Colours that count more than three

Colouring the arcs of a knot diagram with three colours, so that the three arcs at every crossing are either all alike or all different, settles one question completely: the trefoil admits such a colouring and a plain loop does not, so the trefoil is knotted. That is a real theorem obtained by looking at a picture, and it is the first honest proof most people meet that any knot is knotted at all.

It is also nearly the last thing three colours can do.

The figure-eight knot is the fourth entry in every knot table and the first genuine problem. It has four crossings, it cannot be untied, and under three colours it behaves exactly as the unknot does: every colouring uses one colour. So three-colouring reports the same verdict for a knot and for a loop of string, which is to say it reports nothing.

What the rule actually says #

The three-colour rule is usually stated as all the same or all different, which is a condition on a picture. Rewritten as arithmetic it becomes something that can be generalised.

Number the colours 0, 1 and 2. At a crossing, one arc passes over and two arcs end underneath. The rule all the same or all different is exactly the statement that

$$2c_{\text{over}} \equiv c_{\text{under}_1} + c_{\text{under}_2} \pmod 3,$$

which can be checked case by case: three equal colours satisfy it, and so does any assignment of the three distinct colours, while two of one and one of another never does.

Written that way there is nothing special about three. Take p colours numbered 0 to p − 1, ask for twice the over-strand to equal the sum of the two under-strands modulo p, and there is a rule for every p. This is Fox’s p-colouring, and the three-colour case is the one that happens to be drawable with crayons.

Why it survives the three moves #

An invariant is only worth having if it is a property of the knot rather than of the picture, and diagrams of one knot differ from each other. Reidemeister’s theorem makes that manageable: two diagrams show the same knot exactly when a sequence of three local moves turns one into the other, so an invariant needs to survive three checks and no others.

Each check is a short piece of arithmetic on the rule above.

Twisting a strand creates a crossing where the over-arc and both under-arcs are the same arc. The condition becomes 2c ≡ c + c, which is true for every colour, so a colouring of the untwisted diagram extends to the twisted one in exactly one way.

Pulling two strands apart removes two crossings sharing a middle arc. Solving the two conditions shows the middle arc’s colour is forced by the two outer ones, so colourings on either side of the move correspond one to one.

Sliding a strand across a crossing is the only case with any work in it. Three arcs come in and three go out, and writing the three conditions on each side gives two systems of linear equations whose solutions match up — a computation of four lines, and the only place where the assumption that p is odd is used.

So the number of p-colourings is a property of the knot. Two diagrams with different counts are certainly different knots, and that is the whole use of the thing.

Five colours, where three had nothing to say #

Every knot admits the p colourings that give every arc the same colour, since the rule then reads 2c ≡ 2c. Those are the trivial ones and they carry no information; what matters is whether any colouring uses more than one colour.

For the figure-eight and three colours, none does. For the figure-eight and five colours, twenty do — and since an unknotted loop is a single arc, and a single arc can only ever be given one colour, the figure-eight is not the unknot. Three colours could not see that and five colours can.

The counts in the first figure are all of the form p or p × p, and that is not a coincidence. The colourings form the solution set of a system of linear equations over the integers modulo p, so their number is a power of p, and the power is the number of free choices left after the equations are imposed. One free choice is the constant colour everybody has; a second means a genuine colouring exists.

Arithmetic modulo a prime is what makes that argument work, and it is the same reason primes appear wherever a system has to be solved without fractions — in error-correcting codes, in finite fields, and here. Modulo a composite the equations still make sense and the solution set is still a group, but it need not be a vector space, the count need not be a power, and the neat statement about free choices fails. The rule is usually stated for odd primes for that reason and not because the geometry cares.

One number that predicts them all #

Trying every assignment answers the question for one knot and one p. It does not say which p are worth trying, and there is a number that does.

Write the crossing conditions as a matrix: one row per crossing, with 2 in the column of the over-arc and −1 in each under-arc’s column. Delete any one row and any one column, and take the determinant of what is left. Its absolute value is the same whichever row and column were deleted, and it is the determinant of the knot.

The deletion is not a trick to make the matrix square. Every row of the full matrix has entries adding to zero — 2 minus 1 minus 1 — so the columns are dependent and the full determinant is zero whatever the knot. Deleting a column removes exactly that dependence, and deleting a row keeps the thing square; what is left is the object that carries the information, and the fact that the answer does not depend on which row and column were removed is the first thing a careful treatment proves.

The trefoil’s is 3. The figure-eight’s is 5. And the rule is exactly what the figure’s last column suggests: a knot admits a colouring in more than one colour with p colours precisely when p divides its determinant.

That claim is made and checked twice over in the first figure. The counts come from trying all 3³, 5⁴ and 7⁴ assignments and keeping the ones that satisfy every crossing; the determinant comes from elimination on the matrix, done in whole numbers so that no rounding can creep in. The two are required to agree for every knot and every number of colours in the table, and the figure refuses to be drawn if they ever disagree.

The determinant is a considerably cheaper thing to compute than a colouring count. Elimination on an n by n matrix costs about n³ operations; trying every assignment costs p to the power n. For a knot with fifteen crossings and seven colours those are a few thousand operations against about five thousand million.

Where it stops working #

Three facts mark the boundary, and each of them is the reason the subject went on to build something better.

