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. In [13], R.M. Kashaev defined a family of complex valued link invariants indexed by integers N ≥ 2 using the quantum dilogarithm. Later he calculated the asymptotic behavior of his invariant and observed that for the three simplest hyperbolic knots it grows as exp(Vol(K)N/2π) when N goes to the infinity, where Vol(K) is the hyperbolic volume of the complement of a knotK [14]. This amazing result and his conjecture that the same also holds for any hyperbolic knot have been almost ignored by mathematicians since his definition of the invariant is too complicated (though it uses only elementary tools). The aim of this paper is to reveal his mysterious definition and to show that his invariant is nothing but a specialization of the colored Jones polynomial. The colored Jones polynomial is defined for colored links (each component is decorated with an irreducible representation of the Lie algebra sl(2,C)). The original Jones polynomial corresponds to the case that all the colors are identical to the 2-dimensional fundamental representation. We show that Kashaev’s invariant with parameter N coincides with the colored Jones polynomial in a certain normalization with every color the N -dimensional representation, evaluated at the primitive N -th root of unity. (We have to normalize the colored Jones polynomial so that the value for the trivial knot is one, for otherwise it always vanishes). On the other hand there are other colored polynomial invariants, the generalized multivariable Alexander polynomial defined by Y. Akutsu, T. Deguchi and T. Ohtsuki [1]. They used the same Lie algebra sl(2,C) but a different hierarchy of representations. Their invariants are parameterized by c+1 parameters; an integer Date: February 1, 2008. 1991 Mathematics Subject Classification. 57M25, 57M50, 17B37, 81R50.