# Minimal coloring numbers on minimal diagrams of torus links

## TL;DR

This paper determines the minimal number of colors needed for non-trivial integer colorings of minimal diagrams of torus links, providing classifications of such colorings and those using only four colors through rack coloring methods.

## Contribution

It introduces a complete classification of minimal colorings and four-color colorings of torus links using rack colorings, advancing understanding of link colorings.

## Key findings

- Identifies minimal coloring numbers for torus links.
- Classifies all such colorings and four-color colorings.
- Uses rack colorings to achieve classifications.

## Abstract

We determine the minimal number of colors for non-trivial $\mathbb{Z}$-colorings on the standard minimal diagrams of $\mathbb{Z}$-colorable torus links. Also included are complete classifications of such $\mathbb{Z}$-colorings and of such $\mathbb{Z}$-colorings by only four colors, which are shown by using rack colorings on link diagrams.