2015/12/12 by Purcell, Jessica S., Zupan, Alexander
#57M25 #57M27 #57M50 #FOS: Mathematics #Geometric Topology (math.GT)
paper · doi:10.48550/arxiv.1512.03869
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.