Equivalence Classes of Colorings

TL;DR

This paper introduces an equivalence relation on non-trivial m-colorings of links, showing that for prime moduli, the number of classes depends on the modulus and the coloring matrix's rank, providing a new topological invariant.

Contribution

It defines a novel equivalence relation on link colorings and analyzes the number of classes for prime moduli, linking it to the coloring matrix's rank.

Findings

Number of equivalence classes depends on modulus and matrix rank.

Equivalence classes are invariant under topological transformations.

For prime moduli, the classification is explicitly characterized.

Abstract

For any link and for any modulus $m$ we introduce an equivalence relation on the set of non-trivial m-colorings of the link (an m-coloring has values in Z/mZ). Given a diagram of the link, the equivalence class of a non-trivial m-coloring is formed by each assignment of colors to the arcs of the diagram that is obtained from the former coloring by a permutation of the colors in the arcs which preserves the coloring condition at each crossing. This requirement implies topological invariance of the equivalence classes. We show that for a prime modulus the number of equivalence classes depends on the modulus and on the rank of the coloring matrix (with respect to this modulus).