Work

On the nonorientable four-ball genus of torus knots

Year: 2025

Type: article

Abstract: The nonorientable four-ball genus of a knot K in S 3 is the minimal first Betti number of nonorientable surfaces in B 4 bounded by K.By amalgamating ideas from involutive knot Floer homology and unoriented knot Floer homology, we give a new lower bound on the smooth nonorientable four-ball genus 4 of any knot.This bound is sharp for several families of torus knots, including T 4n;.2n˙1/ 2 for even n 2, a family Longo showed were counterexamples to Batson's conjecture.We also prove that, whenever p is an even positive integer and 1 2 p is not a perfect square, the torus knot T p;q does not bound a locally flat Möbius band for almost all integers q relatively prime to p.

Source: Algebraic & Geometric Topology

DOI: 10.2140/agt.2025.25.2209

Authors: Fraser Binns, Sungkyung Kang, Jonathan Simone, Paula Truöl

Institutions: Princeton University, University of Oxford, Georgia Institute of Technology, Max Planck Institute for Mathematics

Language: English

Topic: Geometric and Algebraic Topology

Subfield: Geometry and Topology

Field: Mathematics

Domain: Physical Sciences

Open Access status: hybrid

Funders: National Science Foundation, Schweizerischer Nationalfonds zur Förderung der Wissenschaftlichen Forschung, Institute for Basic Science

Awards: G2583319762, G5387446297, Low-dimensional topology with a view toward the fourth dimension and complex geometry, G670587537