
If there are still several crossing candidates, one should try to imagine how many R1, R2 moves will become available through the switch and try that switch first which seems to allow the most simplifications afterwards.
Another hint to minimize trying switches is to imagine whether after the switch there will definitely be a knot remaining, like a trefoil. If so then that switch is not the correct one in U1 puzzles.
On the worksheet the number of available switches is limited to the minimum that is needed to get the unknot. If you did a switch one can not switch it back because the diagram may have been changed, so one would have to reset the diagram.
Finding U2 Moves
For U2 puzzles the same hint about equivalent switches applies as
for U1 puzzles. Also for U2 puzzles there is just one crossing that
reduces the unknotting number, i.e. makes progress towards the
unknot. Once that unique first crossing is switched and the
resulting diagram is simplified, there may be more than one switch
possible that creates the unknot.
Finding P0U Moves
Research performed by Caribou Contests on unknotting numbers showed
that there exists maximally simplified knot diagrams (with the
minimal number of crossings), which do not have a simplifying
switch. In other words, there are diagrams where switching any
crossings will not make progress to reach the unknot. In that case
one first has to perform one or more P0 moves that change the puzzle
into a U2 puzzle. The good news is that diagrams requiring P0 moves
first are rare and therefore it will be likely that any P0 moves
will change it into a U2 puzzle.
More References about Knots
Mathematical Knot Theory is an old research subject so there
exists a vast amount of literature for it. However, it also is a young
subject as several milestones have only been reached in recent
decades. For example, there is a scientific "Journal of Knot
Theory and Its Ramifications" dedicated to knots which has
a new issue every month.
A great book that we recommend is: Adams, Colin (2004), The Knot Book: An Elementary Introduction to the Mathematical Theory of Knots, American Mathematical Society, ISBN 978-0-8218-3678-1
There are also many websites about knots. A good place to start is the Knot Theory page on Wikipedia.
For videos, check out Numberphile's Playlist of Knot videos on YouTube. They also have a great explanation of Colouring Knots, which is another way to help identify diagrams of the same knot.
Caribou produced two posters on Unknotting and colouring knots.
A great book that we recommend is: Adams, Colin (2004), The Knot Book: An Elementary Introduction to the Mathematical Theory of Knots, American Mathematical Society, ISBN 978-0-8218-3678-1
There are also many websites about knots. A good place to start is the Knot Theory page on Wikipedia.
For videos, check out Numberphile's Playlist of Knot videos on YouTube. They also have a great explanation of Colouring Knots, which is another way to help identify diagrams of the same knot.
Caribou produced two posters on Unknotting and colouring knots.
EN KHMER! :)
This guide will introduce you to the subject of Knot Theory. The first section gives an introduction to important math theory concepts and terminology. If you are only looking for quick tips to solve the unknotting puzzles, skip ahead to the second section. You can always consult the first section if the meaning of some words is unclear.
Introduction with Definitions
Introduction
First things first : what are knots? Knots in mathematics are different from knots in your everyday life.
What kinds of knots do you use every day?
How are these different from mathematical knots?
Are there any mathematical knots in everyday life?
How does the Unknotting game work?
Definitions
From here on, we only talk about mathematical knots. To avoid misunderstandings and to be able to decide whether any statement is true or false we should start with defining the meaning of several words for the rest of this page.
knot line:
knot diagram:
mathematical knot (or simply, knot):
ambient isotopy:
step:
crossing:
pass:
switch:
orientation:
crossing handedness:
writhe number:
knot invariant:
crossing number:
arc:
hole:
over-strand:
under-strand:
Reidemeister moves:
Reidemeister 1 move:
Reidemeister 2 move:
Reidemeister 3 move:
pass move:
P- move:
P0 move:
P+ move:
unknotting number:
How to Simplify Diagrams
Finding R1 Moves
Finding R2 Moves
About the Interface (1)
Finding P- Moves
Finding R3 Moves
About the Interface (2)
Finding P0 Moves
Finding U1 Moves
Finding U2 Moves
Finding P0U Moves
More References about Knots
This guide will introduce you to the subject of Knot Theory. The first section gives an introduction to important math theory concepts and terminology. If you are only looking for quick tips to solve the unknotting puzzles, skip ahead to the second section. You can always consult the first section if the meaning of some words is unclear.
Introduction with Definitions
Introduction
First things first : what are knots? Knots in mathematics are different from knots in your everyday life.
What kinds of knots do you use every day?
Most of us use knots to tie our shoelaces, put on a necktie or scarf, to close up a bag, and so on... You might
know many more knots if you go sailing, camping, fishing, or if you sew, knit, or style hair.
However, none of these are mathematical knots!
How are these different from mathematical knots?
Have a look at the following two drawings of knots. Confusingly, these are both known as 'figure-eight' knots,
because of the figure 8 they contain.
Everyday Figure-eight Knot Mathematical Figure-eight Knot
What big difference can you see? We're sure you can figure it out!
Everyday Figure-eight Knot Mathematical Figure-eight Knot
What big difference can you see? We're sure you can figure it out!
The biggest difference is that the mathematical knot is a closed curve − that is, there are no loose
ends, it's a closed loop. What we call 'knots' in everyday life are known as 'braids' in
mathematics.
Also, while everyday knots might include more than one strand of material, in mathematics knots are a single, closed, continuous strand. Objects that involve more than one knot woven together are known as 'links'.
Also, while everyday knots might include more than one strand of material, in mathematics knots are a single, closed, continuous strand. Objects that involve more than one knot woven together are known as 'links'.
Are there any mathematical knots in everyday life?
Of course! You now know that in mathematics, a knot is a single, closed, continuous strand.
With this definition in mind, what is the simplest mathematical knot?
