Bonn Topology Group - Abstracts

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Talk

December 10, 2024
Paula Truöl (Max Planck Institute for Mathematics): 3-braid knots with maximal topological 4-genus

Abstract

This talk deals with the problem of determining the topological 4-genus for the special case of 3-braid knots. The 4-genus of a knot is the minimal genus of a "nicely" embedded surface in the 4-dimensional ball with boundary the given knot. Asking whether a knot has 4-genus zero, i.e. whether it bounds a disk in the 4-ball, is a natural generalization in dimension 4 of the question whether it is isotopic to the trivial knot. It is one of the curiosities of low-dimensional topology that constructions such as finding these disks can sometimes be done in the topological category, but fail to work smoothly. The first examples of this phenomenon followed Freedman’s famous work on 4-manifolds. Four decades later, the topological 4-genus of knots, even torus knots, remains difficult to determine. In a joint work with S. Baader, L. Lewark and F. Misev, we classify 3-braid knots whose topological 4-genus is maximal (i.e. equal to their 3-dimensional Seifert genus). In the talk, we will define the relevant terms and provide some context for our results.