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On
the Kashaev invariant and the twisted Reidemeister torsion of
two-bridge knots
Tomotada Ohtsuki and Toshie Takata |
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Geometry & Topology 19 (2015) 853–952
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Abstract |
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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. |
Keywords
Kashaev invariant, twisted Reidemeister torsion, two-bridge
knot
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Mathematical Subject Classification 2010
Primary: 57M27
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References |
Publication
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
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Authors |
| Tomotada Ohtsuki | |
| Research Institute for Mathematical
Sciences Kyoto University Sakyo-ku Kyoto, 606-8502 Japan |
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| http://www.kurims.kyoto-u.ac.jp/~tomotada/ | |
| Toshie Takata | |
| Faculty of Mathematics Kyushu University Fukuoka, 819-0395 Japan |
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