Volume 16, issue 1 (2016)

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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

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.

Dehn surgery, bridge number, 3–manifolds, knot theory
Mathematical Subject Classification 2010
Primary: 57M25, 57M27
Received: 20 December 2013
Revised: 11 February 2015
Accepted: 26 April 2015
Published: 23 February 2016
Kenneth L Baker
Department of Mathematics
University of Miami
1365 Memorial Drive
Coral Gables, FL 33146
Cameron Gordon
Department of Mathematics
University of Texas at Austin
2515 Speedway Stop C1200
Austin, TX 78712-1202
John Luecke
Department of Mathematics
University of Texas at Austin
2515 Speedway Stop C1200
Austin, TX 78712-0257