The colored Jones polynomials and the simplicial volume of a knot

Details The colored Jones polynomials and the simplicial volume of a knot The colored Jones polynomials and the simplicial volume of a knot

1999-05-12

arxiv.org/abs/math/9905075v2

Mathematics

Geometric Topology

18 pages. Added Remark 2.2 and improved Remark 5.7

Scientific paper

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.

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