### Abstract

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 K^{n} gotten by twisting K^{n} times along 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 {K^{n}} 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}

Original language | English (US) |
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Pages (from-to) | 1-40 |

Number of pages | 40 |

Journal | Algebraic and Geometric Topology |

Volume | 16 |

Issue number | 1 |

DOIs | |

State | Published - Feb 23 2016 |

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### ASJC Scopus subject areas

- Geometry and Topology

### Cite this

*Algebraic and Geometric Topology*,

*16*(1), 1-40. https://doi.org/10.2140/agt.2016.16.1