Alternating links have representativity 2

Kindred, Thomas

doi:10.2140/agt.2018.18.3339

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Alternating links have representativity $2$

Thomas Kindred

Algebraic & Geometric Topology 18 (2018) 3339–3362

DOI: 10.2140/agt.2018.18.3339

Abstract

We prove that if $L$ is a nontrivial alternating link embedded (without crossings) in a closed surface $F \subset S^{3}$ , then $F$ has a compressing disk whose boundary intersects $L$ in no more than two points. Moreover, whenever the surface is incompressible and $\partial$ –incompressible in the link exterior, it can be isotoped to have a standard tube at some crossing of any reduced alternating diagram.

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

Volume 18, issue 6 (2018)