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Consider all knots with fixed genus $g\ge 2$ (I am considering the classical 3-genus). Do there exist infinite families of genus $g$ knots with arbitrarily large volume?

The answer seems like it should definitely be yes, but I can’t seem to find any references.

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  • $\begingroup$ What is the "3-genus"? $\endgroup$
    – Igor Rivin
    Commented Jun 29, 2018 at 1:42
  • $\begingroup$ @IgorRivin: The usual Seifert genus (as opposed to the $4$-genus, where you allow the surface to go into the $4$-dimensional ball; thus the $4$-genus is $0$ when the knot is slice). $\endgroup$
    – Andy Putman
    Commented Jun 29, 2018 at 1:53
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    $\begingroup$ Is it actually clear whether there have to be infinitely many knots of a given Seifert genus? $\endgroup$
    – ThiKu
    Commented Jun 29, 2018 at 5:33
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    $\begingroup$ @ThiKu: It is for except in genus $0$: just take a small band of the Seifert surface (chosen so that you aren't just pinching off a disc; this uses that the genus is at least $1$) and insert appropriate twists in it. This doesn't change the homeomorphism type of the Seifert surface, but will give you infinitely many knots. $\endgroup$
    – Andy Putman
    Commented Jul 9, 2018 at 15:08
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    $\begingroup$ Please, you really should add the word "hyperbolic" (as in "hyperbolic knot" and "hyperbolic volume") in a few places in your question... $\endgroup$
    – Sam Nead
    Commented Jul 29, 2018 at 13:22

3 Answers 3

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The result of Brittenham for genus 1 knots pointed out by Sam Nead was generalized to all genus in Theorem 8.2 of this paper:

Stoimenow, A., Realizing Alexander polynomials by hyperbolic links, Expo. Math. 28, No. 2, 133-178 (2010). ZBL1196.57009.

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Here is a paper of Mark Brittenham that provides (free!) Seifert genus one knots of arbitrarily large hyperbolic volume:

https://arxiv.org/abs/math/9809143

His construction should generalize to all genera.

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This question is completely answered in the paper of Purcell-Zupan and references therein.

Purcell, Jessica S.; Zupan, Alexander, Independence of volume and genus $g$ bridge numbers, Proc. Am. Math. Soc. 145, No. 4, 1805-1818 (2017). ZBL1364.57009.

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    $\begingroup$ At a cursory glance, it looks like this paper is discussing the Heegaard genus and the “bridge number” genus. I’ll take a better look in the morn. $\endgroup$
    – Rocket Man
    Commented Jun 29, 2018 at 2:01
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    $\begingroup$ Indeed, [PZ] is discussing a different notion of genus. The original question should have older answers (probably even older than the one I offered below). $\endgroup$
    – Sam Nead
    Commented Jul 29, 2018 at 13:20

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