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 $S^3$?

  • $\begingroup$ Unless the link is a boundary link, there will be relations between meridians. Also, longitudes need not necessarily be in the second commutator subgroup of a link group, hence I'd guess they might act non-trivially. So I'd guess no, not in this form anyway. $\endgroup$ – Daniel Moskovich Jun 19 '12 at 0:38

I'm not sure there ought to be a straightforward generalization. One perspective on the $A$-polynomial is that it generates the kernel ideal of the map $R(T)\to R(M)$, where $T=\partial M$ is the torus boundary of a knot complement, and $R(T), R(M)$ are the rings of regular functions on the $SL_2(\mathbb{C})$ character varieties (this is not quite true, but basically up to a reparameterization). When you have a link instead of a knot, then you get a higher-dimensional variety as a product of character varieties of each boundary torus, so one should expect the ideal to be generated by several polynomials, not just one (and with no canonical choice of generators).

From the perspective of Frohman-Gelca-Lofaro, the non-commutative $A$-polynomial may be thought of as a generator of the kernel of the map $K(\partial M \times I)\to K(M)$, where $K(*)$ is the Kauffman bracket skein module (you might have to invert some elements in the rings to get something relating to the Garoufalidis-Le perspective) which is a non-commutative deformation of the character variety. So one probably expects an ideal of non-commutative polynomials lying in the skein module of the boundary of a link which annihilate the colored Jones polynomials associated to that link, instead of just a single non-commutative $A$-polynomial.

For the Hopf link, the $A$-ideal will just equate meridian with longitude, and longitude with meridian of each pair of components of the Hopf link, since the complement is homeomorphic to a torus $\times I$.


I think Ian's answer is right on. This is a comment that is too long to be a comment.

  1. 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 $2\sum_i dl_i\wedge dm_i$.

  2. 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.

  3. 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.

  4. By the argument in FGL that Ian mentions there is a pairing so that $J(L)$ annihilates $\tilde{A}(L)$.

  5. 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 $J(L)$?


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