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

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 |

### Fingerprint

### 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

**Noncharacterizing slopes for hyperbolic knots.** / Baker, Kenneth; Motegi, Kimihiko.

Research output: Contribution to journal › Article

*Algebraic and Geometric Topology*, vol. 18, no. 3, pp. 1461-1480. https://doi.org/10.2140/agt.2018.18.1461

}

TY - JOUR

T1 - Noncharacterizing slopes for hyperbolic knots

AU - Baker, Kenneth

AU - Motegi, Kimihiko

PY - 2018/4/9

Y1 - 2018/4/9

N2 - A nontrivial slope r on a knot K in S3 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 S3 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 86 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.

AB - A nontrivial slope r on a knot K in S3 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 S3 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 86 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.

KW - Characterizing slope

KW - Dehn surgery

UR - http://www.scopus.com/inward/record.url?scp=85045224321&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85045224321&partnerID=8YFLogxK

U2 - 10.2140/agt.2018.18.1461

DO - 10.2140/agt.2018.18.1461

M3 - Article

AN - SCOPUS:85045224321

VL - 18

SP - 1461

EP - 1480

JO - Algebraic and Geometric Topology

JF - Algebraic and Geometric Topology

SN - 1472-2747

IS - 3

ER -