Bridge number and integral Dehn surgery

Baker, Kenneth L¹ ; Gordon, Cameron² ; Luecke, John³

¹Department of Mathematics, University of Miami, 1365 Memorial Drive, Coral Gables, FL 33146, USA
²Department of Mathematics, University of Texas at Austin, 2515 Speedway Stop C1200, Austin, TX 78712-1202, USA
³Department of Mathematics, University of Texas at Austin, 2515 Speedway Stop C1200, Austin, TX 78712-0257, USA

Algebraic and Geometric Topology, Tome 16 (2016) no. 1, pp. 1-40

Résumé

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.

Détail

DOI : 10.2140/agt.2016.16.1

Classification : 57M25, 57M27
Keywords: Dehn surgery, bridge number, 3–manifolds, knot theory

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