Kashaev's Conjecture and the Chern-Simons Invariants of Knots and Links

Abstract

R.M. Kashaev conjectured that the asymptotic behavior of his link invariant, which equals the colored Jones polynomial evaluated at a root of unity, determines the hyperbolic volume of any hyperbolic link complement. We observe numerically that for knots 6 3 , 8 9 and 8 20 and for the Whitehead link, the colored Jones polynomials are related to the hyperbolic volumes and the Chern-Simons invariants and propose a complexification of Kashaev's conjecture.

Three-Dimensional Quantum Gravity, Chern-Simons Theory, and the A-Polynomial

“…Then, the moduli 17 Notice, that our normalization of the Chern-Simons invariant agrees with [52,55], but differs from the normalization used in [36] by a factor of 2π 2 .…”

“…The volume conjecture was extended further in [36], where it was shown that for a large class of knots one can remove the absolute value in (5.3), so that the following limit…”

Optimistic Limits of the Colored Jones Polynomials

“…Although it looks trivial, due to the ambiguity of the optimistic limit, only few results are known. It was numerically confirmed for few examples in [12], actually proved only for the volume part of two bridge links in [13] and for the Chern-Simons part of twist knots in [2]. In a nutshell, the purpose of this paper is to propose a general method to define the optimistic limit of the colored Jones polynomial of a hyperbolic knot K and to prove the following relation:…”

6j–symbols, hyperbolic structures and the volume conjecture

“…The VC in the above form has been formally checked for the Figure Eight knot (Murakami and Murakami [14]), the Borromean link, for torus knots (Kashaev and Tirkkonnen [11], Murakami and Murakami [15]), their Whitehead doubles (Zheng [26]) and for the family of "Whitehead chains" (Van der Veen [25]). Moreover, there is experimental evidence of its validity for the knots 6 3 , 8 9 and 8 20 (Murakami et al [16]). …”