#### 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
##### Abstract

In a $3$–manifold $M$, let $K$ be a knot and $\stackrel{̂}{R}$ be an annulus which meets $K$ transversely. We define the notion of the pair $\left(\stackrel{̂}{R},K\right)$ being caught by a surface $Q$ in the exterior of the link $K\cup \partial \stackrel{̂}{R}$. For a caught pair $\left(\stackrel{̂}{R},K\right)$, we consider the knot ${K}^{n}$ gotten by twisting $K$ $n$ times along $\stackrel{̂}{R}$ and give a lower bound on the bridge number of ${K}^{n}$ with respect to Heegaard splittings of $M\phantom{\rule{0.3em}{0ex}}$; 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 $\left\{{K}^{n}\right\}$ 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