Minimal coloring number on minimal diagrams for $\mathbb{Z}$-colorable links

TL;DR

This paper investigates the minimal number of colors needed for non-trivial $bZ$-colorings on minimal diagrams of $bZ$-colorable links, showing both unbounded minimal color counts and cases with only four colors.

Contribution

It demonstrates that minimal diagrams can require arbitrarily many colors for $bZ$-colorings, yet some torus links have minimal diagrams with only four colors.

Findings

Existence of minimal diagrams with arbitrarily many colors.

Certain torus links admit minimal diagrams with only four colors.

The minimal coloring number can vary significantly among $bZ$-colorable links.

Abstract

It was shown that any $\mathbb{Z}$-colorable link has a diagram which admits a non-trivial $\mathbb{Z}$-coloring with at most four colors. In this paper, we consider minimal numbers of colors for non-trivial $\mathbb{Z}$-colorings on minimal diagrams of $\mathbb{Z}$-colorable links. We show, for any positive integer $N$, there exists a minimal diagram of a $\mathbb{Z}$-colorable link such that any $\mathbb{Z}$-coloring on the diagram has at least $N$ colors. On the other hand, it is shown that certain $\mathbb{Z}$-colorable torus links have minimal diagrams admitting $\mathbb{Z}$-colorings with only four colors.