On the nonorientable four-ball genus of torus knots

Binns, Fraser; Kang, Sungkyung; Simone, Jonathan; Truöl, Paula

doi:10.2140/agt.2025.25.2209

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On the nonorientable four-ball genus of torus knots

Fraser Binns, Sungkyung Kang, Jonathan Simone and Paula Truöl

Algebraic & Geometric Topology 25 (2025) 2209–2251

DOI: 10.2140/agt.2025.25.2209

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 $γ_{4}$ of any knot. This bound is sharp for several families of torus knots, including $T_{4 n, {(2 n \pm 1)}^{2}}$ for even $n \geq 2$ , a family Longo showed were counterexamples to Batson’s conjecture. We also prove that, whenever $p$ is an even positive integer and $\frac{1}{2} p$ 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 $p$ .

Keywords

torus knots, nonorientable 4-ball genus, involutive knot Floer homology, unoriented knot Floer homology

Mathematical Subject Classification

Primary: 57K10

Secondary: 57K18

References

Publication

Received: 13 March 2023

Revised: 6 February 2024

Accepted: 5 June 2024

Published: 11 August 2025

Authors

Fraser Binns
Department of Mathematics Princeton University Princeton, NJ United States

Sungkyung Kang
Mathematical Institute University of Oxford Oxford United Kingdom

Jonathan Simone
School of Mathematics Georgia Institute of Technology Atlanta, GA United States

Paula Truöl
Max Planck Institute for Mathematics Bonn Germany

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Volume 25, issue 4 (2025)