Algebraic & Geometric Topology

Bridge number and integral Dehn surgery

Kenneth L Baker, Cameron Gordon, and John Luecke

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Abstract

In a 3–manifold M, let K be a knot and R̂ be an annulus which meets K transversely. We define the notion of the pair (R̂,K) being caught by a surface Q in the exterior of the link K R̂. For a caught pair (R̂,K), we consider the knot Kn gotten by twisting K n times along R̂ and give a lower bound on the bridge number of Kn with respect to Heegaard splittings of M; as a function of n, the genus of the splitting, and the catching surface Q. As a result, the bridge number of Kn tends to infinity with n. In application, we look at a family of knots {Kn} found by Teragaito that live in a small Seifert fiber space M and where each Kn admits a Dehn surgery giving S3. We show that the bridge number of Kn with respect to any genus-2 Heegaard splitting of M tends to infinity with n. This contrasts with other work of the authors as well as with the conjectured picture for knots in lens spaces that admit Dehn surgeries giving S3.

Article information

Source
Algebr. Geom. Topol., Volume 16, Number 1 (2016), 1-40.

Dates
Received: 20 December 2013
Revised: 11 February 2015
Accepted: 26 April 2015
First available in Project Euclid: 16 November 2017

Permanent link to this document
https://projecteuclid.org/euclid.agt/1510841103

Digital Object Identifier
doi:10.2140/agt.2016.16.1

Mathematical Reviews number (MathSciNet)
MR3470696

Zentralblatt MATH identifier
1339.57005

Subjects
Primary: 57M25: Knots and links in $S^3$ {For higher dimensions, see 57Q45} 57M27: Invariants of knots and 3-manifolds

Keywords
Dehn surgery bridge number 3–manifolds knot theory

Citation

Baker, Kenneth L; Gordon, Cameron; Luecke, John. Bridge number and integral Dehn surgery. Algebr. Geom. Topol. 16 (2016), no. 1, 1--40. doi:10.2140/agt.2016.16.1. https://projecteuclid.org/euclid.agt/1510841103


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