A diagrammatic approach to the AJ Conjecture

Abstract

The AJ Conjecture relates a quantum invariant, a minimal order recursion for the colored Jones polynomial of a knot (known as the hat{A} polynomial), with a classical invariant, namely the defining polynomial A of the {mathrm {PSL}_2(mathbb {C})} character variety of a knot. More precisely, the AJ Conjecture asserts that the set of irreducible factors of the hat{A}-polynomial (after we set q=1, and excluding those of L-degree zero) coincides with those of the A-polynomial. In this paper, we introduce a version of the hat{A}-polynomial that depends on a planar diagram of a knot (that conjecturally agrees with the hat{A}-polynomial) and we prove that it satisfies one direction of the AJ Conjecture. Our proof uses the octahedral decomposition of a knot complement obtained from a planar projection of a knot, the R-matrix state sum formula for the colored Jones polynomial, and its certificate.