Legendrian skein algebras and Hall algebras

We compare two associative algebras which encode the"quantum topology"of Legendrian curves in contact threefolds of product type $S\times\mathbb R$. The first is the skein algebra of graded Legendrian links and the second is the Hall algebra of the Fukaya category of $S$. We construct a natural homomorphism from the former to the latter, which we show is an isomorphism if $S$ is a disk with marked points and injective if $S$ is the annulus.


Introduction
This work relates two different constructions of associative algebras which capture the "quantum topology" of Legendrian curves in contact threefolds.

1) Legendrian skein algebras 2) Hall algebras of Fukaya categories of surfaces
The algebras 1) are defined by imposing linear relations between curves which differ in some small ball, while 2) involves first constructing an A ∞ -category whose structure constants count immersed disks and then passing to its Hall algebra, a kind of decategorification. We relate these by constructing a natural homomorphism from 1) to 2). Two previous works contain evidence of a general connection between (non-Legendrian) skein algebras and Hall algebras. Morton-Samuelson [MS17] show that the HOMFLY-PT skein of the thickened torus is isomorphic to a specialization of Hall algebra of the elliptic curve, and Cooper-Samuelson [CS] give a conjectural presentation of the Hall algebra of the Fukaya category by skein-like relations. Our original motivation was to better understand the general theory behind these results. While we make progress towards this goal, some aspects still remain more mysterious, such as the appearance of the HOMFLY-PT skein relations instead of the legendrian ones.

Skein modules
Let M be a contact threefold with oriented contact distribution, i.e. a smooth threefold equipped with a completely nonintegrable oriented rank two subbundle ξ ⊂ T M . We use a variant of the skein module for graded curves which depends on an additional choice of grading structure on M : a rank one subbundle of the contact distribution ξ. The skein module of M is then defined as the Z[q ± , (q − 1) −1 ]-module generated by isotopy classes of graded embedded closed Legendrian curves In the simplest case, when M is the standard contact R 3 , it follows from work of Rutherford [Rut06, Theorem 3.1] that skein module is freely generated by the empty link. Thus, the class of an arbitrary link L is equal to R L ∅ for some R L ∈ Z[q ± , (q − 1) −1 ], which is, up to change of variables and normalization (described in Subsection 4.1), the graded ruling polynomial of L. This invariant of graded Legendrian links can be defined more directly by counting graded normal rulings of the front projection of L, see Chekanov-Pushkar [PC05]. Thus, for general threefold M the image of a link in the skein can be viewed as the appropriate generalization of a knot polynomial.
The skein module has an algebra structure in the case where M = S × R with contact form p * 1 θ + p * 2 dz where p 1 , p 2 are the projections to S and R respectively, θ is a 1-form on S with dθ = 0 pointwise (a Liouville form), and z is the standard coordinate on R. Also the grading structure should be pulled back from a foliation η on S. The product L 1 L 2 is defined by "stacking L 2 on top of L 1 ", i.e. translating L 2 in sufficiently far in the positive z-direction so that it is entirely above L 1 and then taking the union of the two links.
We also allow the following extension of our setup. Suppose that S has boundary and pick a discrete subset N ⊂ ∂S. Allow links L which are compact Legendrian curves with ∂L ⊂ N × R.

