Bridge number and integral Dehn surgery

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 $\left(\hat{R},K\right)$ being caught by a surface $Q$ in the exterior of the link $K\cup \partial \hat{R}$. For a caught pair $\left(\hat{R},K\right)$, 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 $\left\{K^n\right\}$ 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 $S^3$.