Abstract
The AJ Conjecture relates a quantum invariant, a minimal order recursion for the colored Jones polynomial of a knot (known as the $$\hat{A}$$
A
^
polynomial), with a classical invariant, namely the defining polynomial A of the $${\mathrm {PSL}_2(\mathbb {C})}$$
PSL
2
(
C
)
character variety of a knot. More precisely, the AJ Conjecture asserts that the set of irreducible factors of the $$\hat{A}$$
A
^
-polynomial (after we set $$q=1$$
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 $$\hat{A}$$
A
^
-polynomial that depends on a planar diagram of a knot (that conjecturally agrees with the $$\hat{A}$$
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.