Bridge number and integral Dehn surgery

In a 3–manifold M, let K be a knot and R be an annulus which meets K transversely. We define the notion of the pair (R, K) being caught by a surface Q in the exterior of the link K ∪ ∂ R. For a caught pair (R, K), we consider the knot Kⁿ gotten by twisting Kⁿ times along R 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