The Fukaya category and its Hall algebra
For the purpose of defining the Fukaya category we assume that S is a compact surface with boundary and that the Liouville form θ is chosen so that its dual vector field points outward along ∂S. As above, we have a finite set N ⊂ ∂S and a foliation η on S which provides the grading. Given a choice of ground field K one defines two variants of the Fukaya category, F = F(S, N, θ, η, K) and F ∨ = F ∨ (S, N, θ, η, K), whose objects are compact graded Legendrian curves L with K-linear local system E of finite rank and Maurer-Cartan element (formal deformation) δ ∈ Hom 1 ((L, E), (L, E)) >0 , and where for F we require ∂L ⊂ (∂S \ N ) × R and for F ∨ we require ∂L ⊂ N × R. The setup is described in more detail in Subsection 3.1. Let us make two remarks to relate this to the existing literature. First, in the approach to Fukaya categories of surfaces based on arc systems or ribbon graphs, see for example [STZ14,DK18,HKK17], the category F is defined as a homotopy colimit, while F ∨ is defined as a homotopy limit in the category of dg-categories up to Morita equivalence. Second, in the Legendrian knot theory literature Maurer-Cartan elements and rank one local systems appear in a different guise as augmentations of the Chekanov-Eliashberg DGA, see also Subsection 3.2.2.
Hall algebras were first considered by Steinitz [Ste01] and later rediscovered by Hall [Hal59].
Their definition immediately generalizes to abelian categories satisfying suitable finiteness conditions.
For the case of triangulated dg-categories one needs to modify the naive definition to take into account negative Ext-groups, as was pointed out by Toën [Toe06]. Conceptually, one replaces groupoid cardinality with homotopy (∞-groupoid) cardinality as defined by Baez-Dolan [BD01].
Let us state the definition used here. Assume C is an extension closed A ∞ category over a finite field K = F q such that Ext k (A, B) is finite-dimensional for all A, B ∈ Ob(C), k ∈ Z, and vanishes for k less than some constant depending on A, B. These conditions are satisfied for F ∨ (S, N, θ, η, K) if K is a finite field. Define Hall(C) to be the algebra with underlying Q-vector space with basis the set of isomorphism classes of objects in C and product where [A] denotes the basis element of Hall(C) corresponding to the isomorphism class of the object A. We should note that this is not the formula of Toën in [Toe06], but gives an isomorphic algebra after rescaling the basis vectors. Instead we are following the conventions of Kontsevich-Soibelman [KS08], specialized to the finite setting. The difference comes in regarding the elements of the Hall algebra either as functions or as measures. We adopt the latter view.
As a general remark, there are some limitations to using the version of the Hall algebra based on counting, as opposed to motivic/cohomological variants [KS08]. In the setting of Fukaya categories of punctured surfaces one encounters only union of tori (Artin-Tate motives), so a more sophisticated approach would require introducing a lot of machinery for a rather small payoff. It would however, via the Serre polynomial, give structure constants depending on a formal variable q instead of the number of elements of the finite field.

Main result
We fix a compact surface S with boundary, N ⊂ ∂S, a Liouville form θ on S, a grading structure η, and a finite field K as before. If L is a graded Legendrian link in S × R, then the element Φ(L) ∈ Hall(F ∨ ) we assign to it is given, up to scalar factor, by the sum of all objects supported on L. More precisely, we set Φ(L) = (q − 1) −|π 0 (L)| q −e(L) where e(L) ∈ Z is a self-intersection number, the first sum runs over all rank one K-linear local systems on L and MC(L, E) is the set of Maurer-Cartan elements. In the main text (Subsection 4.2) we give a more conceptual definition and prove the explicit formula above. The following is our central result, see Theorem 4.3 in the main text. An obvious question is whether this homomorphism is injective and/or surjective. The dimension of the Hall algebra in the way it is defined here depends in general on the size |K| of the finite field, while the dimension of the skein algebra does not. In practice, one usually passes to a subalgebra of the Hall-algebra whose dimension is independent of |K|, so perhaps the image of Φ should be viewed as a better behaved substitute for the full Hall algebra. In those cases where all objects of F ∨ are rigid, Φ does have a chance to be an isomorphism and indeed we show: Legendrian satellite invariants and counts of representations of the Chekanov-Eliashberg differential algebra, see [NR13,LR], can presumably be extended or at least given additional justification by our result and its extension to the Z/n-graded context.
More generally, we propose the following conjecture.
Conjecture 1.4. The algebra homomorphism is injective for general surface S.
Work in progress by Ben Cooper and the author aims to prove this using gluing techniques for Hall algebras and skein algebras.

