Fox n-coloring

Definition

Let L be a link, and let ${\displaystyle \pi }$ be the fundamental group of its complement. A representation ${\displaystyle \rho }$ of ${\displaystyle \pi }$ onto ${\displaystyle D_{2n}}$ the dihedral group of order 2n is called a Fox n-coloring (or simply an n-coloring) of L. A link L which admits such a representation is said to be n-colorable, and ${\displaystyle \rho }$ is called an n-coloring of L. Such representations of groups of links had been considered in the context of covering spaces since Reidemeister in 1929. [Actually, Reidemeister fully explained all this in 1926, on page 18 of "Knoten und Gruppen" in Hamburger Abhandlungen 5. The name "Fox coloring" was given to it much later by mathematicians who probably couldn't read German.] Fox's preferred term for so-called "Fox 3-coloring" was "property L"; see Exercise 6 on page 92 of his book "Introduction to Knot Theory" (1963).

The group of a link is generated by paths from a basepoint in ${\displaystyle S^3}$ to the boundary of a tubular neighbourhood of the link, around a meridian of the tubular neighbourhood, and back to the basepoint. By surjectivity of the representation these generators must map to reflections of a regular n-gon. Such reflections correspond to elements ${\displaystyle t{s}^i}$ of the dihedral group, where t is a reflection and s is a generating (${\displaystyle 2\pi /n}$) rotation of the n-gon. The generators of the group of a link given above are in bijective correspondence with arcs of a link diagram, and if a generator maps to ${\displaystyle t{s}^i\in D_{2p}}$ we color the corresponding arc ${\displaystyle i\in \mathbb{Z}/p\mathbb{Z}}$. This is called a Fox n-coloring of the link diagram, and it satisfies the following properties:

• At least two colors are used (by surjectivity of ${\displaystyle \rho }$).
• Around a crossing, the average of the colors of the undercrossing arcs equals the color of the overcrossing arc (because ${\displaystyle \rho }$ is a representation of the group of the link).

A n-colored link yields a 3-manifold M by taking the (irregular) dihedral covering of the 3-sphere branched over L with monodromy given by ${\displaystyle \rho }$. By a theorem of Montesinos and Hilden, any closed oriented 3-manifold may be obtained this way for some knot K and ${\displaystyle \rho }$ some tricoloring of K. This is no longer true when n is greater than three.