On a nonorientable analogue of the Milnor conjecture

Abstract

The nonorientable $4$–genus ${\gamma }_4\left(K\right)$ of a knot $K$ is the smallest first Betti number of any nonorientable surface properly embedded in the $4$–ball and bounding the knot $K$. We study a conjecture proposed by Batson about the value of ${\gamma }_4$ for torus knots, which can be seen as a nonorientable analogue of Milnor’s conjecture for the orientable $4$–genus of torus knots. We prove the conjecture for many infinite families of torus knots, by calculating for all torus knots a lower bound for ${\gamma }_4$ formulated by Ozsváth, Stipsicz and Szabó.