The colored Jones polynomials and the simplicial volume of a knot

Abstract

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/2rr) when N goes to infinity, where Vol(K) is the hyperbolic volume of the complement of a knot K [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 Nth 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, such as the generalized multivariable Alexander polynomial defined by Y. Akutsu, T. Deguehi 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+l parameters: an integer N This research is supported in part by Sumitomo Foundation and Grand-in-Aid for Scientific Research, The Ministry of Education, Science, Sports and Culture. 86 H. MURAKAMI AND J. MURAKAMI and complex numbers pi (i= 1, 2, ..., c) decorating the components, where c is the number of components of the link. In the case where N=2, their invariant coincides with the multivariable Alexander polynomial, and their definition is the same as that of the second author [22]. Using the Akutsu-Deguchi-Ohtsuki invariants we have another coincidence. We will show that if all the colors are 89 then the generalized Alexander polynomial is the same as Kashaev's invariant since it coincides with the specialization of the colored Jones polynomial as stated above. Therefore the set of colored Jones polynomials and the set of generalized Alexander polynomials of Akutsu-Deguchi-Ohtsuki intersect at Kashaev's invariants. The paper is organized as follows. In the first section we recall the definition of the link invariant defined by Yang-Baxter operators. In w we show that the AkutsuDeguchi-Ohtsuki invariant coincides with the colored Jones polynomial when the colors 1 (N-1) by showing that their representation becomes the usual representation are all corresponding to the irreducible N-dimensional representation of sl(2, C). In w weshow that ...

Quantum Invariant for Torus Link and Modular Forms

“…It is known [14] that Kashaev's invariant is given from the colored Jones polynomial at a specific value h → 2 π i/N. By use of a result of Prop.…”

“…19, Zagier further studied a "strange identity" related to the half-derivatives of the Dedekind η-function, and clarified a role of the Eichler integral with half-integral weight. From the viewpoint of the quantum invariant, Zagier's q-series was originally connected with a generating function of an upper bound of the number of linearly independent Vassiliev invariants [17], and later it was found that Zagier's q-series with q being the N-th root of unity coincides with Kashaev's invariant [5,6], which was shown [14] to coincide with a specific value of the colored Jones function, for the trefoil knot. This correspondence was further investigated for the torus knot, and it was shown [3] that Kashaev's invariant for the torus knot T (2, 2 m + 1) also has a nearly modular property; it can be regarded as a limit q being the root of unity of the Eichler integral of the Andrews-Gordon q-series, which is theta series with weight 1/2 spanning m-dimensional space.…”

“…Using the Jones-Wenzl idempotent, we give an explicit formula of the invariant. It is known [14] that Kashaev's invariant coincides with a specific value of the colored Jones polynomial. This correspondence enables us to give an integral form of Kashaev's invariant for the torus link T (2, 2 m) in section 3.…”

Link polynomial calculus and the AENV conjecture

“…This allowed the authors to make a non-trivial prediction about the Ooguri-Vafa partition function of the Whitehead link in the volume conjecture limit [2][3][4][5]. As rightly noted in [1], the parallel recent progress 1 in knot/link polynomial calculus in [12][13][14][15][16][17]- [44] would allow one to test this conjecture by comparison with exact formulas for the corresponding HOM-FLY polynomials.…”

The Colored Jones Polynomial, the Chern–Simons Invariant, and the Reidemeister Torsion of a Twice–Iterated Torus Knot

“…R. Kashaev conjectured [16] that his knot invariant K N ∈ C introduced in [15] would grow exponentially with growth rate the volume of the knot complement S 3 \ K when the integer parameter N goes to the infinity if the knot K is hyperbolic. J. Murakami and the author [34] proved that Kashaev's invariant coincides with J N K; exp(2π √ −1/N ) and generalized Kashaev's conjecture to general knots. …”