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On the asymptotic expansions of the Kashaev invariant of the knots with 6 crossings

Published online by Cambridge University Press:  15 June 2017

TOMOTADA OHTSUKI  and
YOSHIYUKI YOKOTA
TOMOTADA OHTSUKI
Affiliation:
Research Institute for Mathematical Sciences, Kyoto University, Sakyo-ku, Kyoto, 606-8502, Japan. e-mail: [email protected]
YOSHIYUKI YOKOTA
Affiliation:
Department of Mathematics and Information Sciences, Tokyo Metropolitan University, Tokyo, 192-0397, Japan. e-mail: [email protected]
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Abstract

We give presentations of the asymptotic expansions of the Kashaev invariant of the knots with 6 crossings. In particular, we show the volume conjecture for these knots, which states that the leading terms of the expansions present the hyperbolic volume and the Chern--Simons invariant of the complements of the knots. As higher coefficients of the expansions, we obtain a new series of invariants of these knots.

A non-trivial part of the proof is to apply the saddle point method to calculate the asymptotic expansion of an integral which presents the Kashaev invariant. A key step of this part is to give a concrete homotopy of the (real 3-dimensional) domain of the integral in ℂ3 in such a way that the boundary of the domain always stays in a certain domain in ℂ3 given by the potential function of the hyperbolic structure.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 165 , Issue 2 , September 2018 , pp. 287 - 339
Copyright
Copyright © Cambridge Philosophical Society 2017 

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