The Generalized Kauffman-Harary Conjecture is True

TL;DR

This paper proves a generalized version of the Kauffman-Harary conjecture, showing that for prime links with a given determinant, distinct arcs can be distinguished by an appropriate Fox coloring.

Contribution

It extends the Kauffman-Harary conjecture to all prime links with arbitrary determinants, confirming the conjecture's validity in a broader context.

Findings

The conjecture holds for all prime links with prime determinants.

Distinct arcs in reduced alternating diagrams can be distinguished by specific Fox colorings.

The proof generalizes previous results limited to prime determinants.

Abstract

For a reduced alternating diagram of a knot with a prime determinant $p,$ the Kauffman-Harary conjecture states that every non-trivial Fox $p$-coloring of the knot assigns different colors to its arcs. In this paper, we prove a generalization of the conjecture stated nineteen years ago by Asaeda, Przytycki, and Sikora: for every pair of distinct arcs in the reduced alternating diagram of a prime link with determinant $\delta,$ there exists a Fox $\delta$-coloring that distinguishes them.