Categorification
In recent years, Fukaya category-type constructions have been increasingly applied to Legendrian knots [BC14, STZ17, NRS + ]. To a Legendrian knot L in R 3 one assigns its augmentation category C(L) 1 whose objects are augmentations of the Chekanov-Eliasherg DGA. This category has a geometric interpretation as (a rank one part of) the Fukaya-Seidel category with boundary condition L. By a result of Ng-Rutherford-Shende-Sivek [NRSS17] the homotopy cardinality of C(L) 1 is the graded ruling polynomial R L . Thus, the category C(L) 1 is a kind of categorification of the knot polynomial of L. The full category C(L), which was defined in [STZ17] in terms of constructible sheaves, should be thought of as categorifying the satellite invariants of L.
For more general contact threefolds, the generalization of the ruling polynomial is the image of the link in the skein. By our main theorem this element in the skein is categorified by a functor F : C(L) 1 → F ∨ from the category of the link to the Fukaya category in the sense that the pushforward along F of the (weighted) counting measure on C(L) 1 gives the element Φ(L) in Hall(F ∨ ). The following table summarizes this discussion.
Classical Categorical 1.4 Further directions and speculation 1.4.1 Z/n-grading In this work we restrict throughout to Z-graded curves, but it seems plausible that everything extends to the Z/n-graded case. The skein relations (S1), (S2), (S3) make sense for Z/(2n)-graded curves and with some tweaks one can get the odd case as well, see [Rut06]. The boundary skein relations (S1b), (S2b) do not immediately work in the periodic case though, and probably require more radical modification. One the other hand, while Z/(2n)-graded versions of the Fukaya category exist, defining the Hall algebra of say, a Z/2-graded triangulated category is a famous problem.
Approaches of Bridgeland [Bri13] and Kontsevich [Kon18] require additional structure and are thus not intrinsic to the periodic category itself. At least in the cases where N = ∅, i.e. S does not have any marked points on the boundary, the definition of the Z/(2n)-graded skein is clear, and so a good test for any proposed definition of the Hall algebra of the Z/(2n)-graded Fukaya category would be if the analogs of our results hold.
1.4.2 The q = 1 limit Our skein relations assign (q − 1) −1 to the unknot, and so do not immediately specialize to the classical limit q = 1. However, this could be seen as just a defect of our particular choice of model (Z[q ± ]-submodule) and we expect that a more suitable one can be found using ideas of Turaev [Tur91]. Ideally, we would like to have a definition of the "Fukaya category of S over F 1 " and its Hall algebra and compare this to the specialization of the correct model of the skein algebra.
Perhaps the two are isomorphic.
One can be much more precise when replacing triangulated categories by their abelian subcategories. Fukaya categories of surfaces often have bounded t-structures whose hearts are categories of representations of quivers with quadratic monomial relations [HKK17]. Categories of representations of quivers over F 1 as well as their Hall algebras can be defined, see the work of Szczesny [Szc12] and also the very general approach of Dyckerhoff-Kapranov [DK]. The idea is that the category of vector spaces over F 1 is the category of pointed sets (X, x 0 ) and maps (X, x 0 ) → (Y, y 0 ) are functions f : X → Y with f (x 0 ) = y 0 and f is injective away from the preimage of y 0 . In this way one gets a category which has many of the features of Abelian categories but where Hom is just a pointed set. On the skein side one can restrict to those links which give objects in the heart of the chosen t-structure on the Fukaya category. For this submodule one already has the correct model and the q = 1 limit gives the above Hall algebra of the category of representations over F 1 .
Extending this to the full triangulated category remains an intriguing problem. The most basic question is what the right (for our purposes) notion of a "triangulated category over F 1 " is.

Higher dimensions
The construction which assigns to a Legendrian link an element in the Hall

Outline
Section 2 provides background on A ∞ -categories and their Hall algebras. We also discuss filtered A ∞ -categories with curvature and prove some basic results about them which are needed later.

Definitions
We will use the language of A ∞ -categories throughout, as these naturally appear in symplectic topology. The purpose of this subsection is to review some basic definitions and fix notations and sign conventions, adopting those which are common in the Fukaya category literature, e.g. Fukaya- Oh-Ohta-Ono [FOOO09] or Seidel [Sei08b]. For an introduction to A ∞ -categories see Keller [Kel06] and for a more thorough account Lefèvre-Hasegawa [LH03].
All our categories will be small, or at least essentially small, and linear over a fixed field K. An A ∞ -category A over K is given by a set Ob(A) of objects, a Z-graded vector space Hom(A, B) for each pair of objects A, B ∈ Ob(A), and structure maps of degree 2 − n, for each n ≥ 1, satisfying the A ∞ -relations (2.1) i+j+k=n (−1) a k +...+ a 1 m i+1+k (a n , . . . , a n−i+1 , m j (a n−i , . . . , a k+1 ), a k , . . . , a 1 ) = 0 where a := |a| − 1 is the degree in the bar resolution.
We will require A ∞ -categories to be strictly unital for convenience. This means that there is a morphism 1 A ∈ Hom 0 (A, A) for each A ∈ Ob(A) such that Strictly unital A ∞ -categories with m k = 0 for k ≥ 3 correspond to dg-categories via Indeed, the first three A ∞ -relations correspond to d 2 = 0, the Leibniz rule, and associativity of the product.

