## Topology Seminar (Main Talk): Additive Invariants of Knots, Links, and Spatial Graphs in 3-manifolds.

Seminar | April 5 | 4:10-5 p.m. | 3 Evans Hall

Scott Taylor, Colby College

Department of Mathematics

Tunnel number, higher genus bridge number, and Gabai width are classical knot invariants that are additive under connected sum for some classes of knots, but not for others. I'll explain variations of these classical invariants that are defined for almost all (3-manifold, graph) pairs, that detect the unknot in the 3-sphere, and that are additive under connected sum and trivalent vertex sum. The proofs of these facts rely on a new version of thin position for (3-manifold, graph) pairs. Harnessing the relationship between these new invariants and tunnel number, we'll prove a theorem whose statement is a combination of the statements of theorems of Scharlemann-Schultens and Morimoto; it gives a lower bound on the tunnel number of the connected sum of knots in terms of the number and tunnel number of the summands. This is joint work with Maggy Tomova.

hongbins@berkeley.edu