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On the Kashaev invariant and the twisted Reidemeister torsion of two-bridge knots

Tomotada Ohtsuki and Toshie Takata

Geometry & Topology 19 (2015) 853–952

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.

Kashaev invariant, twisted Reidemeister torsion, two-bridge knot
Mathematical Subject Classification 2010
Primary: 57M27
Received: 4 August 2013
Revised: 8 April 2014
Accepted: 27 May 2014
Published: 10 April 2015
Proposed: Shigeyuki Morita
Seconded: Vaughan Jones, Jean-Pierre Otal
Tomotada Ohtsuki
Research Institute for Mathematical Sciences
Kyoto University
Kyoto, 606-8502
Toshie Takata
Faculty of Mathematics
Kyushu University
Fukuoka, 819-0395