How can you make a mathematical knot from a piece of string?
With this definition in mind, what is the simplest mathematical knot?
The simplest mathematical knot is just a single loop or circle, like this:
We will talk more about this knot later, but there are plenty of examples of simple loops like this in real life.
We will talk more about this knot later, but there are plenty of examples of simple loops like this in real life.
How can you make a mathematical knot from a piece of string?
The simplest mathematical knot is a circle. To make it, simply glue the ends of your string together.
What happens if you twist your circle once?
Is this a different knot?
Can all knots be deformed to make a circle?
How different can a knot diagram look from the simplest form?
What happens if you twist your circle once?
If you take your loop of string, twist it, and lay it flat, you might get something like this:
Is this a different knot?
Of course knot! All you did was twist it.
This might seem easy, but finding ways to tell whether two diagrams (pictures) show the same knot is a very important and difficult question for mathematicians who study Knot Theory.
One way to tell that two pictures represent the same knot is whether you can deform one of them to look like the other. For example, here you just need to twist it back to get the loop.
This might seem easy, but finding ways to tell whether two diagrams (pictures) show the same knot is a very important and difficult question for mathematicians who study Knot Theory.
One way to tell that two pictures represent the same knot is whether you can deform one of them to look like the other. For example, here you just need to twist it back to get the loop.
Can all knots be deformed to make a circle?
To answer this question, try this out:
Can you deform this to get a circle?
- take a strand of string
- twist it to form a loop
- pass one end through the loop
Can you deform this to get a circle?
Try as you might, there is no way to deform this knot into a circle. At least, not without cutting
the string and gluing it back together.
This is actually a different mathematical knot, called the trefoil knot because it looks like a three-leaf clover.
This is actually a different mathematical knot, called the trefoil knot because it looks like a three-leaf clover.
How different can a knot diagram look from the simplest form?
Another important property of mathematical knots is that they can be arbirtrarily stretched and bent.
For example, our diagram of the simplest knot looks more like a square than a circle − we could
draw it as a perfect circle, and it would be the same knot. You could take the simplest knot, a circle,
and stretch it out into a long thin ellipse, then use it as a string to tie it into the 'everyday
figure-8 knot'. Mathematically, it is still a circle.
As you might expect, then, the different diagrams of the same underlying knot can look drastically different! For example, the Gordian Knot and this diagram can both be deformed into a circle, with enough patience. They are the diagrams of the same knot as the circle.
The circle is just the easiest way of drawing this knot, but the knot isn't really a circle, it's an abstract mathematical object that we can represent in many different ways. In the same way that "1 car" and "1 apple" is not the number 1, a circle is just one way to represent this knot.
As you might expect, then, the different diagrams of the same underlying knot can look drastically different! For example, the Gordian Knot and this diagram can both be deformed into a circle, with enough patience. They are the diagrams of the same knot as the circle.
The circle is just the easiest way of drawing this knot, but the knot isn't really a circle, it's an abstract mathematical object that we can represent in many different ways. In the same way that "1 car" and "1 apple" is not the number 1, a circle is just one way to represent this knot.
How does the Unknotting game work?
In this game, you deform mathematical knot diagrams to reduce the number of
crossings as much as possible.
While you are allowed to 'cut' the strands by clicking on them, you are only modifying the diagram, not the underlying knot. This is why there is a restriction on how you can reattach the ends you have 'cut'. This restriction guarantees that the mathematical knot is unchanged in any re-routing of strands even if the knot changes its appearance.
When do you undo everyday knots in real life?
Try some other mathematical knot games!
While you are allowed to 'cut' the strands by clicking on them, you are only modifying the diagram, not the underlying knot. This is why there is a restriction on how you can reattach the ends you have 'cut'. This restriction guarantees that the mathematical knot is unchanged in any re-routing of strands even if the knot changes its appearance.
When do you undo everyday knots in real life?
You might be familiar with the struggle of disentangling electronics cables like for earbuds. If you tie
your shoes, you have to untie the laces.
The same property of real knots is what makes them useful, but also harder to undo.
What makes knots in real life so hard to undo?
The same property of real knots is what makes them useful, but also harder to undo.
What makes knots in real life so hard to undo?
The answer is friction! However, mathematical knots have no friction. You can think of them as
'infinitely slippery'.
Try some other mathematical knot games!
Our unknotting game is one way to have fun with knots on a screen, but here are some other games for you to
try:
- 1 player : Eiffel Tower and other string tricks
- 2 players : Cat's cradle
- Group : Human Knot
Definitions
From here on, we only talk about mathematical knots. To avoid misunderstandings and to be able to decide whether any statement is true or false we should start with defining the meaning of several words for the rest of this page.
knot line:
a closed curve in 3-dimensional space which does not
intersect itself and which has a finite thickness (to avoid
infinitely many smaller and smaller knots along the line), Example:
the figure-8 knot.
knot diagram:
a projection of a knot line into 2 dimensions where
different parts of the knot line can cross each other (on this website
lines cross under an angle of 90°), but do not lie on top of each other.
Examples for knot diagrams:
Unknot Trefoil Figure-eight Cinquefoil Three-twist
The 'unknot' is also called the 'trivial knot'.
Which knot does each of these 3 diagrams represent?
Which knot does this diagram represent?
Do you see how to deform this into the earlier trefoil diagram?
Examples for knot diagrams:
Unknot Trefoil Figure-eight Cinquefoil Three-twist
The 'unknot' is also called the 'trivial knot'.
Which knot does each of these 3 diagrams represent?
The unknot. Do you see how they can be deformed to a rectangle?
Which knot does this diagram represent?
The trefoil.
Do you see how to deform this into the earlier trefoil diagram?
