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The colored Jones polynomials and the simplicial volume of a knot
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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.
Forward citations
Cited by 2 Pith papers
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Generalized Minkowski Theorem for Tetrahedra in ${\rm dS}^3$ and ${\rm AdS}^3$
Four based SO+(1,2) holonomies reconstruct a unique strictly convex tetrahedron in dS^3 or AdS^3, with det G selecting the model and recovering Euclidean cases via SU(2).
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The holonomy braiding for $\mathcal{U}_\xi(\mathfrak{sl}_2)$ in terms of geometric quantum dilogarithms
Derives explicit factorization of the holonomy R-matrix for U_ξ(sl₂) at a root of unity into four geometric quantum dilogarithms satisfying a holonomy Yang-Baxter equation.
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