Twisted complexes
There is a canonical way of enlarging an A ∞ -category to include extensions by any sequence of objects. Let A be an A ∞ -category and A 1 , . . . , A n ∈ Ob(A). An upper triangular deformation of A = A 1 ⊕ · · · ⊕ A n or twisted complex is given by morphisms δ ij ∈ Hom 1 (A j , A i ), i < j, forming a strictly upper triangular matrix, δ, such that the A ∞ Maurer-Cartan equation holds. Here, the structure maps m k are extended to matrices in the natural way, i.e.
(m k (δ, . . . , δ)) i 0 ,i k := Twisted complexes form an A ∞ -category, Tw(A), which contains A as a full subcategory by mapping A to its trivial deformation with δ = 0. A morphism, a, of degree d from (A, α) to (B, β) is given by elements a ij ∈ Hom d (A j , B i ). Structure maps are given by "inserting δ's everywhere":

Curved A ∞ -categories
In this subsection we discuss the formalism of curved A ∞ -categories. A general deformation of an A ∞ -category has, in addition to the structure maps m n , n ≥ 1, curvature terms m 0 (A) ∈ Hom 2 (A, A) satisfying a generalization of the usual A ∞ -equations, see (2.4). Such categories typically do not have well defined homotopy categories, since m 2 1 = 0, but there is a way of removing the curvature by "recomputing" the set of objects. The new objects correspond to solutions to the A ∞ Maurer-Cartan equation. Since there are infinitely many terms involved, some topology is needed, and we will consider those coming from R-filtrations.

R-filtrations
By a decreasing R-filtration on a vector space V we mean a collection of subspaces V ≥β ⊂ V for An R-filtered vector space is a vector space V with decreasing R-filtration as above, which gives a topology on V as usual.
Let C be a curved A ∞ -category. A Maurer-Cartan element or bounding cochain for X ∈ Ob(C) where the sum converges since δ ∈ Hom 1 (X, X) ≥ǫ for some ǫ > 0 by our assumptions on R-filtrations, and structure maps are contracting. Denote by MC(X) ⊂ Hom 1 (X, X) >0 the set of Maurer-Cartan elements (which could be empty). Define C to be the (filtered, uncurved) A ∞ -category whose objects are pairs (X, δ) with X ∈ Ob(C) and δ ∈ MC(X), morphisms Hom C ((X, δ), (Y, γ)) := Hom C (X, Y ) and structure maps m k obtained by inserting the Maurer-Cartan elements as in (2.3).
We will also consider the category C ≥0 which has the same objects as C, morphisms and structure maps are restrictions of those of C. Finally, the category C 0 has the same objects as C, morphisms and structure maps induced from C. We then have a diagram of uncurved A ∞ -categories and functors (2.7) The functor F is given on objects by (X, δ) → X, i.e. forgetting the Maurer-Cartan element, and on morphism is the quotient map while the functor G is the identity on objects and the inclusion on morphisms.

Transporting Maurer-Cartan elements
Having defined curved A ∞ -categories, the next goal is to establish some properties of the functor The following proposition is an Inverse Function Theorem-type result.
Proposition 2.1. The functor C ≥0 → C 0 is conservative: A closed map f in C ≥0 is an isomorphism if and only if its reduction modulo Hom >0 in C 0 is an isomorphism.
A note on terminology: A closed morphism f ∈ Hom 0 (X, Y ) in an A ∞ -category is called an isomorphism if its image in the homotopy category, i.e. in Ext 0 (X, Y ), is an isomorphism. This is equivalent to f having an inverse up to homotopy.

