# Minimal coloring number for $\mathbb{Z}$-colorable links II

## TL;DR

This paper proves that all non-splittable $ ext{Z}$-colorable links have a minimal coloring number of four and provides methods to achieve this coloring in specific link configurations.

## Contribution

It establishes that the minimal coloring number for non-splittable $ ext{Z}$-colorable links is always four and offers a technique to construct diagrams with this minimal coloring.

## Key findings

- Non-splittable $ ext{Z}$-colorable links have minimal coloring number four.
- Even parallels of links are $ ext{Z}$-colorable except when two parallels have non-zero linking number.
- A simple method is provided to obtain diagrams with minimal coloring number for even parallels.

## Abstract

The minimal coloring number of a $\mathbb{Z}$-colorable link is the minimal number of colors for non-trivial $\mathbb{Z}$-colorings on diagrams of the link. In this paper, we show that the minimal coloring number of any non-splittable $\mathbb{Z}$-colorable links is four. As an example, we consider the link obtained by replacing each component of the given link with several parallel strands, which we call a parallel of a link. We show that an even parallel of a link is $\mathbb{Z}$-colorable except for the case of 2 parallels with non-zero linking number. We then give a simple way to obtain a diagram which attains the minimal coloring number for such even parallels of links.