On the nonorientable four-ball genus of torus knots

Binns, Fraser¹ ; Kang, Sungkyung² ; Simone, Jonathan³ ; Truöl, Paula⁴

¹Department of Mathematics, Princeton University, Princeton, NJ, United States
²Mathematical Institute, University of Oxford, Oxford, United Kingdom
³School of Mathematics, Georgia Institute of Technology, Atlanta, GA, United States
⁴Max Planck Institute for Mathematics, Bonn, Germany

Algebraic and Geometric Topology, Tome 25 (2025) no. 4, pp. 2209-2251

Résumé

The nonorientable four-ball genus of a knot K in S3 is the minimal first Betti number of nonorientable surfaces in B4 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 γ4 of any knot. This bound is sharp for several families of torus knots, including T4n,(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 2p is not a perfect square, the torus knot Tp,q does not bound a locally flat Möbius band for almost all integers q relatively prime to p.

Détail

DOI : 10.2140/agt.2025.25.2209

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

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