Three moves, and what they cannot undo

Take a piece of string, tangle it, and join the ends. That is a knot — a closed loop sitting in space, with no free ends to pull.

Two of them are the same knot if one can be deformed into the other without cutting the string or passing it through itself. So the question is this knotted? means: can this loop be wiggled until it is a plain circle?

The trefoil above cannot be undone. That is obvious to anyone who has handled string and it is remarkably hard to prove, and the gap between those two facts is what this essay is about.

The picture throws a dimension away #

A knot lives in three dimensions and every figure here is flat, so something has been discarded and something has been kept.

What is discarded is the third coordinate. What is kept is one bit at each crossing: which strand goes over.

That is a very small amount of information and it is enough. A diagram — a closed curve in the plane with finitely many crossings, each labelled over-or-under — determines the knot completely, and every knot has one. So the three-dimensional question has been converted into a question about flat pictures, which is the same trade as reducing a city to a graph: throw away everything except the feature the question is about, and hope the question survives.

It does survive, but at a price. One knot has infinitely many diagrams, and telling whether two diagrams are the same knot is now the problem. The asymmetry in the caption above is the shape of the whole difficulty: a diagram is evidence in one direction only.

The three moves #

Reidemeister’s theorem, 1927: two diagrams represent the same knot if and only if one can be turned into the other by a sequence of three local moves, plus ordinary sliding of the strands around the page.

Each move changes the diagram inside a small disc and leaves everything outside it alone. The first removes or adds a kink. The second separates two strands that overlap. The third slides a strand past a crossing of two others.

That is the whole list, and the finiteness of the list is the theorem. Deformation in three dimensions is a continuous, unbounded thing — infinitely many ways to move a string — and it has been reduced to three pictures. Everything afterwards depends on that reduction, because it converts an impossible search over continuous motions into a combinatorial question about finitely many local rewrites.

Why a number is not enough #

The obvious way to tell knots apart is to count crossings. The unknot has zero, the trefoil three, the figure-eight four.

That fails immediately, because a diagram’s crossing count is not a property of the knot. A circle can be drawn with any number of crossings by adding kinks, and each kink is one application of the first move.

The repair is to take the minimum over all diagrams — the crossing number of the knot — and that is a genuine invariant. It is also close to useless in practice: computing it means searching all diagrams, which is infinite, and there is no procedure that halts. Knowing the trefoil’s crossing number is three requires already knowing it is not the unknot, which is the question.

This is the standing difficulty with the whole subject, and it is worth stating in general terms. Anything defined as a minimum over all diagrams is invariant and uncomputable; anything read off one diagram is computable and not invariant. Every useful tool in knot theory is an attempt to be on both sides at once: something read off a single diagram that happens to be unchanged by all three moves.

The first thing that works #

Colour the arcs of a diagram — the pieces between under-crossings — with three colours, subject to one rule: at every crossing, the three arcs meeting there are all the same colour or all different.

A diagram is tricolourable if it admits such a colouring using more than one colour. The trefoil is: it has three arcs, all three meet at each of its three crossings, so painting them differently satisfies the rule everywhere.

The unknot is not. It has one arc, so every colouring of it uses one colour, and the “more than one” clause fails.

Now the essential step. Tricolourability survives all three moves. Checking that is a finite exercise — each move changes the arcs in a bounded way inside its disc, and in each case a legal colouring outside extends to a legal colouring inside, in both directions. Three cases, each a small diagram.

So tricolourability is a property of the knot rather than of the diagram. The trefoil has it, the unknot does not, therefore they are different knots, therefore the trefoil is knotted.

That is the proof, and it is the first honest one most people meet. It is worth pausing on its shape: nothing was untangled, no deformation was attempted, and no search was made. A quantity was found that cannot change, it takes different values on the two objects, and the conclusion follows.

The shape of every argument in the subject #

The manoeuvre generalises, and it is the same one that runs through the rest of topology.

Euler’s characteristic is a number attached to a surface that does not change when the surface is redrawn, so two surfaces with different characteristics cannot be the same surface. One-sidedness is a property that survives any deformation of a band, so a Möbius band is not a cylinder. Tricolourability is that idea applied to loops in space.

In each case the invariant is easy to compute and the thing it proves is hard. And in each case the invariant is one-directional: equal invariants do not prove the objects are the same. The figure-eight knot is not tricolourable, and neither is the unknot, and they are different knots — so this particular tool cannot tell them apart, and something stronger is needed.

That asymmetry is permanent. An invariant can prove difference and cannot prove sameness, unless it happens to be a complete invariant, and no computable complete invariant for knots is known.

It is the same one-way logic as a counting argument that shows something must exist without producing it, turned around: there, existence was cheap and location was impossible; here, difference is cheap and sameness is impossible. Both are the price of an argument that never inspects the object it is talking about.

What it costs #

Deciding whether a given diagram is the unknot is decidable — Haken showed this in 1961 using normal surface theory — and the algorithm is appalling. The best known bounds put unknot recognition in NP and, since 2011, also in co-NP conditionally, which is evidence that it is not NP-complete and no evidence that it is easy.

In practice the tools used are polynomial invariants. The Alexander polynomial (1928) is computed from a determinant of the crossing data; the Jones polynomial (1984) from a recursive relation on crossings. Both are computable directly from one diagram, both are unchanged by the three moves, and both are strictly stronger than tricolourability — the Jones polynomial distinguishes the trefoil from its own mirror image, which nothing simpler does.

Neither is complete. Whether the Jones polynomial detects the unknot — whether a knot with the unknot’s polynomial must be unknotted — is open, and it has been open since 1984. There are known pairs of distinct knots with the same Jones polynomial, so it is not complete in general; whether it fails at the unknot specifically is the question nobody can settle.

