#### Volume 18, issue 3 (2018)

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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
##### 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.

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