#### 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 ${S}^{3}$ 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
Primary: 57M25