Proof.
A map is an isomorphism if and only if its cone is a zero object. In case Cone(f ) does not exist in C, we can formally add it as a two-step twisted complex. Thus it suffices to show that if X ∈ Ob (C 0 ) is a zero object then (X, δ) ∈ Ob( C ≥0 ) is a zero object for any δ ∈ MC(X). To show this, we construct a series converging to an element h ∈ Hom −1 C (X, X) ≥0 with m 1 (h) = 1 X . Suppose we already have an h ∈ Hom −1 C (X, X) ≥0 such that (2.8) m 1 (h) = 1 X mod Hom ≥β then the goal is to find h ′ ∈ Hom −1 C (X, X) ≥β such that h + h ′ solves (2.8), but modulo terms in Hom ≥2β , i.e.
The right-hand side of the above equation is in Hom ≥β , but also m 1 -closed, so existence of h ′ follows from acyclicity of the complex Hom ≥β /Hom ≥2β . To see this, suppose x ∈ Hom k ≥β such that m 1 (x) ∈ Hom k+1 ≥2β , then (2.9) x = m 2 (x, 1) = m 2 (x, m 1 (h)) = −m 1 (m 2 (x, h)) mod Hom ≥2β so x is a boundary. To finish the proof, an inductive argument and completeness of Hom C (X, X) give the desired h.
The following proposition allows us to transport Maurer-Cartan elements along isomorphisms in C 0 .
Proposition 2.2. The functor F : C ≥0 → C 0 has the isomorphism lifting property: If X, Y ∈ Ob(C), δ ∈ MC(X), and f 0 ∈ Hom C 0 (X, Y ) is an isomorphism, then there exist a γ ∈ MC(Y ) and Proof. The idea is to construct a countable sequence of increasingly better approximations of f and γ and make use of completeness of the filtrations on Hom-spaces. By Proposition 2.1 it suffices to ensure that f is closed and f = f 0 mod Hom >0 -such f is then automatically an isomorphism.
and f is invertible up to terms in Hom ≥β . We want to find γ ′ ∈ Hom 1 (Y, Y ) ≥β and f ′ ∈ Hom 1 (X, X) ≥β such that γ + γ ′ and f + f ′ solve the above equations, but modulo terms in Hom ≥2β , not just Hom ≥β . Modulo Hom ≥2β , the nonlinear terms in γ ′ and f ′ vanish, and we are left to solve By assumption, f has a homotopy inverse g up to terms in Hom ≥β which implies that the induced map is a homotopy equivalence of chain complexes with homotopy inverse y → m 2 (y, g). Now, since the A ∞ -equations hold for m n , we have in particular that m 1 ( m 0 (Y )) = 0 and m 2 ( m 0 (Y ), f ) = m 1 ( m 1 (f )), i.e. ϕ( m 0 (Y )) is a boundary. Hence, since ϕ is a chain homotopy equivalence, there is a γ ′′ which solves (2.10). Furthermore, m 1 (f ) + m 2 (γ ′′ , f ) is then closed mod Hom ≥2β , so again using the fact that ϕ is a chain homotopy equivalence we can find a closed γ ′′′ and f ′ such that thus γ ′ := γ ′′ + γ ′′′ solves both equations (2.10) and (2.11).

Gauge equivalence
Given X ∈ Ob(C) there is a sort of gauge group action on the set of Maurer-Cartan elements MC(X). The analog of the gauge group is on which m 2 gives a not necessarily associative composition. Also consider for δ, δ ′ ∈ Hom 1 (X, X) >0 the set , is the set of isomorphisms (X, δ) → (X, δ ′ ) which map to 1 in C 0 . We say that δ, δ ′ ∈ Hom 1 (X, X) >0 are gauge equivalent if I(δ, δ ′ ) = ∅ and write MC(X)/G X for the set of gauge equivalence classes in MC(X).
Proof. By definition 1 + x ∈ I(δ, δ ′ ) if and only if which we write as ).
This can be used to inductively solve for δ ′ , since x ∈ Hom 0 (X, X) ≥ǫ for some ǫ > 0, thus if δ ′ has been determined mod Hom ≥β , then the right hand side is determined mod Hom ≥β+ǫ . It is also clear that δ ′ is uniquely determined by x and δ.
Note that as a consequence of the above Lemma we have for any δ ∈ MC X , where δ ′ ∼ δ means gauge equivalence.

