#### Volume 20, issue 1 (2020)

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On the genus defect of positive braid knots

### Livio Liechti

Algebraic & Geometric Topology 20 (2020) 403–428
##### Abstract

We show that the difference between the Seifert genus and the topological $4$–genus of a prime positive braid knot is bounded from below by an affine function of the minimal number of strands among positive braid representatives of the knot. We deduce that among prime positive braid knots, the property of having such a genus difference less than any fixed constant is characterised by finitely many forbidden surface minors.

##### Keywords
four-genus, genus defect, positive braid knot, surface minor, well-quasiorder
##### Mathematical Subject Classification 2010
Primary: 57M25, 57M27
Secondary: 06A06
##### Publication
Received: 5 November 2018
Revised: 22 March 2019
Accepted: 18 June 2019
Published: 23 February 2020
##### Authors
 Livio Liechti Department of Mathematics University of Fribourg Fribourg Switzerland