An Improvement of the lower bound for the minimum number of link colorings by quandles

TL;DR

This paper enhances the lower bound for the minimum number of colors needed in linear Alexander quandle knot colorings, relating it to the Alexander polynomial degree and extending results to links.

Contribution

It introduces a refined lower bound based on the Alexander polynomial degree and extends the bound to links with multiple components.

Findings

Lower bound equals k+1 for L-space knots

Bound applies to torus and Pretzel knots

Bound can be attained for specific knots

Abstract

We improve the lower bound for the minimum number of colors for linear Alexander quandle colorings of a knot given in Theorem 1.2 of Colorings beyond Fox: The other linear Alexander quandles (Linear Algebra and its Applications, Vol. 548, 2018). We express this lower bound in terms of the degree k of the reduced Alexander polynomial of the considered knot. We show that it is exactly k + 1 for L-space knots. Then we apply these results to torus knots and Pretzel knots P(-2,3,2l + 1), l>=0. We note that this lower bound can be attained for some particular knots. Furthermore, we show that Theorem 1.2 quoted above can be extended to links with more that one component.