Concordance of Seifert surfaces

TL;DR

This paper demonstrates that any Seifert surface for a knot can be smoothly concordant to one for a hyperbolic knot with arbitrarily large volume, simplifying existing proofs of related knot theory results.

Contribution

It provides new, simpler proofs that every knot is S-equivalent to a hyperbolic knot of large volume and constructs maps inducing epimorphisms on knot groups, expanding understanding of knot concordance.

Findings

Any Seifert surface is concordant to one for a hyperbolic knot of arbitrarily large volume.

Knot Floer homology is not an invariant of Seifert surface concordance.

Hyperbolic 3-manifolds with unbounded Haken numbers have unbounded volumes.

Abstract

This paper proves that every oriented non-disk Seifert surface $F$ for a 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 $f:(S^3,K^{\prime})\rightarrow (S^3,K)$ 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. The paper also proves that a set of finite volume hyperbolic 3-manifolds with unbounded Haken numbers has unbounded volumes.