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.

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Mathematical Subject Classification 2010

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