Categorifying Coloring Numbers

TL;DR

This paper develops a categorification of coloring numbers for knots, links, and tangles, extending algebraic invariants to tangles and deriving matrix representations through decategorification.

Contribution

It introduces a categorification of coloring numbers, enabling their extension to tangles and resulting in matrix representations upon decategorification.

Findings

Categorification extends coloring invariants to tangles.

Decategorification yields matrix representations of tangle categories.

The approach unifies algebraic and topological perspectives on knot invariants.

Abstract

Coloring numbers are one of the simplest combinatorial invariants of knots and links to describe. And with Joyce's introduction of quandles, we can understand them more algebraically. But can we extend these invariants to tangles -- knots and links with free ends? Indeed we can, once we categorify. Starting from the definition of coloring numbers, we will categorify them and establish this extension to tangles. Then, decategorifying will leave us with matrix representations of the monoidal category of tangles.