This talk is based on work with João Faria Martins (Univ. Leeds) [1] and several projects with students. I will describe the construction of an invariant of tangles and framed tangles which takes values in an arbitrary crossed module of finite groups. This involves the fundamental crossed module associated to a natural topological pair coming from a knot diagram, and a suitable class of morphisms from this fundamental crossed module to the chosen finite crossed module. Our construction includes all rack and quandle cohomology (framed) link invariants, as well as the Eisermann invariant of knots [2-3], for which we also find a lifting. The Eisermann invariant detects information about a suitable choice of meridian and longitude in the knot complement boundary.

[1] João Faria Martins and Roger Picken: Link invariants from finite categorical groups, Homology, Homotopy and Applications, 17(2) (2015), 205-233; arXiv:1301.3803v2 [math.GT], arXiv:1612.03501v1 [math.GT]
[2] M. Eisermann: Knot colouring polynomials, Pacific J. Math. 231 (2007), no. 2, 305-336.
[3] M. Eisermann: Homological characterization of the unknot, J. Pure Appl. Algebra 177 (2003), no. 2, 131-157.