Independence of volume and genus 𝑔 bridge numbers

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 ) (g,b) -bridge surface for a knot K K in S 3 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 g -bridge numbers are completely independent. That is, for any g g , we construct explicit sequences of knots with bounded volume and unbounded genus g g -bridge number, and explicit sequences of knots with bounded genus g g -bridge number and unbounded volume.