#### Volume 21, issue 5 (2021)

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On a nonorientable analogue of the Milnor conjecture

### Stanislav Jabuka and Cornelia A Van Cott

Algebraic & Geometric Topology 21 (2021) 2571–2625
##### 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.

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##### Keywords
nonorientable 4–genus, 4–dimensional crosscap number, torus knots
Primary: 57K10
Secondary: 57R58