Quandle Coloring Quivers

TL;DR

This paper introduces a quiver structure on quandle colorings of knots and links, providing a categorification that yields new invariants and enhances existing counting invariants.

Contribution

It presents a novel quiver-based categorification of quandle colorings, leading to new, explicit enhancements of knot and link invariants.

Findings

The quiver structure is invariant under Reidemeister moves.

New enhancements of the quandle counting invariant are derived.

Explicit examples demonstrate the effectiveness of the enhancements.

Abstract

We consider a quiver structure on the set of quandle colorings of an oriented knot or link diagram. This structure contains a wealth of knot and link invariants and provides a categorification of the quandle counting invariant in the most literal sense, i.e., giving the set of quandle colorings the structure of a small category which is unchanged by Reidemeister moves. We derive some new enhancements of the counting invariant from this quiver structure and show that the enhancements are proper with explicit examples.