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 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 p 2 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.

No abstract

We apply sutured Floer homology techniques to study the knot and link Floer homologies of various links with annuli embedded in their exteriors. Our main results include, for large n, characterizations of links with the same link Floer homology as (m, mn)cables of L-space knots, or the same knot Floer homology as (2, 2n)-cables of L-space knots. These yield some new link detection results.