On the nonorientable four-ball genus of torus knots

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;.2n1/ 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 Mbius band for almost all integers q relatively prime to p.