TY - JOUR

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

AU - Ohtsuki, Tomotada

AU - Takata, Toshie

N1 - Publisher Copyright:
© 2015 Mathematical Sciences Publishers. All rights reserved.
Copyright:
Copyright 2015 Elsevier B.V., All rights reserved.

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.

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U2 - 10.2140/gt.2015.19.853

DO - 10.2140/gt.2015.19.853

M3 - Article

AN - SCOPUS:84927779211

VL - 19

SP - 853

EP - 952

JO - Geometry and Topology

JF - Geometry and Topology

SN - 1465-3060

IS - 2

ER -