Homotopy cardinality
We begin with some remarks to put the definitions in this subsection into context. Suppose X is a space with π k (X) finite for all k ≥ 0 and trivial for k ≫ 0. The homotopy cardinality of X is . By the homotopy hypothesis, homotopy types of spaces correspond to equivalence classes of ∞-groupoids. A higher category C has an ∞-groupoid I(C) of isomorphism, so one can, under finiteness conditions, "count" objects of C using (2.12). In particular if C is a dg-or A ∞ -category, then see [TV07] for the case of dg-categories. Since we are not interested here in the actual space I(C) but only its homotopy cardinality, we will simply define everything in terms of Ext-groups.
So let C be an A ∞ -category over a finite field F q . Denote by Iso(C) the set of isomorphism classes of objects in C, and given A ∈ Ob(C) denote its class in Iso(C) by [A]. We say that C is locally In this situation the weighted counting measure on Iso(C) assigns to the singleton {X} ⊂ Iso(C) the rational number where Aut(X) ⊂ Ext 0 (X, X) is the group of automorphisms of X. We think of the vector space QIso(C) of finite Q-linear combinations of elements of Iso(C) as the space of (signed, Q-valued) finite measures on Iso(C). While µ C is in general not finite, we can use it to identify the space of finitely supported functions with the space of finite measures via f → f µ C .
An A ∞ functor F : C → D induces a linear map If furthermore C and D are linear over F q and locally left-finite, and F has the property that for Consider the special case when F : C → * is the functor to the final A ∞ -category, * , with a single object and Hom k = 0, in particular QIso( * ) = Q. In order for F ! to be defined we need C to be locally finite and have only finitely many objects up to isomorphism. Then F ! (1) is the weighted counting measure and F * F ! (1) ∈ Q is the homotopy cardinality of C. We remark that if elements of QIso(C) are interpreted a functions rather then measures, one should instead use F * , which is pullback of functions along the map Iso(C) → Iso(D), and F ! which sends the delta function at , c.f. [Toe06]. Our next goal is to establish a simpler formula for F ! for a special class of functors. Assume as before that C, D are A ∞ -categories over a finite field F q which are locally left-finite and that the induced map Iso(C) → Iso(D) is finite-to-one. Furthermore, we require that: 2) F has the isomorphism lifting property: Given an isomorphism f : By the first assumption on F we have an exact sequence of cochain complexes and thus long exact sequences Lemma 2.5. Let F : C → D be an A ∞ functor satisfying the above conditions, then By the isomorphism lifting property of F there exists an object Y ∈ Ob(C) together with an isomorphism the class of f in Aut(Y ), and we get in this way an action of Aut(Y ) on F Y . The set of orbits is while the stabilizer of X ∈ F Y is the image of the map Aut(X) → Aut(Y ). In particular F Y is a finite set for any Y . Note also the exactness of the sequence where the first map is given by f → 1 X + f and we use the assumption that F reflects isomorphisms.
We conclude that On the other hand, the long exact sequence (2.13) gives Combing all this,

Counting Maurer-Cartan elements
Suppose C is a curved A ∞ -category over a finite field F q and which is locally left-finite. Let F : C ≥0 → C 0 and G : C ≥0 → C be the functors as in (2.7). The pull-push gives a map We want to show that this is well-defined and find a simpler formula. By Proposition 2.1 and Proposition 2.2 we may apply Lemma 2.5 to the functor F to conclude that for any X ∈ Ob(C 0 ).
Proposition 2.6. Let C be a curved A ∞ -category over a finite field F q which is locally left-finite on the chain level, i.e. Hom k (X, Y ) is finite dimensional and vanishes for k ≪ 0, then where F is the canonical functor C ≥0 → C 0 .
Proof. By Lemma 2.3 we have for any δ ∈ MC X , but Lemma 2.4 shows that each of the sets I(δ, δ ′ ) is either empty or isomorphic to I(δ, δ), thus Note that

