Optimistic limits of Kashaev invariants and complex volumes of hyperbolic links

TL;DR

This paper introduces a more versatile and intuitive combinatorial method for calculating the complex volumes of hyperbolic links, extending Yokota's optimistic limit approach to all link diagrams.

Contribution

It develops a modified optimistic limit method that applies to any link diagram, removing previous restrictions and enhancing practicality and geometric understanding.

Findings

The new method works for all link diagrams.

It simplifies the calculation process.

It provides a natural geometric interpretation.

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.