Proof of a stronger version of the AJ conjecture for torus knots

TL;DR

This paper proves a stronger version of the AJ conjecture for all torus knots, linking recurrence relations of colored Jones polynomials to the A-polynomial, advancing understanding in quantum topology.

Contribution

It confirms Sikora's stronger AJ conjecture specifically for all torus knots, establishing a significant case in knot theory.

Findings

Confirmed the stronger AJ conjecture for all torus knots.

Established the relation between recurrence polynomials and A-polynomials for these knots.

Advances the understanding of quantum invariants in knot theory.

Abstract

For a knot $K$ in $S^3$, the $sl_2$-colored Jones function $J_K(n)$ is a sequence of Laurent polynomials in the variable $t$, which is known to satisfy non-trivial linear recurrence relations. The operator corresponding to the minimal linear recurrence relation is called the recurrence polynomial of $K$. The AJ conjecture \cite{Ga04} states that when reducing $t=-1$, the recurrence polynomial is essentially equal to the $A$-polynomial of $K$. In this paper we consider a stronger version of the AJ conjecture, proposed by Sikora \cite{Si}, and confirm it for all torus knots.