On the nonorientable four-ball genus of torus knots

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 p/2 is not a perfect square, the torus knot Tp,q does not bound a locally flat Mobius band for almost all integers q relatively prime to p.