Lemma. Fix $\Lambda \subset T^{\infty, -} R$ for $\Lambda$ a collection of points at minus infinity. Fix a grading on these points, let $\mu$ be the corresponding Maslov potential. Split these into some pairs $(a_i, b_i)$ such that $a_i < b_i$ and some singletons $c_j$. Then
$$\bigoplus_i k_{(a_i, b_i]}[m_i] \oplus \bigoplus_j k_{(c_j, \infty]}[n_j] \in Sh_{\Lambda, \mu}(R, k)_0$$
The above sheaf is acyclic at both $\pm \infty$ iff there are no $c_j$.
Proposition. (Barranikov normal form). If $k$ is a field, then the above is a complete list of isomorphism classes of objects in $Sh_{\Lambda, \mu}(R, k)_0$.
Proposition. (Rutherford): Assume no two singularities of the front projection of $\Lambda$ have the same $x$ coordinate. Then there is a map $R: Sh_{\Lambda, \mu}(\mathbb{R}^2, k) \to NormalRulings(\Lambda)$ such that if two sheaves have different associated rulings, they are not isomorphic.
Proof. A ruling is determined by its restriction to vertical slices of the front projection which do not meet singularities. The ruling here is given by the involution in the Barranikov normal form, which only depends on the isomorphism class of the sheaf restricted to the line, hence the isomorphism class of the sheaf. To check that this indeed extends to a ruling, note that at the cusps, the two strands of the cusp must be paired by this involution, and at a crossing, the two strands of the crossing cannot be paired with each other. Normality is a similar calculation which can be made in the Morse complex category; the corresponding calculation for Morse complex sequences (and the above argument) was given by Dan in some paper.
Corollary. Counting isomorphism classes of objects in the augmentation category can be done for each ruling separately.
Remark. The same result with the same proof holds for knots in $J^1(S^1)$.
The rest of this post is a meditation on how we should think about “rulings” in other situations.
Let $\Lambda$ be a Legendrian in $T^\infty M$, let $\Phi \subset M$ be its front projection.
Definition. We write $\Pi_1(M, \Phi)$ for the group of paths in $M$ which only cross $\Phi$ transversely, considered up to isotopy amongst such paths.
Problem. Compute this groupoid.
Question. Is this actual $\Pi_1$ of something?
By definition, these paths are non-characteristic for any object in $Sh_{\Lambda}(M)$. That is, the conormal to such a path $p: [0,1] \to M$ does not intersect $\Lambda$. It follows that for any $F \in Sh_{\Lambda}(M)$, we have $p^* F \in Sh_{p^* \Lambda}([0,1])$. In fact, something stronger is true, $SS(p^* F) = p^* \Lambda$. Here, $p^* \Lambda$ means the projection to $T^\infty [0,1]$ of $p^* (\Lambda \cap T^\infty M)$.
Definition. We write $\Pi_{1,-}(M, \Phi)$ for the sub-groupoid of $\Pi_1(M, \Phi)$ consisting of paths $p$ such that $p^*\Lambda \subset T^{\infty,-} \mathbb{R}$.
Problem. Compute this groupoid.
Problem. (Something we tried a little while working on [STZ] but without any clear success): intepret the sheaf category in terms of $\Pi_{1,-}(M, \Lambda)$. I.e., give a version of the exit paths formalism for constructible sheaves which cuts down on the number of paths for microlocal reasons.
Question. Does this have anything to do with “transverse loop space” e.g. part 4 here:
Anyway, the point is that for any $p \in \Pi_{1,-}(M, \Lambda)$, we get a transform
$$p*: Sh_{\Lambda, \mu}(M, k) \to Sh_{p^*\Lambda, p^* \mu}([0,1], k)$$.
Definition. A ruling of a collection of points in $T^{\infty, -} \mathbb{R}$ carrying a Maslov potential $\mu$ is just an involution on these points which pairs $a < b$ only if $\mu(a) = \mu(b) – 1$.
Definition. A noncharacteristic homotopy is a map $h: [0,1]_x \times [0,1]_z \to M$ which is non-characteristic with respect to $\Lambda$, and such that the map restricted to every $x \times [0,1]$, and also to the boundary of the square, is also non-characteristic. Note that not all $x \times [0,1]$ need be in $\Pi_1(M, \Lambda)$ — some may pass through cusps and crossings but transverse to the normal cones of these things.
Pulling back along such a map gives a usual knot with front projection $[0,1] \times [0,1]$, and no vertical tangencies.
Example. For the front projection of a knot to $[0,1] \times [0,1]$ the identity map is a noncharacteristic homotopy.
Definition. A (normal) ruling on $\Lambda$ is a family of rulings on $p^* \Lambda$ over paths $p \in \Pi_{1,-}(M, \Lambda)$, compatible with composition, and such that for any noncharacteristic homotopy such that the pulled back front diagram is front-generic, the unique extension of the rulings on the vertical slices on $\Pi_{1, -}(M, \Lambda)$ to the corresponding two-dimensional front is in fact a (normal) ruling. Note this condition can be localized to neighborhoods of cusps and crossings.
Remark. The proof that a normal ruling in the old sense is a normal ruling in the new sense is presumably the same as the proof that normal rulings are an invariant of Legendrian isotopy. I haven’t tried to write such a proof though. Note this is a definition of normal ruling which does not even see planar isotopies. (Remark to DT & EZ: (!))
Question. Is there some more explicit way to say what a ruling / normal ruling is?
Corollary. A sheaf defines a ruling. The ruling of any sheaf in $Sh_{\Lambda, \mu}(M, k)$ is normal.
Proof. Noncharacteristic inverse images behave well, which reduces the question to the 2-d case, where Dan proved it.
Question. What about the converse?