Homology, Homotopy and Applications

Volume 17 (2015)

Number 2

Link invariants from finite categorical groups

Pages: 205 – 233

DOI: https://dx.doi.org/10.4310/HHA.2015.v17.n2.a11


João Faria Martins (Departamento de Matemática, Faculdade de Ciências e Tecnologia, Universidade Nova de Lisboa, Quinta da Torre, Caparica, Portugal)

Roger Picken (Center for Mathematical Analysis, Geometry and Dynamical Systems, Department of Mathematics, Instituto Superior Técnico, Universidade de Lisboa, Portugal)


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.


knot invariant, tangle, peripheral system, quandle, rack, crossed module, categorical group, non-abelian tensor product of groups

2010 Mathematics Subject Classification

18D10, 57M25, 57M27

Published 3 December 2015