Independence of volume and genus 𝑔 bridge numbers

Purcell, Jessica; Zupan, Alexander

doi:10.1090/proc/13327

Independence of volume and genus $g$ bridge numbers
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by Jessica S. Purcell and Alexander Zupan PDF

Proc. Amer. Math. Soc. 145 (2017), 1805-1818 Request permission

Abstract:

A theorem of Jorgensen and Thurston implies that the volume of a hyperbolic 3–manifold is bounded below by a linear function of its Heegaard genus. Heegaard surfaces and bridge surfaces often exhibit similar topological behavior; thus it is natural to extend this comparison to ask whether a $(g,b)$-bridge surface for a knot $K$ in $S^3$ carries any geometric information related to the knot exterior. In this paper, we show that — unlike in the case of Heegaard splittings — hyperbolic volume and genus $g$-bridge numbers are completely independent. That is, for any $g$, we construct explicit sequences of knots with bounded volume and unbounded genus $g$-bridge number, and explicit sequences of knots with bounded genus $g$-bridge number and unbounded volume.

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