On the asymptotic expansions of the Kashaev invariant of the knots with 6 crossings
| Autor: | Tomotada Ohtsuki, Yoshiyuki Yokota |
|---|---|
| Rok vydání: | 2017 |
| Předmět: |
010308 nuclear & particles physics
General Mathematics Homotopy 010102 general mathematics Mathematical analysis Volume conjecture Mathematics::Geometric Topology 01 natural sciences Hyperbolic volume Finite type invariant Hyperbolic set Saddle point 0103 physical sciences 0101 mathematics Invariant (mathematics) Asymptotic expansion Mathematics |
| Zdroj: | Mathematical Proceedings of the Cambridge Philosophical Society. 165:287-339 |
| ISSN: | 1469-8064 0305-0041 |
| DOI: | 10.1017/s0305004117000494 |
| Popis: | 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. |
| Databáze: | OpenAIRE |
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