### Abstract

For a hyperbolic knot, an ideal triangulation of the knot complement corresponding to the colored Jones polynomial was introduced by Thurston. Considering this triangulation of a twist knot, we find a function which gives the hyperbolicity equations and the complex volume of the knot complement, using Zickert's theory of the extended Bloch group and the complex volume. We also consider a formal approximation of the colored Jones polynomial. Following Ohnuki's theory of 2-bridge knots, we define another function which comes from the approximation. We show that this function is essentially the same as the previous function, and therefore it also gives the same hyperbolicity equations and the complex volume. Finally we compare this result with our previous one which dealt with Yokota theory, and, as an application to Yokota theory, present a refined formula of the complex volumes for any twist knots.

Original language | English |
---|---|

Pages (from-to) | 1401-1421 |

Number of pages | 21 |

Journal | Journal of Knot Theory and its Ramifications |

Volume | 19 |

Issue number | 11 |

DOIs | |

Publication status | Published - 2010 Nov |

### Fingerprint

### Keywords

- colored Jones polynomial
- complex volume
- twist knot
- Volume conjecture

### ASJC Scopus subject areas

- Algebra and Number Theory

### Cite this

**The complex volumes of twist knots via colored jones polynomials.** / Cho, Jinseok; Murakami, Jun.

Research output: Contribution to journal › Article

*Journal of Knot Theory and its Ramifications*, vol. 19, no. 11, pp. 1401-1421. https://doi.org/10.1142/S0218216510008443

}

TY - JOUR

T1 - The complex volumes of twist knots via colored jones polynomials

AU - Cho, Jinseok

AU - Murakami, Jun

PY - 2010/11

Y1 - 2010/11

N2 - For a hyperbolic knot, an ideal triangulation of the knot complement corresponding to the colored Jones polynomial was introduced by Thurston. Considering this triangulation of a twist knot, we find a function which gives the hyperbolicity equations and the complex volume of the knot complement, using Zickert's theory of the extended Bloch group and the complex volume. We also consider a formal approximation of the colored Jones polynomial. Following Ohnuki's theory of 2-bridge knots, we define another function which comes from the approximation. We show that this function is essentially the same as the previous function, and therefore it also gives the same hyperbolicity equations and the complex volume. Finally we compare this result with our previous one which dealt with Yokota theory, and, as an application to Yokota theory, present a refined formula of the complex volumes for any twist knots.

AB - For a hyperbolic knot, an ideal triangulation of the knot complement corresponding to the colored Jones polynomial was introduced by Thurston. Considering this triangulation of a twist knot, we find a function which gives the hyperbolicity equations and the complex volume of the knot complement, using Zickert's theory of the extended Bloch group and the complex volume. We also consider a formal approximation of the colored Jones polynomial. Following Ohnuki's theory of 2-bridge knots, we define another function which comes from the approximation. We show that this function is essentially the same as the previous function, and therefore it also gives the same hyperbolicity equations and the complex volume. Finally we compare this result with our previous one which dealt with Yokota theory, and, as an application to Yokota theory, present a refined formula of the complex volumes for any twist knots.

KW - colored Jones polynomial

KW - complex volume

KW - twist knot

KW - Volume conjecture

UR - http://www.scopus.com/inward/record.url?scp=78650381692&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=78650381692&partnerID=8YFLogxK

U2 - 10.1142/S0218216510008443

DO - 10.1142/S0218216510008443

M3 - Article

AN - SCOPUS:78650381692

VL - 19

SP - 1401

EP - 1421

JO - Journal of Knot Theory and its Ramifications

JF - Journal of Knot Theory and its Ramifications

SN - 0218-2165

IS - 11

ER -