On Cornwell’s constructions

I want to record here some ideas on interpretations of the work of Chris Cornwell; these come from discussions with him and others.  Mostly I am concerned with the paper where he constructs representations of $\pi_1(\mathbb{R}^3 \setminus K)$ from augmentations of the Legendrian contact algebra $A(S^*\mathbb{R}^3; S^*_K \mathbb{R}^3)$ of the conormal torus.  This gives an inverse to a construction of Lenny.  The following other papers of Chris are also relevant.

I take an “augmentations are sheaves” or “family Floer” point of view.  In general, this can be done by taking Hom in the augmentation category with the fiber of a front projection.  One thing I want to point out is that just getting the local system can be done with much less work.

 

Theorem.  Let $M$ be a manifold, $N$ a closed submanifold.  There is a functor $Aug(A(S^* M; S^*_N M)) \to Loc(M \setminus N)$.

Proof.  Consider some point $p \in M \setminus N$.  Let $\epsilon: S^*_N M \to End(V)$ be an augmentation, and let $\delta_p$ be the canonical augmentation given by the filling $D^*_p M$ of $S^*_p M$.  A path from $p$ to itself lifts to a Legendrian isotopy from the link $S^*_p M \cup S^*_N M$ to itself (push $S^*_p M$ along the path), hence acts on $Hom(\delta_p, \epsilon)$.  Isotopic paths give homotopic actions, so we get an action of $\pi_1$ on $H^*(Hom(\delta_p, \epsilon))$, i.e. a local system.

Remark. Really one wants the local system of complexes, not its cohomology; to get this one has to know that in fact chains on the loop space give the appropriate higher homotopies.  Presumably this is actually known.

Remark.  A note on coefficients: since we are taking Hom between the cosphere fibre and something else, we probably have to work with coefficients in the group ring of $H_2(S^*M, S^*_N M \cup S^*_p M)$.  In particular, when $M = \mathbb{R}^3$, we can only take augmentations of $A(S^* \mathbb{R}^3; S^*_K \mathbb{R}^3)$ which kill the $H_2$ class of the fibre, i.e., set $Q = 1$.

Remark.  Since we are taking always Hom between different Legendrians, if one just wants the map at the level of sets and not as a functor, it is not necessary to have access to the augmentation category.

 

 

The above is a quick proof that one can get local systems from augmentations, but does not explain how in practice to compute them, i.e., how to compute the continuation maps as $p$ varies of $H^*(Hom(\delta_p, \epsilon))$.  Note that at least the Reeb chords which give a basis for this space have a natural interpretation: they are the straight line segments in $R^3$ from $p$ to $K$ which are normal to $K$.

On the other hand, Chris gives explicit formulas.  I sketch below some ideas about how to match these up.

Consider a knot $K$ which is braided (say with $n$ strands) around the unknot, which we take to be the unit circle in the $x, y$ plane.  Assume all the braiding happens near the negative $y$ axis, and remains very close to the $x, y$ plane.  Consider the $\mathbb{R}^2_{x, z}$ plane, and fix a point $p$ on the $z$ axis and with $z \gg 0$.  Fix an augmentation $\epsilon: A(S^* \mathbb{R}^3; S^*_K \mathbb{R}^3) \to gl(V)$.

We want to describe $L = Hom_{\mathbb{R}^3}(\delta_p, \epsilon)$, together with its action of $\pi_1(\mathbb{R}^3 \setminus K, p)$.  The basis of this vector space is the normal chords from $p$ to $K$; by the way we have chosen $p$ and $K$, these all lie $\mathbb{R}^2_{x, z}$; there are $2n$ of them.  I do not presently see what the holomorphic disks are, or what the “walls” in $\mathbb{R}^3$ where new chords or new disks appear should be.  A guess: these appear where a chord from $p$ to $K$ becomes coplanar with a bi-normal (i.e. projection of Reeb) chord of $K$.

 

 

Remark.  This is already somewhat interesting one dimension down, where instead of $K \subset \mathbb{R}^3$ we study the (conormals of) a collection of points $k_i$ in $\mathbb{R}^2$.  Then the critical moments can be seen to be when the the distinguished point $p$ crosses a line spanned by two of the $k_i$.  Across the Radon transform, the $p$ and the $k_i$ become period 1 sine waves on the annulus, and these critical moments are Reidemeister 3 moves.  There is also a critical behavior corresponding to when a segment connecting $p$ and $k_i$  meets one of these in its basepoint.

6 comments

  1. ericzaslow says:

    In the pure sheaf story, maybe the disks don’t appear directly. The front of a conormal of a point is a two-sphere (round if the point is at the origin), and the corresponding sheaf on $S^2\times \mathbb R$ has rank zero below this sphere and rank one above. As you move the point around, this sphere wobbles, so its homs with the fixed sheaf corresponding to the augmentation will change as well.

    Harder to see what is happening in the Fukaya category, as you say, since we won’t have an explicit Lagrangian to go with the augmentation. Nevertheless, the Reeb chords you describe are vertical lines in the fronts, and maybe one imagines a disk being a lift of a path on $S^2$ from the intersection points above to the base of the Reeb chord (as they appear for Legendrian knots).

    • Vivek Shende says:

      I didn’t mean that the disks appear in the sheaf itself, but rather that in order to compute the sheaf you have to wrestle with a wall crossing question having to do with disks appearing. That is, as you move $p$ around in $\mathbb{R}^3 \setminus K$, the stalk of your sheaf is $Hom(\delta_p, \epsilon)$, which usually will be locally constant as a complex, but in general is only cohomologically locally constant. The moments where the complex is not locally constant are when new Reeb chords and new disks appear.

      • ericzaslow says:

        I phrased my sheaf comment poorly. I didn’t mean to suggest that you had. My second paragraph is more germaine to your construction. I just thought it was cool to think about the Radon-transformed sheaves’ homs would vary. They do, but not in any way that evidences disks.

  2. lennyng says:

    I can write down $Hom_{\mathbb{R}^3}(\delta_p,\epsilon)$ (more precisely, the LCH differential for the link $S_p^*\mathbb{R}^3 \cup S_K^*\mathbb{R}^3$) algebraically. It has $n$ generators corresponding to Reeb chords of degree $0$, and another $n$ corresponding to Reeb chords of degree $1$. The differential (for the degree $1$ generators) involves the matrix $\Phi^R_B$ where $B$ is the braid for $K$.

    Intuitively, the bifurcations that produce holomorphic disks with more than $1$ negative puncture come from chords from $p$ to $K$ that pass through $K$ in their interior. See also my second knot contact homology paper (http://arxiv.org/abs/math/0303343) section 5.1. I think it follows from this that the wall crossings happen at points $p$ when a normal chord from $p$ to $K$ passes through $K$ in its interior. In any case, it should be possible to write down the action of $\pi_1(\mathbb{R}^3\setminus K,p)$ on $Hom_{\mathbb{R}^3}(\delta_p,\epsilon)$ explicitly in terms of the braid $B$.

    • Vivek Shende says:

      When you say “a normal chord from $p$ to $K$ which passes through $K$ in its interior”, are you requiring that this interior passage is normal to $K$, or any sort of crossing will do?

  3. lennyng says:

    Any sort of crossing will do: that way if we consider a 1-parameter family of chords (not normal) beginning at $p$ and ending on $K$, then a codimension-1 subfamily will pass through $K$ in this manner. This isn’t literally what the holomorphic disks look like, but I think that in some appropriate limit, the image of each holomorphic disk projected to $\mathbb{R}^3$ looks like the trace of such a 1-parameter family of chords, with possible bifurcations where a chord passes through $K$ in its interior.

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