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

Abstract

For a knot $K$ in $S^3$, the ${sl}_2$–colored Jones function $J_K\left(n\right)$ is a sequence of Laurent polynomials in the variable $t$ that 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 (see Garoufalidis [Proceedings of the Casson Fest (2004) 291–309]) 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 [arxiv:0807.0943], and confirm it for all torus knots.