Link invariants from finite categorical groups and braided crossed modules

TL;DR

This paper introduces a new class of link invariants based on finite crossed modules and Reidemeister pairs, unifying and extending existing invariants like rack, quandle cohomology, and the Eisermann invariant.

Contribution

It defines a broad framework for link invariants using finite crossed modules and Reidemeister pairs, encompassing many known invariants and introducing new liftings.

Findings

Includes all rack and quandle (framed) link invariants.

Provides a lifting of the Eisermann invariant using braided crossed modules.

Establishes a unified approach to various link invariants.

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, 2-crossed modules and braided crossed modules. We prove that our construction includes all rack and quandle cohomology (framed) link invariants, as well as the Eisermann invariant of knots, for which we also find a lifting by using braided crossed modules.