TITLE: AJ conjecture for links
QUESTION [8 upvotes]: Garoufalidis proposed a conjecture on $q$-difference equations for the colored Jones polynomials of knots.
\begin{equation}
\hat{A}_K(\hat{l},\hat{m};q)J_n(K;q)=0
\end{equation}
where the actions of the operators $\hat l$, $\hat m$ are defined by 
\begin{equation}
\hat{l}J_n(K;q)=J_{n+1}(K;q) , \quad \hat{m}J_n(K;q)=q^{n/2}J_n(K;q) \ .
\end{equation}
Are there analogous $\hat{A}$-polynomials which define $q$-difference equations for the colored Jones polynomials of links? If there are, they should be polynomials with $2n$ variables $\hat{l}_i, \hat{m}_i$ ($i=1,\cdots,n$) for a link with $n$ components. Are their actions to the colored Jones polynomials of links known?
More simply, is there a paper which expresses the classical $A$-polynomial $A(l,m)$ or the character variety for the Hopf link in $S^3$?

REPLY [9 votes]: I think Ian's answer is right on.  This is a comment that is too long to be a comment.

Let $L\subset S^3$ be a link of $k$-components.  We use $X(L)$ to denote the $SL(2,\mathbb{C})$ character variety of the fundamental group of $S^3-L$.  Let $X(T^2)$ denote the $SL(2,\mathbb{C})$ character variety of the torus.  There is a restriction map
$$X(K)\rightarrow X(T^2)^k $$  the image is an algebraic set, with ideal $A(L)$.  You can lift it to a $2^k$ fold branched cover by $\mathbb{C}^{2k}$.  Which corresponds to extending an ideal to get $\tilde{A}(L)\subset \mathbb{C}[l_i^{\pm 1},m_i^{\pm 1}]$.  The ideal $\tilde{A}(L)$ is not necessarily principle, so no $A$-polynomial, just an $A$-ideal. By Poincare-Lefschetz duality applied to cohomology with coefficents in the adjoint representation, away from the singular points the image of $X(L)$ in the character variety of the torus is complex Lagrangian with respect to the complex symplectic form $2\sum_i dl_i\wedge dm_i$.
The procedure outlined by Ian yields an ideal in the exponentiated Weyl algebra $W_k$ quantizing $\mathbb{C}[l_i,m_i]$ which would be the noncommutative $A$-ideal.
Define a function 
$$ J(L): \mathbb{N}^k \rightarrow \mathbb{Z}[t_i^{\pm 1}]$$ 
which assigns to each tuple of integers the Kauffman bracket of the framed link $L$ colored by the Jones-Wenzl idempotents.
By the argument in FGL that Ian mentions there is a pairing so that $J(L)$ annihilates $\tilde{A}(L)$.
On the other hand the exponetiated Weyl algebra $W_k$ acts on the space of functions $f:\mathbb{N}^k\rightarrow \mathbb{Z}[t_i^{\pm 1}]$ as explained in Garoufalidis and Le's first paper on the subject, and their proof shows $J(L)$ is holonomic.  Hence it is annihilated by an ideal that acts like the quantization of the ideal of a complex Lagrangian subvariety of the cotangent bundle of $\mathbb{C}^k$.

Modulo normalization the noncommutative $A$-ideal lives in the annihilator.  The content of the AJ conjecture is, how close does the noncommutative $A$-ideal come to being all of the annhilator of $J(L)$?