A polynomial behind the colourings

Colourings modulo a prime settle a great deal. A knot can be coloured with $p$ colours in more than one colour exactly when $p$ divides a single whole number, the determinant, and that number is read off the diagram by elimination. The trefoil’s is 3, the figure-eight’s is 5, and between them they account for everything the colourings of those two knots can say.

Which is also the limitation. Two knots with the same determinant have the same colourings for every prime, and no colouring can distinguish them. The figure-eight knot has four crossings and the cinquefoil — the knot that winds twice round a torus while wobbling five times — has five. They are different knots. Both have determinant 5.

Twenty-five and twenty-five. With seven colours both counts are seven, with three both are three, and so on for every prime. The colourings see a single number, the number is the same, and the two knots are invisible to each other. Something finer is needed, and it was found in 1928 by James Alexander, a decade before anybody thought of colouring a knot at all.

The rule with a variable in it #

The colouring rule at a crossing says $2x_k = x_i + x_j$, where $x_k$ is the colour of the over-arc and $x_i$, $x_j$ those of the two under-arcs. Written as a row of a matrix it is $2$ in the over-arc’s column and $-1$ in each under-arc’s.

Alexander’s matrix has the same shape with a variable $t$ in it. Orient the knot, so that at each crossing one under-arc is arriving and the other leaving. The row for the crossing puts $1 - t$ in the over-arc’s column, and $t$ and $-1$ in the columns of the two under-arcs, in an order fixed by whether the crossing is right-handed or left-handed. Set $t = -1$ and the row becomes $2$, $-1$, $-1$: the colouring rule is Alexander’s rule at one value of the variable.

Every row adds up to nought when $t = 1$, since $(1 - t) + t - 1 = 0$, so the full matrix is singular and the interesting quantity is a minor: strike out one row and one column and take the determinant of what is left. The result is a polynomial in $t$, and it is well defined only up to multiplication by $\pm t^k$ — striking a different row or column, or relabelling, changes it by such a factor and by nothing else. Normalised so that it reads the same backwards and takes the value 1 at $t = 1$, it is the Alexander polynomial $\Delta(t)$.

For the trefoil it is $t - 1 + t^{-1}$. For the figure-eight it is $-t + 3 - t^{-1}$.

The same determinant, two polynomials #

The figure-eight gives $-t + 3 - t^{-1}$; the cinquefoil gives $t^2 - t + 1 - t^{-1} + t^{-2}$. At $t = -1$ both are 5 — that is the determinant, and it had to agree. Everywhere else they differ. The polynomials are invariants of the knots, so the knots are different, and the invariant that separated them is the one the colourings were a single value of.

The difference can be turned back into colourings, and doing so shows exactly what the old colourings were missing. Instead of setting $t = -1$, choose a prime $p$ and any other non-zero value $a$ modulo $p$, and ask for labels of the arcs by numbers modulo $p$ satisfying $(1 - a)x_k + a x_i - x_j = 0$ at every crossing. The ordinary colourings are the case $a = -1$. Labellings beyond the constant ones exist exactly when $\Delta(a) \equiv 0 \pmod p$.

Take $p = 11$ and $a = 2$. The cinquefoil’s polynomial, cleared of negative powers, is $t^4 - t^3 + t^2 - t + 1$, which at $t = 2$ is $16 - 8 + 4 - 2 + 1 = 11$ — nought modulo 11, so the cinquefoil has non-constant labellings of this twisted kind. The figure-eight’s is $-t^2 + 3t - 1$, which at $t = 2$ is 1, so it has none. A colouring rule with the two replaced by a different number separates the two knots, and the polynomial is the bookkeeping that says which rules will.

The table also shows a pattern in the torus knots that is worth pausing on. The trefoil, the cinquefoil and the seven-crossing torus knot have polynomials $t - 1 + t^{-1}$, then two more terms, then two more, with the signs alternating and every coefficient $\pm 1$. They are the knots that wind twice round a torus, and their polynomials are

$$\frac{(t^{2q} - 1)(t - 1)}{(t^2 - 1)(t^q - 1)}$$

up to a power of $t$, a quotient of cyclotomic factors — which is why their roots are all roots of unity. The figure-eight’s polynomial has a coefficient of 3 and roots that are not on the unit circle at all, which is a first sign that it is a different kind of knot.

Why the matrix does not care which picture it came from #

A knot has infinitely many diagrams, and the matrix is built from one of them. For the polynomial to be a property of the knot, it must survive each of Reidemeister’s three moves, since every change of diagram is a sequence of those.

