Algebraic & Geometric Topology

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

Stanislav Jabuka and Tynan Kelly

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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

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

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

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

knots nonorientable 4-genus crosscap number slicing number


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.

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