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!

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.
What big difference can you see? We're sure you can figure it out!

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?

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!

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.

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:
The 'unknot' is also called the 'trivial 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.

the mathematical term when one knot line can be continuously distorted to another one.

On this website knot diagrams are drawn using only 6 tiles which we call steps:

the place in a diagram where two steps cross, one on top of the other:

a step that is part of a crossing, there are over-passes (fully visible) and under-passes (partially covered).

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?

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 8₁₇ to lose symmetry (not identical anymore to its mirror image).

The figure-eight knot mentioned above has crossing number 4, so less than 8, and must therefore be invertible.
How could the figure-eight knot be deformed to leave its diagram unchanged, but with the orientation reversed?
In an earlier example, a figure-eight knot was transformed via a simple deformation into its mirror image. This deformation also changed the orientation− please verify it for yourself! So, if one wanted to deform the diagram to its mirror image without reverting orientation, then one could simply combine both sequences.

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'.

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?

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?

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.

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?

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?

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?

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.

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 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.

removes or adds a hole surrounded by one arc: Which diagram shows a left-handed crossing and which shows a right-handed crossing?

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?

removes and adds a hole surrounded by 3 arcs.
Which 2 types of holes surrounded by 3 arcs can you think of?

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.

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:

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:

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'.

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?

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:

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.

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.

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.

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:

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.

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?
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.

A P+ move increases the number of crossings in a diagram. Why could such a move be useful for anything if the goal is usually to simplify diagrams? To see their purpose remind yourself about crossing numbers, handedness of crossings, writhe numbers and how the different Reidemeister moves affect both numbers. Then have a look at the following (difficult) challenge.

For a long time since the late 19^th century two diagrams named 10₁₆₁ and 10₁₆₂ in the Dale Rolfsen Knot Table were thought to belong to two different knots. In 1973 Kenneth Perko realized that both represent the same knot. Since then the pair of entries in classical knot tables that actually represent the same knot is called Perko pair.
These two diagrams represent the same knot 10₁₆₁. How can they be deformed into each other? How can one find such a sequence of P+ and P- and possibly P0 moves that accomplish such a deformation?
Here is another challenge. How can these two diagrams be deformed into each other?

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.

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.

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.

The Unknotting challenges on this site are specially selected. Knowing how they are special can help to solve them.
R3 Moves
U1 and U2 Moves
P0 Moves
P0U Moves