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.
Let K be a knot in the 3-sphere S3, N(K) the regular neighbourhood of K and E(K) = cl(S3−N(K)) the exterior of K. The tunnel number t(K) is the minimum number of mutually disjoint arcs properly embedded in E(K) such that the complementary space of a regular neighbourhood of the arcs is a handlebody. We call the family of arcs satisfying this condition an unknotting tunnel system for K. In particular, we call it an unknotting tunnel if the system consists of a single arc.
In this paper, we give a formula of the complex volumes of hyperbolic knot complements, which is related to the volume conjecture for hyperbolic knots.
We calculate limits of the colored Jones polynomials of the figureeight knot and conclude that in most cases they determine the volumes and the Chern-Simons invariants of the three-manifolds obtained by Dehn surgeries along it.
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