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 [Kashaev 951, 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 63, 89 and 820 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.