Concordance of Seifert surfaces

Robert Myers

Vol. 298 (2019), No. 2, 429–444

DOI: 10.2140/pjm.2019.298.429

Abstract

We prove that every oriented nondisk Seifert surface $F$ for an oriented knot  $K$ in $S^3$ is smoothly concordant to a Seifert surface $F^{\prime }$ for a hyperbolic knot $K^{\prime }$ 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 $K^{\prime }$ of arbitrarily large volume and a map of pairs $f:\left(S^3,K^{\prime }\right)\to \left(S^3,K\right)$ 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