### Abstract

Let X be a norm curve in the SL(2, ℂ)-character variety of a knot exterior M. Let t = ||β||/||α|| be the ratio of the Culler-Shalen norms of two disαtinct non-zero classes α,β ∈ H _{1}(∂M,Z). We demonstrate that either X has exactly two associated strict boundary slopes ±t, or else there are strict boundary slopes r_{1} and r_{2} with |r_{1}| > t and |r^{2}| < t. As a consequence, we show that there are strict boundary slopes near cyclic, finite, and Seifert slopes. We also prove that the diameter of the set of strict boundary slopes can be bounded below using the Culler-Shalen norm of those slopes.

Original language | English |
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Pages (from-to) | 807-821 |

Number of pages | 15 |

Journal | Osaka Journal of Mathematics |

Volume | 43 |

Issue number | 4 |

Publication status | Published - 2006 Dec 1 |

Externally published | Yes |

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### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Osaka Journal of Mathematics*,

*43*(4), 807-821.

**Exceptional surgery and boundary slopes.** / Ishikawa, Masaharu; Mattman, Thomas W.; Shimokawa, Koya.

Research output: Contribution to journal › Article

*Osaka Journal of Mathematics*, vol. 43, no. 4, pp. 807-821.

}

TY - JOUR

T1 - Exceptional surgery and boundary slopes

AU - Ishikawa, Masaharu

AU - Mattman, Thomas W.

AU - Shimokawa, Koya

PY - 2006/12/1

Y1 - 2006/12/1

N2 - Let X be a norm curve in the SL(2, ℂ)-character variety of a knot exterior M. Let t = ||β||/||α|| be the ratio of the Culler-Shalen norms of two disαtinct non-zero classes α,β ∈ H 1(∂M,Z). We demonstrate that either X has exactly two associated strict boundary slopes ±t, or else there are strict boundary slopes r1 and r2 with |r1| > t and |r2| < t. As a consequence, we show that there are strict boundary slopes near cyclic, finite, and Seifert slopes. We also prove that the diameter of the set of strict boundary slopes can be bounded below using the Culler-Shalen norm of those slopes.

AB - Let X be a norm curve in the SL(2, ℂ)-character variety of a knot exterior M. Let t = ||β||/||α|| be the ratio of the Culler-Shalen norms of two disαtinct non-zero classes α,β ∈ H 1(∂M,Z). We demonstrate that either X has exactly two associated strict boundary slopes ±t, or else there are strict boundary slopes r1 and r2 with |r1| > t and |r2| < t. As a consequence, we show that there are strict boundary slopes near cyclic, finite, and Seifert slopes. We also prove that the diameter of the set of strict boundary slopes can be bounded below using the Culler-Shalen norm of those slopes.

UR - http://www.scopus.com/inward/record.url?scp=33847749342&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=33847749342&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:33847749342

VL - 43

SP - 807

EP - 821

JO - Osaka Journal of Mathematics

JF - Osaka Journal of Mathematics

SN - 0030-6126

IS - 4

ER -