On the Annihilating polynomial of the Colored Jones Polynomial for Some Links

TL;DR

This paper explores polynomials derived from the colored Jones polynomial for links, relating them to the $A$-polynomial and $A_q$-polynomial, and formulates a link version of the AJ conjecture.

Contribution

It introduces link versions of the $A$-polynomial and $A_q$-polynomial through ideals from the colored Jones polynomial and proposes a link AJ conjecture.

Findings

Identifies polynomials as link analogs of the $A$- and $A_q$-polynomials.

Formulates the link version of the AJ conjecture.

Abstract

In this paper, we consider polynomials and ideals obtained from the colored Jones polynomial in both commutative and noncommutative cases. In the commutative case, this ideal contains polynomials that can be regarded as the link version of the $A$-polynomial; in the noncommutative case, it consists of annihilating polynomials of the colored Jones polynomial and can be regarded as the link version of the $A_q$-polynomial. Moreover, we formulate the link version of the AJ conjecture.