Volume 18, issue 6 (2018)

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Alternating links have representativity $2$

Thomas Kindred

Algebraic & Geometric Topology 18 (2018) 3339–3362
Abstract

We prove that if L is a nontrivial alternating link embedded (without crossings) in a closed surface F S3, then F has a compressing disk whose boundary intersects L in no more than two points. Moreover, whenever the surface is incompressible and –incompressible in the link exterior, it can be isotoped to have a standard tube at some crossing of any reduced alternating diagram.

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Keywords
alternating knot, alternating link, representativity, closed surface, compressing disk
Mathematical Subject Classification 2010
Primary: 57M25
Secondary: 57M50
References
Publication
Received: 24 August 2017
Revised: 4 December 2017
Accepted: 11 April 2018
Published: 18 October 2018
Authors
Thomas Kindred
Department of Mathematics
University of Nebraska-Lincoln
Lincoln, NE
United States
http://www.thomaskindred.com