The determinant is the whole story. Every p-colouring count is determined by one integer, so the entire family of invariants — three colours, five, seven, eleven — carries no more information than that integer does. Two knots with the same determinant are indistinguishable by any of them, and plenty of different knots share a determinant. The invariant is a single number wearing several hats.

It cannot detect a mirror image. Reflecting a diagram swaps every over-strand for an under-strand, and the crossing conditions come out identical. So the left-handed and right-handed trefoils have exactly the same colouring counts, and they are genuinely different knots — no deformation turns one into the other. An invariant blind to handedness will never prove that.

A determinant of 1 proves nothing. The unknot’s determinant is 1, and so is that of several genuinely knotted knots. For those, no p at all gives a non-constant colouring, and the invariant reports the same verdict it reports for a loop of string.

That third failure is the standing shape of the subject and it is worth recognising as such. The Euler characteristic does the same thing to surfaces — a torus and a Klein bottle both give zero — and the count of loop classes does it to a Möbius band and an ordinary ring. Every invariant is a summary, every summary loses something, and the useful question about a new one is never whether it loses information but which information it loses.

The older invariant, and what changed #

The three-colour rule was published by Ralph Fox in the 1950s as a way of explaining knot groups to undergraduates, and the general p version came from the same lectures. Both are shadows of an older object: the Alexander polynomial, from 1928, whose value at −1 is the determinant used above.

That relationship is worth stating in the direction that makes it useful. The polynomial is a whole function’s worth of information and the determinant is one of its values, so the polynomial separates strictly more knots — but the determinant can be computed by hand from a drawing in a minute, and the colouring version can be checked by a reader with three pencils. The cheaper invariant survives because the expensive one is not always needed.

What it costs #

The counts in these figures are taken by brute force over every assignment of colours to arcs, which is p to the power of the number of arcs. For the figure-eight and seven colours that is 7⁴ = 2401 assignments, each checked against four conditions — nothing at all. For a knot with twenty crossings it would be 7²⁰, which is a number with seventeen digits, and the brute force stops being an option.

The linear-algebra route does not care. The colouring count is p raised to the nullity of the crossing matrix modulo p, and the nullity comes from elimination in a few hundred operations. So the honest description of the computation is: brute force is used here because the knots are small and because a count obtained by looking at every case is better evidence than a count obtained by an argument, and the argument is what would be used at any size that mattered.

There is a third cost worth naming. Deciding whether two diagrams show the same knot is decidable but expensive; deciding whether a diagram shows the unknot is decidable too, and the best known bounds are far from comfortable. An invariant that answers no quickly is valuable precisely because the general question is so hard.

The asymmetry between the two answers is worth dwelling on, because it recurs whenever a hard question is attacked by a cheap test. A non-constant colouring is a certificate: it can be handed to a sceptic, who checks each crossing in turn and is finished. The absence of one is not a certificate of anything, since some other invariant may still separate the knots. That is exactly the shape of a search that can only report success, and it is why a table of invariant values is read down the columns for disagreements rather than across the rows for agreements.

What the picture cannot show #

A colouring is drawn on one diagram, and the theorem is about all diagrams of the knot at once. The invariance argument is what closes that gap, and the argument is arithmetic — three cases, each a line of algebra — which no drawing contains.

Nor can the figures show the failure that matters most. When a p-colouring count agrees for two knots, the picture looks exactly as it does when two diagrams show the same knot, and there is no way to tell from the drawing whether the invariant has proved something or merely failed to distinguish. Only disagreement is evidence, and a page of agreement is a page of silence.

The diagrams themselves are also a choice. A knot has infinitely many diagrams; these are drawn at the minimum crossing number because that is the tidiest one, and nothing in the figures shows that no tidier one exists.

There is one more thing missing from every drawing on this page, and it is the third dimension. A knot is a closed curve in space and a diagram is its shadow with the over-strand marked, so each picture has thrown away everything except one bit of information per crossing. That the thrown-away information can be recovered — that the shadow plus the crossing data determines the knot up to deformation — is a theorem, and it is what licenses the whole practice of arguing about knots by drawing them flat.

The ladder from here #

Rungs above: the Alexander polynomial, with the determinant as its value at −1. The Jones polynomial, computed from the bracket by summing over every way of smoothing the crossings, which separates knots the Alexander polynomial cannot and detects handedness. The knot group — the fundamental group of the complement — which is a complete invariant and impossible to compare. The Seifert surface and the genus it minimises. Unknotting number, and the reason nobody knows it for some small knots. Skein relations, which compute a polynomial by changing one crossing at a time. And the counting of knots by crossing number, where the tables were built by hand for a century and by computer since.

The shape of the idea #

An invariant is a machine that takes a drawing and returns a number, together with a proof that the number is unmoved by the ways the drawing can change without the object changing. Everything else about it is negotiable.

The colouring rule is a good place to see that clearly, because the machine is so nearly trivial: colour the arcs, check three conditions, count. The whole weight of the thing sits in the invariance proof — three moves, three short calculations — and once that is in hand, a child’s colouring exercise settles a question about three-dimensional space that no amount of pulling on string could settle.

What the essay above adds to its own first rung is the observation that the exercise has a dial on it. Turning three into five turns a blind invariant into a sighted one, and turning it further eventually runs out of information altogether, because the whole dial was always reporting a single integer.