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Given a knot $K$, a Fox $n$-coloring is a map $\pi_1(S^3-K)\to D_{2n}$ such that a meridian is sent to some reflection (this excludes the case of the image being contained in $C_n\subset D_{2n}$). Equivalently, a Fox $n$-coloring of a knot diagram is a labeling of the arcs with elements of $\mathbb{Z}/n\mathbb{Z}$ with the property that $2b\equiv a+c\pmod{n}$ when $b$ is the label of the overstrand and $a,c$ are the labels of the arcs of the understrand. (The equivalence comes from sending a loop around the arc labeled $a$, a Wirtinger generator, to the element $\sigma^a\tau\in D_{2n}$.)

Crossing with arc labels

A trivial coloring is one where all the arcs are given the same color. There are $n$ trivial colorings.

Question: For every knot, is there some $n$ such that the knot has a nontrivial Fox $n$-coloring?


Idea 1. Each crossing corresponds to an equation $a+c-2b\equiv 0\pmod{n}$. Construct a square matrix $A$ whose rows represent these equations, and show that the determinant of $A$ when restricted to the subspace orthogonal to $(1,1,\dots,1)$ is not $\pm 1$. Then by letting $n$ equal this determinant, there must be a nontrivial solution within this subspace, and therefore a nontrivial Fox $n$-coloring.

Idea 2. Put the knot into braid form and let $\sigma\in B_k$ be the corresponding braid. We can think of this as an operator $\mathbb{Z}^k\to\mathbb{Z}^k$ by observing the labeling of the arcs at the top of the crossing is a linear function of the labels at the bottom of the crossing. The right-hand crossing has the matrix $\begin{pmatrix}2&-1\\1&0\end{pmatrix}$ when arcs are read left-to-right. If $A$ is the matrix of $\sigma$, then the problem is whether $A-I_k$ modulo some $n>1$ has nullspace vector which is not a multiple of $(1,1,\dots,1)$. (Or, whether $A$ has an eigenvector with an integer eigenvalue that is $1$ modulo some $n$.)


While formulating this question, I ran into a paper by Kauffman and Lopes which cites Crowell and Fox's Introduction to Knot Theory as giving a partial solution to the problem (though I couldn't find the referent in the book!)

Idea 1 ends up working by taking an $(m-1)\times(m-1)$ minor of the matrix, where $m$ is the number of crossings. (This corresponds to setting a particular arc color to $0$.) The determinant of this minor is an odd integer (because it corresponds to the Alexander polynomial evaluated at $-1$). If the determinant is not $\pm 1$, then $n$ can be the absolute value of the determinant. Unfortunately for finding Fox $n$-colorings, there are knots where this determinant is $\pm 1$.

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If the knot has a nontrivial $n$-coloring, then the $(m-1)\times (m-1)$ minor has a nontrivial nullspace vector, so its determinant is divisible by $n$. It follows that if the determinant is $\pm 1$, there is only the trivial coloring.

In particular, knot $10_{124}$ has $\lvert \Delta(-1)\rvert=1$, and one can check it only has a trivial coloring no matter the modulus.

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