On the Complex Volume of Hyperbolic Knots

Abstract

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.

Optimistic Limits of the Colored Jones Polynomials

“…6 + log wm wj log wk wj , 1 We remark that the Kashaev invariant of a knot K defined in [17] is the one of the mirror image K defined in [11]. This paper follows the definition of [17].…”

“…However, Yokota made a very useful way to determine the optimistic limit of a hyperbolic knot K in [17] and [18] by defining a potential function V (z 1 , . .…”

“…This paper follows the definition of [17]. For negative crossings: If no horizontal edge is collapsed at the positive nor the negative crossing, we assign any of P 1 , .…”

“…Now we describe the meaning of (17). Let n 1 be the endpoint of I, n 2 be the previous over-crossing point and n 3 be the previous under-crossing point.…”

“…This implies the colored Jones polynomial of a knot K here is the one of the mirror image K in [8]. This choice is natural to [17] and Theorem 1.3.…”