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Abstract
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We prove that every oriented nondisk Seifert surface
for an oriented
knot
in
is smoothly concordant
to a Seifert surface
for a hyperbolic knot
of arbitrarily large volume. This gives a new and simpler proof
of the result of Friedl and of Kawauchi that every knot is
-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
there is a hyperbolic
knot
of arbitrarily large
volume and a map of pairs
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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Keywords
knot, Seifert surface, concordance
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Mathematical Subject Classification 2010
Primary: 57M25
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Milestones
Received: 4 January 2017
Revised: 18 May 2018
Accepted: 6 June 2018
Published: 8 March 2019
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