Three dimensions of knot coloring

TL;DR

This survey explores three types of knot colorings—Fox, Dehn, and Alexander-Briggs—detailing their rules, relationships, and how they assign labels to different knot components using modular arithmetic.

Contribution

It provides a comprehensive overview of the three main types of knot colorings and clarifies their interrelations and underlying rules.

Findings

Explains the rules for each coloring type.

Details the relationships among the three colorings.

Highlights the role of modular arithmetic in knot colorings.

Abstract

This survey article discusses three aspects of knot colorings. Fox colorings are assignments of labels to arcs, Dehn colorings are assignments of labels to regions, and Alexander-Briggs colorings assign labels to vertices. The labels are found among the integers modulo n. The choice of n depends upon the knot. Each type of coloring rules has an associated rule that must hold at each crossing. For the Alexander Briggs colorings, the rules hold around regions. The relationships among the colorings is explained.