The colored Jones polynomials and the simplicial volume of a knot

Abstract

We show that the set of colored Jones polynomials and the set of generalized Alexander polynomials defined by Akutsu, Deguchi and Ohtsuki intersect non-trivially. Moreover it is shown that the intersection is (at least includes) the set of Kashaev's quantum dilogarithm invariants for links. Therefore Kashaev's conjecture can be restated as follows: The colored Jones polynomials determine the hyperbolic volume for a hyperbolic knot. Modifying this, we propose a stronger conjecture: The colored Jones polynomials determine the simplicial volume for any knot. If our conjecture is true, then we can prove that a knot is trivial if and only if all of its Vassiliev invariants are trivial.