Minimal coloring number for Z-colorable links

TL;DR

This paper investigates the minimal number of colors needed for non-trivial Z-colorings of Z-colorable links, providing conditions under which certain links achieve the lowest possible coloring number.

Contribution

It introduces sufficient conditions for non-splittable Z-colorable links to attain the minimal coloring number, advancing understanding of link colorings.

Findings

Identifies conditions for minimal coloring number in Z-colorable links

Provides criteria for non-splittable Z-colorable links to have minimal coloring number

Enhances classification of links based on coloring properties

Abstract

For a link with zero determinants, a Z-coloring is defined as a generalization of Fox coloring. We call a link having a diagram which admits a non-trivial Z-coloring a Z-colorable link. The minimal coloring number of a Z-colorable link is the minimal number of colors for non-trivial Z-colorings on diagrams of the link. We give sufficient conditions for non-splittable Z-colorable links to have the least minimal coloring number.