Each move changes the matrix in a controlled way. A twist that adds a crossing adds an arc and a crossing, so it adds a row and a column; the new row can be used to eliminate the new column, leaving the old matrix bordered by a unit, which changes the minor by $\pm t$. The second move adds two crossings and two arcs, and the four new entries eliminate in pairs. The third move rearranges three crossings without changing their count, and its rows are related by a sequence of row operations that do not change a determinant at all. In every case the minor changes by at most a factor of $\pm t^k$, which the normalisation removes.

That is the same argument the colourings rest on, carried out with a variable instead of a number. The colouring argument is the special case $t = -1$, and the reason the colourings survive the moves is that the polynomial does.

What the variable means #

The matrix looks like a trick — a coefficient replaced by a symbol, and a theorem that the symbol is harmless — and it is not one. The variable has a geometric meaning, and it connects the polynomial to a construction drawn elsewhere.

The complement of a knot in space — everything except the string — has a fundamental group, and every loop in the complement has a linking number with the knot: how many times it winds round the string, with sign. That number defines a map from the group onto the integers, and every such map gives a covering space whose deck transformations are the integers: the infinite cyclic cover of the complement, an infinite stack of copies of the complement cut open along a surface and glued end to end, like a spiral staircase built out of the space round a knot.

The variable $t$ is the deck transformation — the step one storey up the staircase. The first homology of the cover is a module over polynomials in $t$, because $t$ acts on it by moving cycles up a storey, and Alexander’s matrix is a presentation of that module: arcs are generators, crossings are relations, and $t$ appears exactly where a relation passes from one storey to the next. The polynomial is the module’s order. So $\Delta(t)$ records how the homology of the staircase behaves under climbing it, and the determinant is what that structure records about a two-storey quotient — the branched double cover, whose homology has order $|\Delta(-1)|$.

That is also why colourings modulo $p$ count what they count. A $p$-colouring is a map from the double branched cover’s homology to the integers modulo $p$, and there are such maps beyond the trivial ones exactly when $p$ divides the order of that homology, which is the determinant.

The polynomial on a circle #

A polynomial that reads the same backwards takes real values on the unit circle: at $t = e^{i\theta}$ the terms $t^k$ and $t^{-k}$ pair into $2\cos k\theta$.

The picture puts the whole story into one frame. The colourings see only the right-hand end of each curve, and there the figure-eight and the cinquefoil coincide. The polynomial sees the whole curve, and the cinquefoil dips below nought and turns back while the figure-eight climbs steadily. Every point of the curve is an invariant, and two knots with the same curve are indistinguishable to the polynomial.

The places where a curve crosses nought are roots of $\Delta$ on the unit circle, and they are meaningful rather than incidental. The trefoil’s curve is $2\cos\theta - 1$, which vanishes at $\theta = \pi/3$ — a sixth root of unity, as the cyclotomic formula predicts. Roots of this kind mark where a finer family of invariants, the Tristram–Levine signatures, is allowed to jump, and the figure-eight’s curve, which never reaches nought, is why all of its signatures vanish.

Changing one crossing at a time #

Matrices are one way to compute the polynomial. John Conway found another in 1969, and it turns the invariant into a recursion on pictures.

Take a diagram and fix one crossing. There are three diagrams that agree everywhere else: $L_+$, where that crossing is right-handed; $L_-$, where it is left-handed; and $L_0$, where it is removed by the smoothing that respects the orientation — the two strands cut and reconnected so that they no longer cross. Normalised in Conway’s way, the polynomials of the three satisfy

$$\Delta(L_+) - \Delta(L_-) = \big(t^{1/2} - t^{-1/2}\big)\,\Delta(L_0).$$

The relation is local: it is a statement about one crossing and says nothing about the rest of the diagram. That makes it an algorithm. Changing a crossing of any knot diagram can eventually unknot it, and smoothing a crossing reduces the number of crossings, so repeated use of the relation writes any knot’s polynomial in terms of unknots and unlinks, whose polynomials are 1 and 0.

The trefoil takes two steps. Change one of its crossings and it becomes the unknot, with polynomial 1; smooth that crossing instead and it becomes the Hopf link — two circles linked once. Change a crossing of the Hopf link and it becomes two unlinked circles, with polynomial 0; smooth it and it becomes a single unknot, polynomial 1. So the Hopf link’s polynomial is $t^{1/2} - t^{-1/2}$, and the trefoil’s is $1 + (t^{1/2} - t^{-1/2})^2 = t - 1 + t^{-1}$, which is the matrix’s answer.

