Volume Conjecture for Knots

• Main Authors: Murakami, Hitoshi (Author, http://id.loc.gov/vocabulary/relators/aut), Yokota, Yoshiyuki (http://id.loc.gov/vocabulary/relators/aut)
• Corporate Author: SpringerLink (Online service)
• Format: Electronic eBook
• Language: English
• Published: Singapore : Springer Singapore : Imprint: Springer, 2018.
• Edition: 1st ed. 2018.
• Series: SpringerBriefs in Mathematical Physics, 30
• Subjects: Mathematical physics.; Topology.; Hyperbolic geometry.; Mathematical Physics.; Hyperbolic Geometry.

The volume conjecture states that a certain limit of the colored Jones polynomial of a knot in the three-dimensional sphere would give the volume of the knot complement. Here the colored Jones polynomial is a generalization of the celebrated Jones polynomial and is defined by using a so-called R-mat...

Table of Contents:

• 1. Preliminaries (knots and links, braids, hyperbolic geometry)
• 2. R-matrix, the Kashaev invariant and the colored Jones polynomimal
• 3. Volume conjecture
• 4. Triangulation of a knot complement and hyperbolicity equation
• 5. Idea of the "proof"
• 6. Representations of a knot group into SL(2;C) and their Chern-Simons invariant
• 7. Generalization of the volume conjecture.