Fox $p$-Colorings as Fixed Points of Braid Representations

Abstract

Fox $p$-colorings of knots and links admit a linear-algebraic description in terms of braid representations. For each braid $\beta \in B_n$ we associate a representation $M:B_n\to \mathrm{GL}(n,{\mathbb{F}}_p)$ such that Fox $p$-colorings of the closed braid $\hat{\beta }$ correspond to fixed points of $M(\beta )$. As a consequence, $${\mathrm{Col}}_p(\hat{\beta })=p^{\dim \ker (M(\beta )-I)},$$ yielding an explicit formula for the number of Fox $p$-colorings of arbitrary knots and links. We apply the method to torus links and spiral links, obtaining explicit descriptions of their Fox 3-coloring spaces.