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

Tomotada Ohtsuki, Toshie Takata

    Research output: Contribution to journalArticlepeer-review

    16 Citations (Scopus)

    Abstract

    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.

    Original languageEnglish
    Pages (from-to)853-952
    Number of pages100
    JournalGeometry and Topology
    Volume19
    Issue number2
    DOIs
    Publication statusPublished - Apr 10 2015

    All Science Journal Classification (ASJC) codes

    • Geometry and Topology

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