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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 β∈ Bn we associate a representation M : Bn → GL(n,𝔽p) such that Fox p-colorings of the closed braid β̂ correspond to fixed points of M(β). As a consequence, Colp(β̂) = pdim ker(M(β)−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
@misc{goh2026pcolorings,
title = {{Fox $p$-Colorings as Fixed Points of Braid Representations}},
author = {Neungjoo Goh and Suah Jeong and Seungbin Oh and Hyungkee Yoo},
howpublished = {arXiv},
year = {2026},
month = aug,
doi = {10.48550/arxiv.2608.29046},
eprint = {2608.29046},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.29046},
}
RIS
TY - UNPB
TI - Fox p-Colorings as Fixed Points of Braid Representations
AU - Neungjoo Goh
AU - Suah Jeong
AU - Seungbin Oh
AU - Hyungkee Yoo
PY - 2026
DA - 2026/08/29
PB - arXiv
DO - 10.48550/arxiv.2608.29046
UR - https://arxiv.org/abs/2608.29046
AB - Fox p-colorings of knots and links admit a linear-algebraic description in terms of braid representations. For each braid β∈ B_n we associate a representation M : B_n → GL(n,𝔽_p) such that Fox p-colorings of the closed braid β̂ correspond to fixed points of M(β). As a consequence, Col_p(β̂) = p^dim ker(M(β)−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.
ER -