## Algebraic & Geometric Topology

### Bridge number and integral Dehn surgery

#### 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
Revised: 11 February 2015
Accepted: 26 April 2015
First available in Project Euclid: 16 November 2017

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

#### 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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