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The nonorientable $4\mskip-1.5mu $–genus for knots with $8$ or $9$ crossings

Stanislav Jabuka and Tynan Kelly

Algebraic & Geometric Topology 18 (2018) 1823–1856

The nonorientable 4–genus of a knot in the 3–sphere is defined as the smallest first Betti number of any nonorientable surface smoothly and properly embedded in the 4–ball with boundary the given knot. We compute the nonorientable 4–genus for all knots with crossing number 8 or 9. As applications we prove a conjecture of Murakami and Yasuhara and compute the clasp and slicing number of a  knot. An errata was submitted on 18 August 2020 and posted online on 1 December 2020.

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knots, nonorientable 4-genus, crosscap number, slicing number
Mathematical Subject Classification 2010
Primary: 57M25, 57M27
Supplementary material


Received: 29 August 2017
Revised: 30 November 2017
Accepted: 24 December 2017
Published: 3 April 2018

Correction posted: 11 November 2020

Stanislav Jabuka
Department of Mathematics and Statistics
University of Nevada
Reno, NV
United States
Tynan Kelly
Department of Mathematics and Statistics
University of Nevada
Reno, NV
United States