An Introduction to the Volume Conjecture

TL;DR

This paper introduces the Volume Conjecture, linking colored Jones polynomials to hyperbolic volume and invariants of knot complements and Dehn surgeries, providing foundational understanding for nonexperts.

Contribution

It offers an accessible overview of the Volume Conjecture, its generalizations, and elementary examples, bridging knot invariants with hyperbolic geometry.

Findings

Colored Jones polynomial limits relate to hyperbolic volume

Deformations of the polynomial connect to 3-manifold invariants

Elementary examples illustrate the conjecture's concepts

Abstract

This is an introduction to the Volume Conjecture and its generalizations for nonexperts. The Volume Conjecture states that a certain limit of the colored Jones polynomial of a knot would give the volume of its complement. If we deform the parameter of the colored Jones polynomial we also conjecture that it would also give the volume and the Chern-Simons invariant of a three-manifold obtained by Dehn surgery determined by the parameter. I start with a definition of the colored Jones polynomial and include elementary examples and short description of elementary hyperbolic geometry.