Independence of volume and genus $g$ bridge numbers

Jessica S. Purcell; Alexander Zupan

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arXiv · 1512.03869

Independence of volume and genus $g$ bridge numbers

Jessica S. Purcell·Alexander Zupan

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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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BibTeX RIS

Jessica S. Purcell, Alexander Zupan. 2016-03-29. Independence of volume and genus $g$ bridge numbers. https://arxiv.org/abs/1512.03869

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