Optimistic limits of Kashaev invariants and complex volumes of hyperbolic links

Abstract

Yokota suggested an optimistic limit method of the Kashaev invariants of hyperbolic knots and showed it determines the complex volumes of the knots. His method is very effective and gives almost combinatorial method of calculating the complex volumes. However, to describe the triangulation of the knot complement, he restricted his method to knot diagrams with certain conditions. Although these restrictions are general enough for any hyperbolic knots, we have to select a good diagram of the knot to apply his theory. In this article, we suggest more combinatorial way to calculate the complex volumes of hyperbolic links using the modified optimistic limit method. This new method works for any link diagrams, and it is more intuitive, easy to handle and has natural geometric meaning.

30 pages, 24 figures, 3 tables. Final version. To appear in J. Knot Theory Ramifications

References in corpus (5)

• On the volume conjecture for hyperbolic knots
• Volume Optimization, Normal Surfaces and Thurston's Equation on Triangulated 3-Manifolds
• Thurston's Spinning Construction and Solutions to the Hyperbolic Gluing Equations for Closed Hyperbolic 3-Manifolds
• Yokota theory, the invariant trace fields of hyperbolic knots and the Borel regulator map
• Optimistic limits of the colored Jones polynomials