### 抄録

It is conjectured that, in the asymptotic expansion of the Kashaev invariant of a hyperbolic knot, the first coefficient is represented by the complex volume of the knot complement, and the second coefficient is represented by a constant multiple of the square root of the twisted Reidemeister torsion associated with the holonomy representation of the hyperbolic structure of the knot complement. In particular, this conjecture has been rigorously proved for some simple hyperbolic knots, for which the second coefficient is presented by a modification of the square root of the Hessian of the potential function of the hyperbolic structure of the knot complement. In this paper, we define an invariant of a parametrized knot diagram as a modification of the Hessian of the potential function obtained from the parametrized knot diagram. Further, we show that this invariant is equal (up to sign) to a constant multiple of the twisted Reidemeister torsion for any two-bridge knot.

元の言語 | 英語 |
---|---|

ページ（範囲） | 853-952 |

ページ数 | 100 |

ジャーナル | Geometry and Topology |

巻 | 19 |

発行部数 | 2 |

DOI | |

出版物ステータス | 出版済み - 4 10 2015 |

### Fingerprint

### All Science Journal Classification (ASJC) codes

- Geometry and Topology

### これを引用

*Geometry and Topology*,

*19*(2), 853-952. https://doi.org/10.2140/gt.2015.19.853

**On the kashaev invariant and the twisted reidemeister torsion of two-bridge knots.** / Ohtsuki, Tomotada; Takata, Toshie.

研究成果: ジャーナルへの寄稿 › 記事

*Geometry and Topology*, 巻. 19, 番号 2, pp. 853-952. https://doi.org/10.2140/gt.2015.19.853

}

TY - JOUR

T1 - On the kashaev invariant and the twisted reidemeister torsion of two-bridge knots

AU - Ohtsuki, Tomotada

AU - Takata, Toshie

PY - 2015/4/10

Y1 - 2015/4/10

N2 - It is conjectured that, in the asymptotic expansion of the Kashaev invariant of a hyperbolic knot, the first coefficient is represented by the complex volume of the knot complement, and the second coefficient is represented by a constant multiple of the square root of the twisted Reidemeister torsion associated with the holonomy representation of the hyperbolic structure of the knot complement. In particular, this conjecture has been rigorously proved for some simple hyperbolic knots, for which the second coefficient is presented by a modification of the square root of the Hessian of the potential function of the hyperbolic structure of the knot complement. In this paper, we define an invariant of a parametrized knot diagram as a modification of the Hessian of the potential function obtained from the parametrized knot diagram. Further, we show that this invariant is equal (up to sign) to a constant multiple of the twisted Reidemeister torsion for any two-bridge knot.

AB - It is conjectured that, in the asymptotic expansion of the Kashaev invariant of a hyperbolic knot, the first coefficient is represented by the complex volume of the knot complement, and the second coefficient is represented by a constant multiple of the square root of the twisted Reidemeister torsion associated with the holonomy representation of the hyperbolic structure of the knot complement. In particular, this conjecture has been rigorously proved for some simple hyperbolic knots, for which the second coefficient is presented by a modification of the square root of the Hessian of the potential function of the hyperbolic structure of the knot complement. In this paper, we define an invariant of a parametrized knot diagram as a modification of the Hessian of the potential function obtained from the parametrized knot diagram. Further, we show that this invariant is equal (up to sign) to a constant multiple of the twisted Reidemeister torsion for any two-bridge knot.

UR - http://www.scopus.com/inward/record.url?scp=84927779211&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84927779211&partnerID=8YFLogxK

U2 - 10.2140/gt.2015.19.853

DO - 10.2140/gt.2015.19.853

M3 - Article

VL - 19

SP - 853

EP - 952

JO - Geometry and Topology

JF - Geometry and Topology

SN - 1465-3060

IS - 2

ER -