Independence of volume and genus $g$ bridge numbers
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- by Jessica S. Purcell and Alexander Zupan PDF
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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.References
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Additional Information
- Jessica S. Purcell
- Affiliation: School of Mathematical Sciences, 9 Rainforest Walk, Monash University, Victoria 3800, Australia
- MR Author ID: 807518
- ORCID: 0000-0002-0618-2840
- Email: jessica.purcell@monash.edu
- Alexander Zupan
- Affiliation: Department of Mathematics, University of Nebraska Lincoln, Lincoln, Nebraska 68588
- MR Author ID: 863648
- Email: zupan@unl.edu
- 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
- © Copyright 2016 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 145 (2017), 1805-1818
- MSC (2010): Primary 57M25, 57M27, 57M50
- DOI: https://doi.org/10.1090/proc/13327
- MathSciNet review: 3601570