You can deform this diagram into the other by flipping down the top arc.
→
→
mathematical knot (or simply, knot):
the abstract object behind
a set of (infinitely many) knot diagrams that can all be deformed,
stretched, and shifted into each other without being cut. Example:
the knot 31 also called 'trefoil' which is the simplest
non-trivial knot.
ambient isotopy:
the mathematical term
when one knot line can be continuously distorted to another one.
step:
On this website knot diagrams are drawn using only 6 tiles which we call steps:
crossing:
the place in a diagram where two steps cross, one on top of
the other:
pass:
a step that is part of a crossing, there are over-passes (fully
visible) and under-passes (partially covered).
switch:
swapping over- and under-pass of a crossing, i.e. switching between these two crossings:
If a crossing is switched, the old and new diagram in general
represent different knots. Switching all crossings is equivalent to
changing a knot to its mirror image.
Some knots are identical to their mirrored version, that means there is an
ambient isotopy between them. These are called 'achiral'. For example, the figure-8 knot is achiral.
Others can not be deformed to their mirrored version, like the
trefoil. They are called 'chiral'.
Can this knot be deformed into its mirror image?
Can this knot be deformed into its mirror image?
Yes! This diagram represents the figure-eight knot which is achiral and can be deformed into its mirror
image as follows:
Notice that the only difference between the first diagram and the last one is that all of the crossings are switched!
Initial position
A 180° rotation
Moving a strand
The mirror image
Notice that the only difference between the first diagram and the last one is that all of the crossings are switched!
orientation:
This is not a property of the knot line, nor of the knot.
It is a question of how to move along the knot line. One can move in 2
directions, also called 2 orientations.
A knot that can be deformed via an ambient isotopy into itself but with the
orientation reversed is called 'invertible' otherwise it is called
noninvertible. The smallest noninvertible knot is 817 which is achiral but
if an orientation is added it becomes chiral (find more on
the Invertible Knot Wikipedia page). Adding more
structure (here
an orientation) causes it to lose symmetry (not identical anymore to its mirror
image).
A chiral knot (a knot that can not be deformed to its mirror image) can still be invertible (symmetric against change of orientation). Such knots are called 'reversible'.
A chiral knot (a knot that can not be deformed to its mirror image) can still be invertible (symmetric against change of orientation). Such knots are called 'reversible'.
crossing handedness:
For a given knot diagram, crossings are either right- or left-handed.
In the following we will explore two types of crossings and determine which is right- or left-handed.
How many different crossings are there if we consider
which pass is an over-/under-pass and consider both orientations?
If one keeps the diagram unchanged and only switches one crossing,
does the handedness of that crossing change?
By using one's hands, how can one remember whether a crossing is
right- or left-handed?
In the following we will explore two types of crossings and determine which is right- or left-handed.
How many different crossings are there if we consider
which pass is an over-/under-pass and consider both orientations?
In total there are 8 cases:
If the horizontal pass is the over-pass then there are 4 options:
Similarly if the vertical pass is the over-pass then there are 4 more options:
The crossings in group 1, 7, 4, 6 are called right-handed crossings and
the crossings in group 2, 5, 3, 8 are called left-handed crossings.
If the horizontal pass is the over-pass then there are 4 options:
1
2
3
4
5
6
7
8
- Principle: Handedness should not depend on the orientation (direction of stepping through the knot line), so reversing both arrows we identify 4 pairs of crossings: 1 = 4, 2 = 3, 5 = 8, 6 = 7. Therefore, whichever groups we end up with, crossings 1 and 4 should be in the same group and so on.
- Principle: The group that a crossing belongs to should not change if we rotate the whole knot. We therefore identify crossings 1 = 7 = 4 = 6 and 2 = 5 = 3 = 8.
The crossings in group 1, 7, 4, 6 are called right-handed crossings and
the crossings in group 2, 5, 3, 8 are called left-handed crossings.
If one keeps the diagram unchanged and only switches one crossing,
does the handedness of that crossing change?
Yes. Try switching any of the eight crossings,
then check which one it has become and check whether it is
still in the same handedness group. For example, switching crossing 1
gives crossing 5, both are in different handedness groups.
Statement: The question whether a crossing is right-handed or left-handed does not only depend on the crossing itself but on the diagram around it.
Proof: Whether the horizontal pass is above or below does not alone decide the handedness. Both cases can be right- and left-handed (see the 8 crossings above). If one rotates the knot so that the over-pass is horizontal and if one then leaves the crossing through the overhead pass to the right (to the East) then it depends on the rest of the knot whether one returns to the crossing from the South (then the crossing is right-handed) or from the North (then the crossing is left-handed).
The fact that the two groups of crossings are called left- and right-handed gives a hint that one can distinguish crossings with the left and right hands.
Statement: The question whether a crossing is right-handed or left-handed does not only depend on the crossing itself but on the diagram around it.
Proof: Whether the horizontal pass is above or below does not alone decide the handedness. Both cases can be right- and left-handed (see the 8 crossings above). If one rotates the knot so that the over-pass is horizontal and if one then leaves the crossing through the overhead pass to the right (to the East) then it depends on the rest of the knot whether one returns to the crossing from the South (then the crossing is right-handed) or from the North (then the crossing is left-handed).
The fact that the two groups of crossings are called left- and right-handed gives a hint that one can distinguish crossings with the left and right hands.
By using one's hands, how can one remember whether a crossing is
right- or left-handed?
Stretch out your fingers so that all are in one plane, and your
thumb is at a right angle to all others which are parallel to
each other. Rotate your hand so that you can see your palm and
your thumb points towards the outgoing direction of the
over-pass and your fingers point towards the outgoing direction of
the under-pass. The hand that can do that decides the
handedness.
