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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

The nonorientable 4–genus γ4(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 γ4 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 γ4 formulated by Ozsváth, Stipsicz and Szabó.

nonorientable 4–genus, 4–dimensional crosscap number, torus knots
Mathematical Subject Classification
Primary: 57K10
Secondary: 57R58
Received: 22 April 2020
Revised: 31 August 2020
Accepted: 8 December 2020
Published: 31 October 2021
Stanislav Jabuka
Department of Mathematics and Statistics
University of Nevada, Reno
Reno, NV
United States
Cornelia A Van Cott
Department of Mathematics
University of San Francisco
San Francisco, CA
United States