The nonorientable 4–genus for knots with 8 or 9 crossings

Jabuka, Stanislav; Kelly, Tynan

doi:10.2140/agt.2018.18.1823

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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

DOI: 10.2140/agt.2018.18.1823

Abstract

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.

Keywords

knots, nonorientable 4-genus, crosscap number, slicing number

Mathematical Subject Classification 2010

Primary: 57M25, 57M27

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Publication

Received: 29 August 2017

Revised: 30 November 2017

Accepted: 24 December 2017

Published: 3 April 2018

Correction posted: 11 November 2020

Authors

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

Volume 18, issue 3 (2018)