Alexander invariants for ribbon tangles
- Date: Tuesday 10 March 2020, 16:00 – 17:00
- Location: Mathematics Level 8, MALL 1, School of Mathematics
- Type: Seminars, Pure Mathematics, Algebra
- Cost: Free
Celeste Damiani, University of Leeds. Part of the Algebra Seminar series.
Ribbon tangles are proper embeddings of tori and annuli in the 4-dimensional ball, bounding 3-manifolds with only ribbon singularities. We construct an Alexan- der 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 Florens-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 com- mutes 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.