Alexander invariants for ribbon tangles
Celeste Damiani (University of Caen)
Wednesday 1st June, 2016 15:00-16:00 Maths 522
Ribbon tangles are proper embeddings of tori and annuli in the 4-dimensional ball, bounding 3-manifolds with only ribbon singularities. We construct an Alexander invariant for these objects that induces a functorial generalisation of the Alexander polynomial. This functor is an extension of the Alexander functor for usual tangles defined by Bigelow-Cattabriga-Florens and studied by Flores-Massuyeau. If considered on braid-like ribbon tangles, this functor coincides with the exterior powers of the Burau-Gassner representation. On one hand, we observe that the action of cobordisms on ribbon tangles endows them with a circuit algebra structure over the operad of cobordisms, and we show that the Alexander invariant commutes with the circuit algebra’s composition. On the other hand, ribbon tangles can be represented by welded tangle diagrams: this allows to give a combinatorial description of the Alexander invariant.