For example, for the crossing below, you would stretch your hand out like this:
Since you can only do this with your left hand, this is a left-handed crossing.
Right- and left-handed crossings are also referred to as positive or negative crossings.
Since you can only do this with your left hand, this is a left-handed crossing.
Right- and left-handed crossings are also referred to as positive or negative crossings.
writhe number:
the difference between the number of left- and right-handed
crossings in one diagram. The writhe number characterizes a diagram, not a knot as there can be 2
different diagrams of the same knot with different writhe numbers.
What is the writhe number of this diagram?
What is the writhe number of this diagram?
The right-handed crossings in the diagram are highlighted in red and the left-handed crossings in green.
To get the writhe number, we can count the number of left- and right-handed crossings, then subtract the
number of right-handed crossings
from the number of left-handed crossings.
This diagram has 2 left-handed crossings and 4 right-handed crossings, so its writhe number is 2 − 4 =
−2.
knot invariant:
a number or a polynomial or a feasibility statement that is characteristic for all (infinitely many)
diagrams of a knot. The properties of a knot being chiral/achiral, invertible/noninvertible,
reversible are knot invariants.
crossing number:
the minimal number of crossings that any diagram of
this knot can have after deformation, this is a characteristic of each knot and therefore a knot
invariant.
How many crossings does this diagram have?
What is the crossing number of the knot represented through the above diagram?
What are the two lowest crossing numbers that a knot can have?
How many crossings does this diagram have?
This diagram has 5 crossings.
What is the crossing number of the knot represented through the above diagram?
Zero! The crossing number is a property of the abstract
mathematical knot, it is not the property of a diagram. The
diagram above can be deformed to get the unknot
which has zero crossings.
Do you see how?
Because the crossing number is the minimal number of crossings of any diagram of a knot and because one cannot have less than zero crossings, the crossing number of the knot represented by the above diagram is zero.
which has zero crossings. Do you see how?
Because the crossing number is the minimal number of crossings of any diagram of a knot and because one cannot have less than zero crossings, the crossing number of the knot represented by the above diagram is zero.
What are the two lowest crossing numbers that a knot can have?
The lowest crossing number belongs to the unknot which is 0.
A knot diagram with 1 crossing would look like:
and could be deformed to the unknot. A knot diagram with 2 crossings would look like
and could also be deformed to the unknot.
The trefoil diagram shown further above has 3 crossings and can not be deformed to the unknot, so the lowest crossing numbers are 0 and 3.
and could be deformed to the unknot. A knot diagram with 2 crossings would look like
and could also be deformed to the unknot.The trefoil diagram shown further above has 3 crossings and can not be deformed to the unknot, so the lowest crossing numbers are 0 and 3.
arc:
The part of the knot line in a diagram from one crossing to the next.
How many arcs does a diagram with N crossings have?
How many arcs does a diagram with N crossings have?
Each crossing has 4 ends of arcs. Each arc has 2 ends, so there are 4/2 = 2
times as many arcs as crossings, so 2N arcs.
hole:
empty space in a diagram that is surrounded by arcs. The whole
empty space outside the diagram is also one hole.
How many holes does a diagram with N crossings have?
How many holes does a diagram with N crossings have?
One could draw several knots and guess a formula but one can derive
it too. Euler's formula says that for any drawing in the plane where
m lines (here m=2N arcs) each connect 2 out of n points (here n=N
crossings)
then the number f of faces (here holes) is f = 2 + m − n. That gives
for the number of holes of a knot: 2 + 2N − N = N + 2.
over-strand:
A sequence of consecutive arcs in a diagram
(i.e. arcs following one another) which starts and ends at an
under-pass and otherwise involves 0, 1 or more over-passes.
under-strand:
A sequence of consecutive arcs in a diagram
(i.e. arcs following one another) which starts and ends at an
over-pass and otherwise involves 0, 1 or more under-passes.
(In literature, a 'strand' is often used for what we call an over-strand. For us the number of over-passes of an over-strand matters as well as the number of under-passes of an under-strand. We therefore consider under-strands as well as over-strands.)
What kind of strand is the shown horizontal line consisting of five arcs?
How many over-strands does a diagram with N crossings have?
(In literature, a 'strand' is often used for what we call an over-strand. For us the number of over-passes of an over-strand matters as well as the number of under-passes of an under-strand. We therefore consider under-strands as well as over-strands.)
What kind of strand is the shown horizontal line consisting of five arcs?
This is an under-strand with 4 under-passes.
How many over-strands does a diagram with N crossings have?
At each crossing there are 2 ends of strands (either one end of
2 different strands or both ends from one strand). On the other hand,
each strand has 2 ends which are at a crossing. Therefore the number
of crossings is equal to the number of over-strands and because of
symmetry also equal to the number of under-strands, so there are
N of each.
Reidemeister moves:
In 1927 the German mathematician Kurt
Reidemeister and, independently, James Waddell Alexander and Garland
Baird Briggs (1926), proved that any two diagrams that represent the
same knot can be deformed into each other through a sequence of only 3
different types of moves. The problem is that during the deformation
the number of crossings may temporarily rise and a sharp upper bound for
this increase is unknown as well as the number of needed moves.
Reidemeister 1 move:
removes or adds a hole surrounded by one arc:
Which diagram shows a left-handed crossing and which
shows a right-handed crossing?
Which diagram shows a left-handed crossing and which
shows a right-handed crossing?
The left diagram shows a right-handed crossing and the right diagram shows a
left-handed crossing. A Reidemeister 1 move therefore changes the number
of right- or left-handed crossings by 1 and thus changes the
writhe number of the diagram.
