  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.
