The strong AJ conjecture for the figure eight knot

TL;DR

This paper proves the strong AJ conjecture for the figure eight knot, linking quantum invariants and classical knot invariants, thereby extending the conjecture's validation beyond torus knots.

Contribution

The paper provides the first verification of the strong AJ conjecture for the figure eight knot, a significant non-torus knot, using quantum and classical invariant relations.

Findings

Confirmed the strong AJ conjecture for the figure eight knot.

Established connections between quantum A-ideals and recurrence relations.

Extended the class of knots for which the conjecture is verified.

Abstract

Motivated by the theory of quantum A-ideals of Frohman-Gelca-LoFaro, the theory of q-holonomicity of quantum invariants of Garoufalidis-Le and the AJ conjecture of Garoufalidis, Sikora formulated the strong AJ conjecture which relates the A-ideal and recurrence ideal of a knot in the 3-sphere. This conjecture has been verified for all torus knots and most of their cables. In this paper, we verify the strong AJ conjecture for the figure eight knot.