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

Abstract

R. M. Kashaev conjectured that the asymptotic behavior of the link invariant he introduced, 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 {\small $6_3$, $8_9$ 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.