Optimistic limits of Kashaev invariants and complex volumes of hyperbolic links

Abstract

Yokota suggested an optimistic limit method of the Kashaev invariants of hyper- bolic knots and showed it determines the complex volumes of the knots. His method is very eective and gives almost combinatorial method of calculating the complex vol- umes. 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 modied optimistic limit method. This new method works for any link diagrams, and it is more intuitive, easy to handle and has natural geometric meaning.