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

Autor:	Tomotada Ohtsuki, Toshie Takata
Rok vydání:	2015
Předmět:	Knot complement Pure mathematics two-bridge knot Quantum invariant Skein relation Mathematical analysis Volume conjecture Tricolorability Mathematics::Geometric Topology Knot theory Knot invariant 57M27 twisted Reidemeister torsion Kashaev invariant Geometry and Topology Trefoil knot Mathematics
Zdroj:	Geom. Topol. 19, no. 2 (2015), 853-952
ISSN:	1364-0380 1465-3060
DOI:	10.2140/gt.2015.19.853
Popis:	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. 57M27
Databáze:	OpenAIRE
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