Hyperbolic Structure Arising from a Knot Invariant

Details Hyperbolic Structure Arising from a Knot Invariant Hyperbolic Structure Arising from a Knot Invariant

2001-05-28

arxiv.org/abs/math-ph/0105039v1

Int. J. Mod. Phys. A 16 (2001) 3309--3333

Physics

Mathematical Physics

30 pages, 18 figures

Scientific paper

10.1142/S0217751X0100444X

We study the knot invariant based on the quantum dilogarithm function. This invariant can be regarded as a non-compact analogue of Kashaev's invariant, or the colored Jones invariant, and is defined by an integral form. The 3-dimensional picture of our invariant originates from the pentagon identity of the quantum dilogarithm function, and we show that the hyperbolicity consistency conditions in gluing polyhedra arise naturally in the classical limit as the saddle point equation of our invariant.

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