Vol. 298, No. 2, 2019

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Concordance of Seifert surfaces

Robert Myers

Vol. 298 (2019), No. 2, 429–444
Abstract

We prove that every oriented nondisk Seifert surface F for an oriented knot  K in S3 is smoothly concordant to a Seifert surface F for a hyperbolic knot K of arbitrarily large volume. This gives a new and simpler proof of the result of Friedl and of Kawauchi that every knot is S-equivalent to a hyperbolic knot of arbitrarily large volume. The construction also gives a new and simpler proof of the result of Silver and Whitten and of Kawauchi that for every knot K there is a hyperbolic knot K of arbitrarily large volume and a map of pairs f : (S3,K) (S3,K) which induces an epimorphism on the knot groups. An example is given which shows that knot Floer homology is not an invariant of Seifert surface concordance. We also prove that a set of finite volume hyperbolic 3-manifolds with unbounded Haken numbers has unbounded volumes.

Keywords
knot, Seifert surface, concordance
Mathematical Subject Classification 2010
Primary: 57M25
Milestones
Received: 4 January 2017
Revised: 18 May 2018
Accepted: 6 June 2018
Published: 8 March 2019
Authors
Robert Myers
Department of Mathematics
Oklahoma State University
Stillwater
United States