On a Nonorientable Analogue of the Milnor Conjecture

Stanislav Jabuka, Cornelia A. Van Cott

Published 2018-09-06Version 1

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 Szabo. As a side product we obtain new closed formulas for the signature of torus knots. We provide a comparison between $\gamma_4$ and $\gamma_3$ for torus knots, where $\gamma_3(K)$ is the cross-cap number of $K$.