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

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 [L. H. Kauffman and P. Lopes, Colorings beyond Fox: The other linear Alexander quandles, Linear Algebra Appl. 548 (2018) 221–258]. We express this lower bound in terms of the degree [Formula: see text] of the reduced Alexander polynomial of the knot. We show that it is exactly [Formula: see text] for L-space knots. Then we apply these results to torus knots and Pretzel knots [Formula: see text], [Formula: see text]. 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 than one component.