On a nonorientable analogue of the Milnor conjecture

Jabuka, Stanislav; Van Cott, Cornelia A

doi:10.2140/agt.2021.21.2571

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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

DOI: 10.2140/agt.2021.21.2571

Abstract

The nonorientable $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mn>4</mn></math>$ –genus $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mrow><mi>γ</mi></mrow><mrow><mn>4</mn></mrow></msub><mrow><mo class="MathClass-open">(</mo><mrow><mi>K</mi></mrow><mo class="MathClass-close">)</mo></mrow></math>$ of a knot $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>K</mi></math>$ is the smallest first Betti number of any nonorientable surface properly embedded in the $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mn>4</mn></math>$ –ball and bounding the knot $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>K</mi></math>$ . We study a conjecture proposed by Batson about the value of $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mrow><mi>γ</mi></mrow><mrow><mn>4</mn></mrow></msub></math>$ for torus knots, which can be seen as a nonorientable analogue of Milnor’s conjecture for the orientable $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mn>4</mn></math>$ –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 $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mrow><mi>γ</mi></mrow><mrow><mn>4</mn></mrow></msub></math>$ formulated by Ozsváth, Stipsicz and Szabó.

Keywords

nonorientable 4–genus, 4–dimensional crosscap number, torus knots

Mathematical Subject Classification

Primary: 57K10

Secondary: 57R58

References

Publication

Received: 22 April 2020

Revised: 31 August 2020

Accepted: 8 December 2020

Published: 31 October 2021

Authors

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

Volume 21, issue 5 (2021)