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

Neungjoo Goh; Suah Jeong; Seungbin Oh; Hyungkee Yoo

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arXiv · 2608.29046

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

Neungjoo Goh·Suah Jeong·Seungbin Oh·Hyungkee Yoo

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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 $\widehat{\beta}$ correspond to fixed points of $M(\beta)$. As a consequence, $$ \mathrm{Col}_p(\widehat{\beta}) = p^{\dim \ker(M(\beta)-I)}, $$ 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.

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BibTeX RIS

Neungjoo Goh, Suah Jeong, Seungbin Oh, Hyungkee Yoo. 2026-08-29. Fox $p$-Colorings as Fixed Points of Braid Representations. https://arxiv.org/abs/2608.29046

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