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.

Keywords:

• Volume conjecture
• Kashaev's conjecture
• colored Jones polynomial
• Chern-Simons invariant
• volume