Algebraic & Geometric Topology

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

Stanislav Jabuka and Tynan Kelly

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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.

Article information

Source
Algebr. Geom. Topol., Volume 18, Number 3 (2018), 1823-1856.

Dates
Received: 29 August 2017
Revised: 30 November 2017
Accepted: 24 December 2017
First available in Project Euclid: 26 April 2018

Permanent link to this document
https://projecteuclid.org/euclid.agt/1524708107

Digital Object Identifier
doi:10.2140/agt.2018.18.1823

Mathematical Reviews number (MathSciNet)
MR3784020

Zentralblatt MATH identifier
06866414

Subjects
Primary: 57M25: Knots and links in $S^3$ {For higher dimensions, see 57Q45} 57M27: Invariants of knots and 3-manifolds

Keywords
knots nonorientable 4-genus crosscap number slicing number

Citation

Jabuka, Stanislav; Kelly, Tynan. The nonorientable $4$–genus for knots with $8$ or $9$ crossings. Algebr. Geom. Topol. 18 (2018), no. 3, 1823--1856. doi:10.2140/agt.2018.18.1823. https://projecteuclid.org/euclid.agt/1524708107


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