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Independence of volume and genus $ g$ bridge numbers


Authors: Jessica S. Purcell and Alexander Zupan
Journal: Proc. Amer. Math. Soc. 145 (2017), 1805-1818
MSC (2010): Primary 57M25, 57M27, 57M50
DOI: https://doi.org/10.1090/proc/13327
Published electronically: December 30, 2016
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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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Additional Information

Jessica S. Purcell
Affiliation: School of Mathematical Sciences, 9 Rainforest Walk, Monash University, Victoria 3800, Australia
Email: jessica.purcell@monash.edu

Alexander Zupan
Affiliation: Department of Mathematics, University of Nebraska Lincoln, Lincoln, Nebraska 68588
Email: zupan@unl.edu

DOI: https://doi.org/10.1090/proc/13327
Received by editor(s): March 29, 2016
Published electronically: December 30, 2016
Additional Notes: The first author was partially supported by NSF grants DMS–1252687 and DMS-1128155, and ARC grant DP160103085
The second author was partially supported by NSF grant DMS–1203988
Communicated by: David Futer
Article copyright: © Copyright 2016 American Mathematical Society