An introduction to the volume conjecture and its generalizations

TL;DR

This paper introduces the volume conjecture and explores its generalizations, analyzing how the asymptotic behavior of colored Jones polynomials relates to knot group representations, with examples like the figure-eight and torus knots.

Contribution

It provides an accessible overview of the volume conjecture and extends its discussion to various generalizations involving different parameters and knot types.

Findings

Asymptotic behaviors of colored Jones polynomials relate to fundamental group representations.

Illustrates these relations using the figure-eight and torus knots.

Highlights the significance of the special linear group over complex numbers in these relations.

Abstract

In this paper we give an introduction to the volume conjecture and its generalizations. Especially we discuss relations of the asymptotic behaviors of the colored Jones polynomials of a knot with different parameters to representations of the fundamental group of the knot complement at the special linear group over complex numbers by taking the figure-eight knot and torus knots as examples.