Periodicity Properties of the Colored Jones Polynomial

TL;DR

This paper uncovers periodic patterns in the colored Jones polynomial at specific roots of unity, revealing new algebraic properties and relationships with link determinants.

Contribution

It identifies novel periodicity patterns of the colored Jones polynomial at roots of unity, expanding understanding of its algebraic structure.

Findings

Periodic pattern at second and third roots of unity

Value alternates between 1 and link determinant at -1

New periodic behavior at primitive third root of unity

Abstract

The "color" in the colored Jones polynomial is an integer parameter. In this paper, a periodic pattern of the values of the colored Jones polynomial at the second and the third roots of unity is found. If we substitute -1 to the colored Jones polynomial, the value is alternately 1 or the determinant of the given link. When it comes to substituting the primitive third root of unity, another periodic pattern appears.