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.

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Keywords
four-genus, genus defect, positive braid knot, surface minor, well-quasiorder
Mathematical Subject Classification 2010
Primary: 57M25, 57M27
Secondary: 06A06
References
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