Topology Seminar: Joshua Batson “Nonorientable Four-ball Genus Can be Arbitrarily Large”

Introduction: This talk provides background for my 1 p.m. talk on the nonorientable four-ball genus. First, I’ll give some background on the topology of surfaces in four-space–how to visualize them and their normal bundles, and the Gordon-Litherland construction of the knot signature. Then I’ll talk about gradings in Heegaard-Floer homology (an invariant of 3-manifolds and cobordisms between them). Those gradings can be used to define the d-invariants, which will be the key technical ingredient in my second talk.
[box, type=”download”]Watch the Introduction video.[/box]

Abstract: The nonorientable four-ball genus of a knot K is the smallest first Betti number of any smoothly embedded, nonorientable surface F in B^4 bounding K. In contrast to

the orientable four-ball genus, which can be bounded using algebraic topology, Heegaard-Floer homology, and Khovanov homology, the best lower bound in the literature

on the nonorientable four-ball genus for any K is 3. We find a lower bound in terms of the signature of K and the Heegaard-Floer d-invariant of the integer homology

sphere given by -1 surgery on K. In particular, we prove that the nonorientable four-ball genus of the torus knot T(2k,2k-1) is k-1.

[box, type=”download”]Watch the video.[/box]