### Abstract

A nontrivial slope r on a knot K in S^{3} is called a characterizing slope if whenever the result of r-surgery on a knot K’ is orientation-preservingly homeomorphic to the result of r-surgery on K, then K’ is isotopic to K. Ni and Zhang ask: for any hyperbolic knot K, is a slope r=p/q with |p|+|q| sufficiently large a characterizing slope? In this article, we prove that if we can take an unknot c so that (0,0)-surgery on K∪c results in S^{3} and c is not a meridian of K, then K has infinitely many noncharacterizing slopes. As the simplest known example, the hyperbolic, two-bridge knot 8_{6} has no integral characterizing slopes. This answers the above question in the negative. We also prove that any L-space knot never admits such an unknot c.

Original language | English (US) |
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Pages (from-to) | 1461-1480 |

Number of pages | 20 |

Journal | Algebraic and Geometric Topology |

Volume | 18 |

Issue number | 3 |

DOIs | |

State | Published - Apr 9 2018 |

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### Keywords

- Characterizing slope
- Dehn surgery

### ASJC Scopus subject areas

- Geometry and Topology

### Cite this

*Algebraic and Geometric Topology*,

*18*(3), 1461-1480. https://doi.org/10.2140/agt.2018.18.1461