The colored Jones polynomials and the simplicial volume of a knot
Hitoshi Murakami, Jun Murakami
TL;DR
This work establishes a deep link between colored Jones polynomials, Akutsu–Deguchi–Ohtsuki generalized Alexander polynomials, and Kashaev invariants by showing that specialized AKO data reproduce $J_N$ at roots of unity and that Kashaev’s $R$-matrix is equivalent to the Jones $R$-matrix up to a scalar. Through a Vandermonde-type transformation, it derives an explicit form for the transformed $R$-matrix and demonstrates that Kashaev’s invariant and $J_N$ coincide, implying a unified framework for these quantum invariants. The authors then generalize the volume conjecture to simplicial volume, proposing $ig ig(Kig)= rac{2\,\pi}{v_3}\lim_{N o\infty} rac{ \, ig|J_N(K)ig|}{N}$, with implications for mutation invariance, connect-sum additivity, and Vassiliev invariants. If true, this program would imply that a knot is trivial iff all Vassiliev invariants vanish, connecting quantum invariants to classical knot-theoretic properties and offering a path to classify knots via asymptotic quantum data.
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.