Reidemeister 2 move:
removes or adds a hole surrounded by 2 arcs:
What can one say about the handedness of the two crossings that are
added or removed in a Reidemeister 2 move?
What can one say about the handedness of the two crossings that are
added or removed in a Reidemeister 2 move?
One of the two crossings is right-handed and one is left-handed.
A Reidemeister 2 move therefore does not change the writhe
number of a diagram.
Reidemeister 3 move:
removes and adds a hole surrounded by 3 arcs.
Which 2 types of holes surrounded by 3 arcs can you think of?
Which 2 types of holes surrounded by 3 arcs can you think of?
Either:
Verify that the result of the 3 moves is always the same.
Does the handedness of the 3 crossings change in a Reidemeister 3 move?
What have we learned?
- 1) each arc has 1 over-pass and 1 under-pass:
then none of the 3 arcs can be moved over/under/through the other crossing
- 2) one arc has 2 over-passes, one has 1 over- and 1
under-pass, and one has 2 under-passes:
then there are 3 moves. One can move the over-over-strand which stays an over-over-strand:
or move the under-over-strand which becomes an over-under-strand:
or move the under-under-strand which stays an under-under-strand:
Verify that the result of the 3 moves is always the same.
When comparing the right-hand sides of the above moves it is easy to see that
all 3 moves produce identical results. Therefore, if there is a Reidemeister 3 move
then there is only one. All that changes is that for all 3 arcs the other two arcs are
now crossed in the reverse order. This means that for the middle arc the order of over-pass
and under-pass is reversed.
Does the handedness of the 3 crossings change in a Reidemeister 3 move?
No. To see that, pick any orientation for each strand and use the
hand rule above.
What have we learned?
We learned:
- how to spot holes with 3 arcs that allow a Reidemeister 3 move,
- that for such a hole it does not matter which arc is moved,
- that the handedness of the 3 crossings does not change,
- that the order of over- and under-pass is reversed for the middle arc.
pass move:
This has nothing to do with a 'pass' defined above.
A pass move replaces an over-(under-)strand with another
over-(under-)strand where both strands have the same ends. For examples, please see P-, P0 and P+ moves below.
P- move:
a pass move where the new strand has less passes than the
old strand.
Find a P- move replacing the green strand in this diagram:
Find a P- move replacing the green strand in this diagram:
In this diagram the new red strand has fewer passes than the old green strand. Therefore this diagram shows
a P- move.
P0 move:
a pass move where the new strand has the same number of passes
as the old strand.
Find a P0 move replacing the green strand in this diagram:
Find a P0 move replacing the green strand in this diagram:
In this diagram the new red strand has the same number of passes as the old green strand. Therefore this
diagram shows a P0 move.
P+ move:
a pass move where the new strand has more passes
than the old strand.
Find a P+ move replacing the green strand in this diagram:
P+ moves become necessary if one wants to change the writhe number of a diagram. More about that is described further below under 'Finding P0 Moves'.
Find a P+ move replacing the green strand in this diagram:
In this diagram the new red strand has one more pass than the old green strand, therefore this is a P+ move.
P+ moves become necessary if one wants to change the writhe number of a diagram. More about that is described further below under 'Finding P0 Moves'.
unknotting number:
The unknotting number is the property of a knot, not the property of a diagram and is therefore
a knot invariant.
Starting with a knot diagram it is the minimal number of times that one or more crossings need to be switched in order to obtain the unknot. Before the first switch and inbetween switches the diagram can be arbitrarily deformed. The unknotting number is therefore not an easy number to determine because any deformation is allowed.
Why does the trefoil have unknotting number 1?
Starting with a knot diagram it is the minimal number of times that one or more crossings need to be switched in order to obtain the unknot. Before the first switch and inbetween switches the diagram can be arbitrarily deformed. The unknotting number is therefore not an easy number to determine because any deformation is allowed.
Why does the trefoil have unknotting number 1?
The trefoil cannot have the unknotting number 0 because it cannot
be deformed into the unknot (this needs to be and can be proven). The unknot has unknotting number 0. So the
trefoil has unknotting number ≥1. On the other hand one can easily
see that switching any one crossing of the trefoil diagram
shown further above produces the unknot, so the unknotting
number of the trefoil is ≤1. If it is ≥1 and ≤1 then it must be =1.
How to Simplify Diagrams
Finding R1 Moves
Simple cases of R1 moves, like here:
where one can flip a loop 4 times and instantly get the unknot
are easy to spot by following the knot line and looking for an arc
with both ends at the same crossing. The order of performing R1 moves
does not matter.
But there are more general cases of applying Reidemeister 1 moves. If an over-strand starts at a crossing of which it is an overpass but otherwise lies completely on top of other arcs (therefore being called 'over-strand') then this loop can surely be shortcut and thus removed. For example, at first the loop on top in the center can be removed and then the other ones, one by one:
This loop can also be removed when lying entirely underneath:
where one can flip a loop 4 times and instantly get the unknot
are easy to spot by following the knot line and looking for an arc
with both ends at the same crossing. The order of performing R1 moves
does not matter. But there are more general cases of applying Reidemeister 1 moves. If an over-strand starts at a crossing of which it is an overpass but otherwise lies completely on top of other arcs (therefore being called 'over-strand') then this loop can surely be shortcut and thus removed. For example, at first the loop on top in the center can be removed and then the other ones, one by one:
This loop can also be removed when lying entirely underneath:
Finding R2 Moves
Similarly to R1 moves it is easy to spot prototype R2 moves like
here where two R2 moves need to be done before an R1 move yields
the unknot:
In the following example one has to perform an R2 move with the same strand twice, once pulling this strand from underneath and once from above:
The last step is not an R2 move. It is only added to show that the knot is the sum of two trefoil knots.
