Maybe you guys understood this already, but it just clicked for me today: the Poisson structure I was claiming had to exist on the knotty character varieties is exactly given by the Sabloff duality.

Recall the setup.  Let $\mathrm{\Lambda }$ be a Legendrian, $F,G$ objects in the sheaf or whatever else category.  We write $F_{+}$, etc, to mean what happens when you Reeb flow $F$ forward by $+ϵ$, and $F_{-}$ for flowing it backward.  [Recall that this makes sense: contact transformations act on the sheaf category.  Actually just at this moment I have this picture where the real place the microlocal monodromy lives is not on the Legendrian, but on what you get when you “follow it along the integral kernel” in the space where that integral kernel is defined.] Microlocal considerations (either in the old KS style or via the microlocalization equivalence) imply that

$$Hom(F,F_{+})=Hom(F,F)=Hom(F_{-},F)$$

In particular, there is a canonical morphism $F_{-}\to F$.  We can take the cone

$$F_{-}\to F\to Cone(F_{-}\to F)\to$$

Translating our knowledge from NRSSZ (which strictly speaking we know only in the case where the front plane is $R^2$, and know at the moment only by augmentation-theoretic arguments), we expect that

$$Hom(G,Cone(F_{-}\to F))=C^{\ast }(\mathrm{\Lambda })$$

for any $G\in C_1(\mathrm{\Lambda })$.  [  We need to give a sheaf theoretic proof of this!]

We expect this because we expect, for any $G\in C_1(\mathrm{\Lambda })$, the following exact triangle:

$$Hom(G,F_{-})\to Hom(G,F)\to C^{\ast }(\mathrm{\Lambda })\to$$

In addition, we expect the so-called “Sabloff duality”, which I at least don’t *really* understand from the sheaf-theoretic point of view, [  although probably I can and should translate Dan’s proof of it in the generating function setting into the sheaf world; alternatively it follows from NZ]:

$$Hom(F,G_{-})\simeq Hom(G,F{)}^{†}[-1–dim\mathrm{\Lambda }]$$

But taking these together, we have a map (now I set $F=G$)

$$Hom(F,F{)}^{†}[-1–dim\mathrm{\Lambda }]\to Hom(F,F)$$

In the case at hand, note this specializes to

$$(Ex{t}^1(F,F){)}^{\ast }\to Ex{t}^1(F,F)$$

i.e. a map from the cotangent space at $F$ to the tangent space at $F$.

Conjecture: This map gives a Poisson structure.  

(Worry: maybe being a Poisson structure has to do with the commutativity of the product on $Hom(F,F)$.)

Note that the degeneracy of this map has to do with the cohomology of the Legendrian.

Note also that in higher dimensions, this looks more or less like a shifted Poisson structure, at least if PTVV ever get around to defining what that should be.  (I think “Sabloff duality + the duality exact sequence” are more or less what the definition of a “CY category with boundary” should look like, and the moduli of objects in such a thing should be (shifted) Poisson, just like the moduli of objects in a CY category should be symplectic.)