Bridge number and integral Dehn surgery

Baker, Kenneth L; Gordon, Cameron; Luecke, John

doi:10.2140/agt.2016.16.1

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Bridge number and integral Dehn surgery

Kenneth L Baker, Cameron Gordon and John Luecke

Algebraic & Geometric Topology 16 (2016) 1–40

DOI: 10.2140/agt.2016.16.1

Abstract

In a $3$ –manifold $M$ , let $K$ be a knot and $\hat{R}$ be an annulus which meets $K$ transversely. We define the notion of the pair $(\hat{R}, K)$ being caught by a surface $Q$ in the exterior of the link $K \cup \partial \hat{R}$ . For a caught pair $(\hat{R}, K)$ , we consider the knot $K^{n}$ gotten by twisting $K$ $n$ times along $\hat{R}$ and give a lower bound on the bridge number of $K^{n}$ 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 $K^{n}$ tends to infinity with $n$ . In application, we look at a family of knots ${K^{n}}$ found by Teragaito that live in a small Seifert fiber space $M$ and where each $K^{n}$ admits a Dehn surgery giving $S^{3}$ . We show that the bridge number of $K^{n}$ 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 $S^{3}$ .

Keywords

Dehn surgery, bridge number, 3–manifolds, knot theory

Mathematical Subject Classification 2010

Primary: 57M25, 57M27

References

Publication

Received: 20 December 2013

Revised: 11 February 2015

Accepted: 26 April 2015

Published: 23 February 2016

Authors

Kenneth L Baker
Department of Mathematics University of Miami 1365 Memorial Drive Coral Gables, FL 33146 USA

Cameron Gordon
Department of Mathematics University of Texas at Austin 2515 Speedway Stop C1200 Austin, TX 78712-1202 USA

John Luecke
Department of Mathematics University of Texas at Austin 2515 Speedway Stop C1200 Austin, TX 78712-0257 USA

Volume 16, issue 1 (2016)