In the following example one has to perform an R2 move with the same strand twice, once pulling this strand from underneath and once from above:
The last step is not an R2 move. It is only added to show that the knot is the sum of two trefoil knots.
About the Interface (1)
The example above is suitable to demonstrate the optimal use of the
interface. After intercepting the knot line:
One can NOT retreat one end all the way; and then retreat the other end all the way because one would first remove an under-pass and have both ends in the under-pass status and in this status it is not possible to remove an over-pass. Instead, one removes one under-pass, jumps to the other end, removes the other under-pass which brings both ends into the same hole, which changes the ends into neutral status and then allows to remove over-passes and then to re-unite the ends. In short, one jumps between ends to remove all under-passes then all over-passes and so on.
This implementation of a status of ends is not a weakness of the program but it guarantees that interactive modification of the diagram does not change the mathematical knot.
One can NOT retreat one end all the way; and then retreat the other end all the way because one would first remove an under-pass and have both ends in the under-pass status and in this status it is not possible to remove an over-pass. Instead, one removes one under-pass, jumps to the other end, removes the other under-pass which brings both ends into the same hole, which changes the ends into neutral status and then allows to remove over-passes and then to re-unite the ends. In short, one jumps between ends to remove all under-passes then all over-passes and so on.
This implementation of a status of ends is not a weakness of the program but it guarantees that interactive modification of the diagram does not change the mathematical knot.
Finding P- Moves
P- moves replace an over-strand with one having less over-passes
or an under-strand with one having less under-passes. In both
cases the number of crossings is reduced.
To find such moves one steps through the knot line and looks for as many consecutive over-passes as possible or for as many consecutive under-passes as possible, at least two. If one found such a strand, for example, an over-strand then one tries to find an alternative route with less over-passes that links the same two under-pass ends.
In the following example a strand with 4 consecutive under-passes is replaced by a strand without underpasses and then this strand with 3 consecutive over-passes is replaced by a strand with one over-pass. Two more P- moves remove each 2 more crossings. The resulting diagram can be simplified further by two more R1 moves, as seen below.
The more consecutive passes of one sort one finds, the higher the
chance to find a different route that needs less passes.
To find such moves one steps through the knot line and looks for as many consecutive over-passes as possible or for as many consecutive under-passes as possible, at least two. If one found such a strand, for example, an over-strand then one tries to find an alternative route with less over-passes that links the same two under-pass ends.
In the following example a strand with 4 consecutive under-passes is replaced by a strand without underpasses and then this strand with 3 consecutive over-passes is replaced by a strand with one over-pass. Two more P- moves remove each 2 more crossings. The resulting diagram can be simplified further by two more R1 moves, as seen below.
Finding R3 Moves
R1, R2 and P- moves change the number of crossings. An R3
move does not change the number of crossings, therefore we place
its description after the P- move. The following example shows how
R3 moves can still be useful by making P- moves possible.
As described in the first section, an R3 hole is surrounded by a top arc
with 2 over-pass ends (here A,B), a middle arc with 1 over-pass end (C) and 1
under-pass end (B) and a bottom arc with 2 under-pass ends (here
A,C).
(unknot taken from the Unknot Wikipedia page)
We found in the above section that an R3 move reverses for the middle strand (here through B,C) the order of under- and over-pass. An R3 move is beneficial if the R3 move leads for the continuation of the middle arc to an increase of consecutive over-passes and/or consecutive under-passes and thus increases the chance to find a P- move. This is the case if in the continuation of the middle arc after the under-pass B comes an over-pass (D,E are over-passes) and/or after the over-pass (C) follows an under-pass (F,G,H are under-passes). The R3 move slides the middle arc BC between the top arc and bottom arc at A:
Before, there were only 2 consecutive over-passes at D and E,
now there are 3 at C, D and E. This longer over-strand can now be
re-routed in a P- move:
reducing the number of crossings by 2 from 13 to 11.
Also, the other side of the strand can be re-routed in a P- move. Before, there were 3 consecutive under-passes at F,G and H, now there are 4 at B, F, G, and H. This P- move leads to:
and is also reducing the number of crossings by 2. Both diagrams can be
simplified further through P- and R1 moves resulting finally in the unknot.
Can you see how? Just follow the hints on how to spot P- moves given above.
Let us practise that with an example.
How many R3 moves are possible in this diagram:
We found in the above section that an R3 move reverses for the middle strand (here through B,C) the order of under- and over-pass. An R3 move is beneficial if the R3 move leads for the continuation of the middle arc to an increase of consecutive over-passes and/or consecutive under-passes and thus increases the chance to find a P- move. This is the case if in the continuation of the middle arc after the under-pass B comes an over-pass (D,E are over-passes) and/or after the over-pass (C) follows an under-pass (F,G,H are under-passes). The R3 move slides the middle arc BC between the top arc and bottom arc at A:
Also, the other side of the strand can be re-routed in a P- move. Before, there were 3 consecutive under-passes at F,G and H, now there are 4 at B, F, G, and H. This P- move leads to:
Let us practise that with an example.
How many R3 moves are possible in this diagram:
Three R3 moves are possible. For each one we show in light blue the three arcs that are
involved. What is easily overlooked is the third one where the hole
is the whole outside space which is 'surrounded' by only 3 arcs.
Perform the 1. R3 move and find out whether it is beneficial:
Perform the 2. R3 move and find out whether it is beneficial:
Perform the 3. R3 move and find out whether it is beneficial:
1. R3 move
2. R3 move
3. R3 move
Perform the 1. R3 move and find out whether it is beneficial:
This R3 move is beneficial. It allows afterwards a P- moves as shown in a sequence of moves further
below.
