Link invariants from finite categorical groups

TL;DR

This paper introduces a new class of link invariants derived from finite categorical groups and Reidemeister pairs, unifying and extending existing invariants like rack, quandle cohomology, and the Eisermann invariant.

Contribution

It defines a novel invariant framework using finite crossed modules and Reidemeister pairs, encompassing and surpassing previous invariants such as rack, quandle, and Eisermann invariants.

Findings

Includes all rack and quandle cohomology invariants

Contains the Eisermann invariant as a special case

Constructs a stronger invariant than Eisermann through a lifting class

Abstract

We define an invariant of tangles and framed tangles given a finite crossed module and a pair of functions, called a Reidemeister pair, satisfying natural properties. We give several examples of Reidemeister pairs derived from racks, quandles, rack and quandle cocycles, and central extensions of groups. We prove that our construction includes all rack and quandle cohomology (framed) link invariants, as well as the Eisermann invariant of knots. We construct a class of Reidemeister pairs which constitute a lifting of the Eisermann invariant, and show through an example that this class is strictly stronger than the Eisermann invariant itself.