Modul: MAT772 Geometrie-Seminar

## Satellite operations and knot genera

Dr. Allison N. Miller talk

**Date:** 25.09.19 **Time:** 15.45 - 16.45 **Room:** ETH HG G 43

The classical 3-genus of knots behaves predictably under the satellite operation: a formula of Schubert states that a winding number w pattern P has g3(P(K))=g3(P)+|w|g3(K), for an appropriately defined g3(P) bounded below by g3(P(U)). In this talk, I’ll discuss the extent to which analogous formulae, particularly the intuition that the genus of P(K) should be roughly |w| times the genus of K, hold in the context of the slice genus. In the smooth setting the winding number continues to play a pivotal role, but in new joint work with Peter Feller and Juanita Pinz{‘o}n-Caicedo we give surprising evidence that in the topological category the winding number of a pattern is almost irrelevant in its effect on the 4-genus!