Our definition for an R3 move to be beneficial it is not
neccessarily to allow a P- move but to increase the number of consecutive
over- or under-passes and that is easy to see even without
performing all these moves. In the following diagram the middle arc
of the R3 hole has an
over-pass at A, an under-pass at B followed by two over-passes at C and D.
In an R3 move the order of over- and under-pass is reversed for the middle arc
as shown in the Diagram 3 with now 3 consecutive over-passes. This is enough
to find a P- move needing less than 3 passes in Diagram 5.
About the sequence of diagrams below: In Dia 1 we make space to prepare the R3 move in Dia 2 (here by
moving
the top arc) with the result in Dia 3. In Dia 4 we make space to prepare
the P- move in Dia 5 where the green strand with 3 over-passes is replaced
by the red strand with only 1 over-pass in Dia 6. In Dia 7 we shift a strand
to make space for the next P- move in Dia 9 with result in Dia 10 and Dia 11
after shrinking which is easily identified as knot 51.
1. Widening
2. The R3 move
3. After the R3 move
4. Widening
5. A P- move
6. After the P- move
7. For the next P- move
8. Before the P- move
9. The 2nd P- move
10. Afterwards
11. Contracted
Perform the 2. R3 move and find out whether it is beneficial:
The sequence shows that the R3 move is beneficial.
1. Widening
2. The R3 move
3. After the R3 move
4. A P- move
5. After the P- move
6. Another P- move
7. After the P- move
8. Contracted
Perform the 3. R3 move and find out whether it is beneficial:
The 3rd R3 move is also beneficial. To execute this move, one follows the same principle: the middle
strand cuts the two other strands, which this time
'surround' the outside hole, in reverse order.
The result is that travelling upwards in the Dia 2 along the light blue strand one first gets to an overpass and then an underpass. If after the R3 move one travels along the red strand then one passes the other 2 strands in reverse order and therefore first gets to an underpass and then an overpass. As one can see in Dia 5, the sequence of now 3 consecutive underpasses of the light blue strand there enables a P- move.
The result is that travelling upwards in the Dia 2 along the light blue strand one first gets to an overpass and then an underpass. If after the R3 move one travels along the red strand then one passes the other 2 strands in reverse order and therefore first gets to an underpass and then an overpass. As one can see in Dia 5, the sequence of now 3 consecutive underpasses of the light blue strand there enables a P- move.
1. Widening
2. An R3 move
3. After the R3 move
4. Widening
5. A P- move
6. Widening
7. A 2nd P- move
8. After the P- move
9. Shortening
10. After shortening
11. Shortening
12. Straightening
13. ↻90° rotation
About the Interface (2)
The example above is suitable to demonstrate how to perform an R3
move with our interface.
As described in the 'Introduction with Definitions' > 'Reidemeister 3 move' there are 3 ways to perform a Reidemeister 3 move: moving the bottom strand, moving the middle strand, or, moving the top strand. As shown there, all 3 ways have the same result, they perform the same Reidemeister 3 move.
Our interface only allows one to complete an R3 move by moving the bottom strand or the top strand but not the middle strand. The reason is the feature of our interface that BOTH ends can only add/remove over-passes or BOTH ends can only add/remove under-passes at a time. But that does not prevent us from performing R3 moves, as moving any one of the 3 strands gives the same result.
As described in the 'Introduction with Definitions' > 'Reidemeister 3 move' there are 3 ways to perform a Reidemeister 3 move: moving the bottom strand, moving the middle strand, or, moving the top strand. As shown there, all 3 ways have the same result, they perform the same Reidemeister 3 move.
Our interface only allows one to complete an R3 move by moving the bottom strand or the top strand but not the middle strand. The reason is the feature of our interface that BOTH ends can only add/remove over-passes or BOTH ends can only add/remove under-passes at a time. But that does not prevent us from performing R3 moves, as moving any one of the 3 strands gives the same result.
Finding P0 Moves
P0 moves are pass moves that do not change the number of
crossings, just like R3 moves which are special versions of
P0 moves. Like R3 moves, a P0 move may be beneficial and
enable a P- move. Because P0 moves are less useful on
average, they occur more frequently but it is more
challenging to see whether they enable a P- move.
To find a P0 move one looks for an over-strand or an under-strand just like for a P- move. To check whether a P0 move may be beneficial and enable a P- move one proceeds like in the case of the R3 move. One looks whether the removal of the strand increases the number of consecutive over- or under-passes of strands that were crossed and one checks whether after re-routing the new strand increases the number of consecutive over- or under-passes of strands that are crossed now. In either one of these cases one checks whether the strands with an increased number of consecutive over- or under-passes can be re-routed with less crossings.
Let us look at this example:
We label the crossings:
and find a beneficial P0 move step by step.
How many over-strands with at least 2 over-passes
and how many under-strands with at least 2
under-passes do you see?
It is not difficult to see that the IE strand has a P0 move that re-locates it to cross over the GC and BH strand:
but the question is whether this P0 move is beneficial.
Did moving the IE strand increase the number of consecutive over-/under-passes
of the previously crossed DF or HJ strands?
Did more consecutive over-/under-passes get created when placing the strand on top of the
2 strands GC and BH?
Can this BH strand be re-routed in a P- move to reduce the number of crossings?
In the original diagram this P- move could have been done first as a P0 move and the P0 move mentioned above afterwards as P- move with the same total saving of 2 crossings.
In difficult problems it may be necessary to perform several P0 moves before a P- move becomes possible.
A diagram is maximally simplified if the number of crossings is the crossing number (see first section). In that case P0 moves will never enable a P- move.