Conway’s version is the one that generalised. The Jones polynomial satisfies a relation of exactly the same shape with different coefficients, and it was the existence of such a relation, not its geometric meaning, that made it computable before anybody knew what it meant.

Knots added together, polynomials multiplied #

Two knots can be combined by cutting each open and joining the loose ends — the connected sum, the knot a piece of string makes when two separate knots are tied in it one after the other. The complement of the sum is assembled from the two complements, the staircases stack in the same way, and the polynomial of the sum is the product of the polynomials.

So a string with two trefoils tied in it has polynomial $(t - 1 + t^{-1})^2$. It does not matter whether the two trefoils have the same handedness — the granny knot — or opposite handedness — the square knot — because the polynomial cannot see handedness, and the granny and square knots, which are different, have the same polynomial.

The multiplication has a consequence worth noticing. The span of a product is the sum of the spans, so the span of the polynomial adds under connected sum. The trefoil’s span is two, the double trefoil’s is four, and no amount of tying can make the span shrink. A quantity that adds when knots are combined and is nought for the unknot is a measure of complexity, and the span turns out to bound a geometric one: half of it is at most the smallest number of handles on a surface whose edge is the knot, which is the subject of the surface a knot bounds.

What the polynomial cannot see #

The polynomial is far stronger than the determinant, and it has three blind spots that matter.

It does not see mirror images. The trefoil and its reflection are different knots — no motion in space turns one into the other — and they have the same polynomial, because reflecting a diagram changes every crossing’s handedness, which swaps the positions of $t$ and $-1$ in every row and replaces $\Delta(t)$ by $\Delta(t^{-1})$, which for a symmetric polynomial is the same thing. Telling the trefoil from its mirror image had to wait for the Jones polynomial in 1984.

It does not see every knot. There are non-trivial knots whose Alexander polynomial is 1, the same as the unknot’s. The Kinoshita–Terasaka knot and the Conway knot, each with eleven crossings, are the first examples, and there are infinitely many. For such a knot the matrix minor is exactly the one an unknotted loop would give, and the polynomial reports nothing.

It does not separate every pair. Different knots can share a polynomial, and there are pairs of knots with the same polynomial, the same determinant and the same colourings that are nonetheless distinct. The Kinoshita–Terasaka and Conway knots are such a pair — mutants of each other, related by cutting out a tangle and turning it over — and the Alexander polynomial is provably unable to tell mutants apart.

What the drawn diagrams leave out #

The matrices here are computed from one diagram of each knot, and the invariance argument is described rather than drawn. Nothing in the figures shows a Reidemeister move being performed and the minor changing by a power of $t$; the claim that the polynomial belongs to the knot rests on that argument, and the figures only compute its value.

The covering space is not drawn at all. The infinite staircase built from the complement of a knot is a three-dimensional object with infinitely many storeys, and the polynomial is a fact about its homology — a quantity whose meaning is invisible in any diagram of the knot, and which the matrix computes without ever constructing the space it describes.

And the knots drawn are all alternating torus knots and the figure-eight, the simplest cases, where the polynomial behaves best. The knots it fails on have eleven crossings or more and cannot be drawn legibly at this size, so the blind spots above are stated rather than shown.

Still open: whether a later polynomial sees the unknot #

The Alexander polynomial fails to detect the unknot, and that failure is completely understood: its blind knots are known in infinite families. The Jones polynomial, discovered in 1984 by a route through operator algebras that had nothing to do with covering spaces, is stronger in every direction tested — it separates mirror images, and it distinguishes many pairs the Alexander polynomial cannot.

Whether the Jones polynomial detects the unknot is not known. No non-trivial knot with Jones polynomial 1 has ever been found, and computer searches have checked every knot with up to twenty-four crossings. Non-trivial links with the same Jones polynomial as an unlinked pair of circles do exist, which is part of why the question for knots is taken seriously as open rather than as a formality. A negative answer would need an example nobody has found; a positive one would need an understanding of the Jones polynomial’s geometric meaning at the level Alexander’s already has, and that understanding has been sought for forty years.

A number that was a value all along #

The determinant looked, from inside the colourings, like a number with a meaning of its own — the thing a prime has to divide. It was a single value of a function, taken at the one point where the function’s variable is invisible. Replace the two in the colouring rule by $1 - t$ and the $-1$’s by $t$ and $-1$, and the number opens into a polynomial that carries the structure of the whole infinite staircase round the knot.

That is a common shape in this subject, and in the determinant’s own theory: a count that looks final turns out to be one evaluation of a richer object. The figure-eight knot and the cinquefoil were never the same to the polynomial. They were only the same at $t = -1$, which is where the colourings happened to be looking.