Independence of volume and genus g bridge numbers

Jessica Purcell, Alexander Zupan

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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 S3 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.

Original languageEnglish
Pages (from-to)1805-1818
Number of pages14
JournalProceedings of the American Mathematical Society
Issue number4
Publication statusPublished - 2017

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