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Bridge number, Heegaard genus and non-integral Dehn surgery


Authors: Kenneth L. Baker, Cameron Gordon and John Luecke
Journal: Trans. Amer. Math. Soc. 367 (2015), 5753-5830
MSC (2010): Primary 57M27
DOI: https://doi.org/10.1090/S0002-9947-2014-06328-9
Published electronically: October 22, 2014
MathSciNet review: 3347189
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Abstract: We show there exists a linear function $ w \colon \mathbb{N} \to \mathbb{N}$ with the following property. Let $ K$ be a hyperbolic knot in a hyperbolic $ 3$-manifold $ M$ admitting a non-longitudinal $ S^3$ surgery. If $ K$ is put into thin position with respect to a strongly irreducible, genus $ g$ Heegaard splitting of $ M$, then $ K$ intersects a thick level at most $ 2w(g)$ times. Typically, this shows that the bridge number of $ K$ with respect to this Heegaard splitting is at most $ w(g)$, and the tunnel number of $ K$ is at most $ w(g) + g-1$.


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Additional Information

Kenneth L. Baker
Affiliation: Department of Mathematics, University of Miami, Coral Gables, Florida 33146
Email: kb@math.miami.edu

Cameron Gordon
Affiliation: Department of Mathematics, The University of Texas at Austin, Austin, Texas 78712
Email: gordon@math.utexas.edu

John Luecke
Affiliation: Department of Mathematics, The University of Texas at Austin, Austin, Texas 78712
Email: luecke@math.utexas.edu

DOI: https://doi.org/10.1090/S0002-9947-2014-06328-9
Received by editor(s): February 1, 2012
Received by editor(s) in revised form: September 6, 2013
Published electronically: October 22, 2014
Additional Notes: In the course of this work the first author was partially supported by NSF Grant DMS-0239600, by the University of Miami 2011 Provost Research Award, and by a grant from the Simons Foundation (#209184). The first author would like to thank the Department of Mathematics at the University of Texas at Austin for its hospitality during his visits. These visits were supported in part by NSF RTG Grant DMS-0636643
The second author was partially supported by NSF Grant DMS-0906276
Article copyright: © Copyright 2014 American Mathematical Society