On a Nonorientable Analogue of the Milnor Conjecture

The nonorientable 4-genus ${\gamma }_4(K)$ 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 relying on a lower bound for ${\gamma }_4$ formulated by Ozsv'ath, Stipsicz, and Szab'o. As a side product we obtain new closed formulas for the signature of torus knots.