The computational cost is also worth stating: the Jones polynomial is #P-hard to evaluate exactly at most points, so the practical algorithms are exponential in the crossing number. Knot tables were produced by hand into the tens of crossings and by machine to sixteen, where there are $1{,}388{,}705$ prime knots, and the growth beyond that is what stops the enumeration rather than any lack of interest.

Where it needs a condition #

The whole subject depends on being in exactly three dimensions, and the failures on either side are instructive.

In four dimensions every knot comes undone. A loop can be lifted into the fourth coordinate, passed over itself with no intersection, and put back — so there are no knotted circles in four-space at all. Knotting requires exactly enough room to have crossings and not enough room to avoid them, and three is the only number of dimensions with that property for circles. In four dimensions the knotted objects are surfaces, which is a different and harder subject.

The moves also require the string to be tame. A curve can be pathological enough that no diagram exists — with infinitely many crossings accumulating at a point — and such wild knots have properties no argument here touches. Every result in this essay assumes tameness, which is a hypothesis usually left silent.

And the second move, which pulls two strands apart, is where an orientation subtlety hides: two strands can overlap in two ways, and the move as drawn covers both only because the crossings involved are of opposite sign. For invariants that track crossing sign — which the polynomial ones do — the moves have to be stated more carefully than the pictures suggest.

Where it came from, and what it was for #

Knot theory has an origin story that is unusually specific and unusually wrong.

In the 1860s William Thomson, later Lord Kelvin, proposed that atoms are knotted vortices in the ether, with different elements corresponding to different knots. It is a genuinely elegant idea — it explains why elements are discrete, why they are stable, and why there are a limited number of them — and it motivated Peter Guthrie Tait to spend years tabulating knots, which is the origin of the tables still in use.

The ether did not exist and the theory was abandoned by about 1890. The tables survived, the subject survived, and for eighty years it was a corner of topology with no application whatever.

Then in the 1980s it turned out that DNA is knotted, that the enzymes which unknot it can be studied by measuring which knots appear, and that the Jones polynomial has an interpretation in quantum field theory. A subject built for a wrong physics and pursued for its own sake for a century became applicable to two others.

That sequence is worth recording without drawing too large a moral from it. The useful thing to notice is not that pure mathematics eventually pays; it is that the tables Tait made were correct even though his reason for making them was not, because what he was tabulating was a mathematical object rather than a physical hypothesis.

What the picture cannot show #

Every figure here is a diagram, and a diagram is a projection. The essay’s whole subject is the loop in space, and no figure on this page shows one — the reader is looking at shadows with annotations.

The three moves are drawn as before-and-after pairs, and a move is a motion. What happens between the two frames is exactly the content, and a static figure has nowhere to put it. This is the same gap as the shear in Euclid’s proof, and it is worse here because the whole theorem is about motions.

The tricolouring figure shows one colouring of the trefoil and one of the unknot. What it cannot show is the argument’s essential step — that the property survives all three moves — because that is a claim about every diagram of every knot, and it is verified by three case analyses rather than by any picture.

Nor can any figure here show a knot that is the unknot in disguise. Those exist and are common: diagrams with dozens of crossings that untangle completely, some of which require the crossing number to increase before it can decrease. A drawing of one would be a drawing of an ordinary tangle, which is exactly why the problem is hard.

What tricolourability is really counting #

There is a reformulation that explains why the rule looks arbitrary and is not.

Label the three colours $0$, $1$ and $2$, and read them as residues on a dial of three. The crossing rule — all the same or all different — is exactly the condition that the two under-arcs and the over-arc satisfy

$$a + b \equiv 2c \pmod 3$$

at every crossing, where $c$ is the over-strand. All-same and all-different are precisely the solutions of that equation, and no other triple satisfies it.

So a tricolouring is a solution to a system of linear equations modulo three, one equation per crossing, one unknown per arc. Tricolourability is the statement that the system has a solution other than the constant ones — that is, that the corresponding matrix is singular over the integers modulo three.

That reformulation does three things at once. It explains why the rule survives the moves, since each move changes the system by operations that do not alter its solution space. It makes the invariant computable by linear algebra rather than by inspection. And it generalises immediately: replace three by any modulus $p$ and the same construction gives $p$-colourability, a whole family of invariants, of which the three-colour case is only the first and the easiest to draw.

The determinant of that system, taken over the integers instead of modulo anything, is the knot determinant — and it is the Alexander polynomial evaluated at $-1$. So the schoolroom colouring rule and the first serious polynomial invariant are the same object at two levels of resolution.

The ladder from here #

Rungs above this one: the proof that tricolourability survives each move, drawn move by move. Mirror images and chirality — the trefoil is not the same as its reflection, and showing it needs more than three colours. The knot group, and Wirtinger’s presentation read off a diagram. The Alexander polynomial from the crossing determinant. The Jones polynomial from the skein relation, drawn as a recursion on crossings. Seifert surfaces, where a knot bounds an orientable surface and the surface’s genus becomes an invariant — which connects this anchor directly to the Euler characteristic. Connected sums and prime knots. And the unknotting problem’s complexity.

The value of a short list #

The single most consequential thing in this essay is the number three.

Before Reidemeister, “can this be deformed into that” was a question about continuous motions in space, with no way to enumerate the possibilities and therefore no way to prove a negative. Afterwards it is a question about sequences of three local rewrites, and proving a negative becomes routine: find a quantity unchanged by each of three pictures, and compute it twice.

The pattern — replace an unbounded notion of sameness with a finite list of generating moves — recurs wherever equivalence is the hard part. It is why presentations of groups are useful, why rewriting systems are studied at all, and why the classification of surfaces can be stated. The list is never the interesting part and it is always the part that makes the rest possible.