To find a P0 move one looks for an over-strand or an under-strand just like for a P- move. To check whether a P0 move may be beneficial and enable a P- move one proceeds like in the case of the R3 move. One looks whether the removal of the strand increases the number of consecutive over- or under-passes of strands that were crossed and one checks whether after re-routing the new strand increases the number of consecutive over- or under-passes of strands that are crossed now. In either one of these cases one checks whether the strands with an increased number of consecutive over- or under-passes can be re-routed with less crossings.
Let us look at this example:
How many over-strands with at least 2 over-passes
and how many under-strands with at least 2
under-passes do you see?
We get three over-strands with at least 2 over-passes:
AB, GC, IE and three under-strands with at least 2
under-passes: EF, BH, DJ.
It is not difficult to see that the IE strand has a P0 move that re-locates it to cross over the GC and BH strand:
Did moving the IE strand increase the number of consecutive over-/under-passes
of the previously crossed DF or HJ strands?
Yes, the HJ strand now has 2 over-passes but this strand can not be re-routed
to link the same two holes with less over-passes.
Did more consecutive over-/under-passes get created when placing the strand on top of the
2 strands GC and BH?
Yes, the BH under-strand had 2 under-passes and now has 3 under-passes.
Can this BH strand be re-routed in a P- move to reduce the number of crossings?
Yes: The new route of the BH strand links the same holes but with only 1 under-pass instead of
3 under-passes.
The fact that the new strand is longer (involves more steps) in this
diagram than the replaced strand does not matter. All that matters
is the reduction of the number of crossings from 10 to 8 which now
allows to identify this knot as knot 817. In the original diagram this P- move could have been done first as a P0 move and the P0 move mentioned above afterwards as P- move with the same total saving of 2 crossings.
In difficult problems it may be necessary to perform several P0 moves before a P- move becomes possible.
A diagram is maximally simplified if the number of crossings is the crossing number (see first section). In that case P0 moves will never enable a P- move.
Finding U1 Moves
A U1 move switches a crossing which afterwards allows to simplify
the diagram to remove all crossings and show that the switch
produced the unknot.
In general nice puzzles should have unique solutions so our U1 and U2 puzzles show diagrams where the switch of only one crossing results in the unknot. This hint allows you to cut down the search.
If the diagram includes a twist of the knot line like this:
would it matter which one of the crossings is switched?
If there are still several crossing candidates, one should try to imagine how many R1, R2 moves will become available through the switch and try that switch first which seems to allow the most simplifications afterwards.
Another hint to minimize trying switches is to imagine whether after the switch there will definitely be a knot remaining, like a trefoil. If so then that switch is not the correct one in U1 puzzles.
On the worksheet the number of available switches is limited to the minimum that is needed to get the unknot. If you did a switch one can not switch it back because the diagram may have been changed, so one would have to reset the diagram.
In general nice puzzles should have unique solutions so our U1 and U2 puzzles show diagrams where the switch of only one crossing results in the unknot. This hint allows you to cut down the search.
If the diagram includes a twist of the knot line like this:
It will not matter which crossing is switched. Both results are equivalent:
Therefore either both of these crossings are unknotting switches or
none of them. Because our puzzles have only one unknotting switch,
these two crossings can be ignored.
=
=
If there are still several crossing candidates, one should try to imagine how many R1, R2 moves will become available through the switch and try that switch first which seems to allow the most simplifications afterwards.
Another hint to minimize trying switches is to imagine whether after the switch there will definitely be a knot remaining, like a trefoil. If so then that switch is not the correct one in U1 puzzles.
On the worksheet the number of available switches is limited to the minimum that is needed to get the unknot. If you did a switch one can not switch it back because the diagram may have been changed, so one would have to reset the diagram.
Finding U2 Moves
For U2 puzzles the same hint about equivalent switches applies as
for U1 puzzles. Also for U2 puzzles there is just one crossing that
reduces the unknotting number, i.e. makes progress towards the
unknot. Once that unique first crossing is switched and the
resulting diagram is simplified, there may be more than one switch
possible that creates the unknot.
Finding P0U Moves
Research performed by Caribou Contests on unknotting numbers showed
that there exists maximally simplified knot diagrams (with the
minimal number of crossings), which do not have a simplifying
switch. In other words, there are diagrams where switching any
crossings will not make progress to reach the unknot. In that case
one first has to perform one or more P0 moves that change the puzzle
into a U2 puzzle. The good news is that diagrams requiring P0 moves
first are rare and therefore it will be likely that any P0 moves
will change it into a U2 puzzle.
More References about Knots
Mathematical Knot Theory is an old research subject so there
exists a vast amount of literature for it. However, it also is a young
subject as several milestones have only been reached in recent
decades. For example, there is a scientific "Journal of Knot
Theory and Its Ramifications" dedicated to knots which has
a new issue every month.
A great book that we recommend is: Adams, Colin (2004), The Knot Book: An Elementary Introduction to the Mathematical Theory of Knots, American Mathematical Society, ISBN 978-0-8218-3678-1
There are also many websites about knots. A good place to start is the Knot Theory page on Wikipedia.
For videos, check out Numberphile's Playlist of Knot videos on YouTube. They also have a great explanation of Colouring Knots, which is another way to help identify diagrams of the same knot.
Caribou produced two posters on Unknotting and colouring knots.
A great book that we recommend is: Adams, Colin (2004), The Knot Book: An Elementary Introduction to the Mathematical Theory of Knots, American Mathematical Society, ISBN 978-0-8218-3678-1
There are also many websites about knots. A good place to start is the Knot Theory page on Wikipedia.
For videos, check out Numberphile's Playlist of Knot videos on YouTube. They also have a great explanation of Colouring Knots, which is another way to help identify diagrams of the same knot.
Caribou produced two posters on Unknotting and colouring knots.
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