### 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) |
---|---|

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

**Bridge number and integral Dehn surgery.** / Baker, Kenneth; Gordon, Cameron; Luecke, John.

Research output: Contribution to journal › Article

*Algebraic and Geometric Topology*, vol. 16, no. 1, pp. 1-40. https://doi.org/10.2140/agt.2016.16.1

}

TY - JOUR

T1 - Bridge number and integral Dehn surgery

AU - Baker, Kenneth

AU - Gordon, Cameron

AU - Luecke, John

PY - 2016/2/23

Y1 - 2016/2/23

N2 - 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

AB - 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

UR - http://www.scopus.com/inward/record.url?scp=84960089337&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84960089337&partnerID=8YFLogxK

U2 - 10.2140/agt.2016.16.1

DO - 10.2140/agt.2016.16.1

M3 - Article

AN - SCOPUS:84960089337

VL - 16

SP - 1

EP - 40

JO - Algebraic and Geometric Topology

JF - Algebraic and Geometric Topology

SN - 1472-2747

IS - 1

ER -