Hall algebra
Let C be a locally left-finite A ∞ -category over a finite field F q . Assume furthermore that C is closed under extensions and has a zero object. Then we have a diagram of categories and functors where C A 2 is the category of exact triangles in C, whose objects can be concretely represented by , which exists in C by assumption. Passing to QIso(C), the pull-push along the diagram gives a product map Using Lemma 2.5 one derives the following explicit formula for the product, which can also be deduced from [Toe06]. (2.14) The vector space QIso(C) together with this product is called the Hall algebra of C, denoted Hall(C). This is an associative algebra (see below) with unit [0], where 0 ∈ Ob(C) is a zero object.
As noted above, we think of elements of QIso(C) as measures, following the convention of Kontsevich-Soibelman [KS08, Section 6.1]. Toën [Toe06] uses instead F * and G ! , consistent with the point of view that elements of the Hall algebra are functions. Multiplication by the weighted counting measure, f → f µ C , defines an isomorphism between the two Hall algebras.
Proposition 2.7. The Hall algebra is associative.
Proof. The proof below is adapted from [KS08] with some simplifications. Passing to a quasiequivalent category, we may assume that local left-finiteness holds on the chain level, i.e. each Hom i (X, Y ) is finite-dimensional and vanishes for i ≪ 0, then Fix a triple of objects A 1 , A 2 , A 3 ∈ Ob(C) and consider the set X 123 of upper triangular deforma- triples a 12 , a 13 , a 23 , a ij ∈ Hom 1 (A j , A i ), with m 1 (a 12 ) = 0, m 1 (a 23 ) = 0, m 1 (a 13 ) + m 2 (a 12 , a 23 ) = 0, c.f. (2.6). Since C is assumed to be closed under extensions, each element of X 123 gives an object in C up to isomorphism. We have and similarly which completes the proof.
The idea in the above proof generalizes to give a formula for the product [A 1 ] · · · [A n ] in terms of twisted complexes. Other proofs of various flavors appear in [Toe06], [XX08], and [DK].
Example 2.8. Let C be the category with only the zero object, then Hall(C) = Q.
Example 2.9. Let C = Perf(F q ) be the category of finite-dimensional complexes of vector spaces over F q . Then Hall(C) has generators The first is obtained from and similarly for the second.

Slicings
The underlying vector space of the Hall algebra often admits a tensor product decomposition coming from a slicing. This notion was introduced by Bridgeland [Bri07] and generalizes that of a tstructure. More precisely, a slicing of a triangulated category C is given by a collection of full additive subcategories C φ such that 3) Every E ∈ C has a Harder-Narasimhan filtration: A tower of triangles The HN-filtrations are unique as a consequence of the other axioms. As an example, any bounded t-structure can be interpreted as a slicing with C φ = 0 for φ / ∈ Z. A slicing is part of the data of a Bridgeland stability condition, however most slicings do not come from stability conditions.
We consider slicings which satisfy the additional condition that For example, if A is a hereditary abelian category (Ext ≥2 = 0) and C = D b (A), then the slicing defined by the standard bounded t-structure on C has this property. The condition (2.15) implies that all Harder-Narasimhan filtrations are split, so for some scalar c, where the product is taken in the Hall algebra. We can conclude that, as a vector where the natural map from the right-hand side to the left-hand side is given by where A i ∈ C φ i and φ 1 > . . . > φ n . What is really new, to our knowledge, is the general relation between smoothing of intersections and Maurer-Cartan elements which we discuss in Subsection 3.3. Certain foundational issues in defining Fukaya categories of surfaces will not be addressed in detail here and we instead refer the reader to the reference given above.

Setup and conventions
For our purposes it will be essential to have a version of the Fukaya category which is defined over arbitrary base field and Z-graded, so that its Hall algebra is defined. To provide an overview we start by listing the data which enters into the definition. This is essentially the setup from [HKK17], except that we also want an explicit choice of Liouville 1-form, which was suppressed there.
is outward pointing along ∂S (c.f. see [Sei08a]). We can find such a θ provided S is orientable and every component has non-empty boundary. The 1-form θ provides a contact form α = p * 1 θ + p * 2 dz on S × R where z denotes the standard coordinate on the second factor and p 1 : S × R → S, p 2 : S × R → R are the projection maps.
The grading structure on S is needed to define the Z-grading on morphisms of the Fukaya category and is given by a section η of the projectivized tangent bundle P(T S), i.e. a foliation on S. The section η provides each fiber of P(T S) with a basepoint, so there is a well-defined fiberwise universal cover which we denote by P(T S). Given such a choice, there is a notion of a graded curve, which is an immersed curve γ : I → S together with a sectionγ of γ * P(T S) such thatγ(t) is a lift of the tangent space to the curve at γ(t). Thus, locally, there is a Z-torsor of choices of gradings.
An immersed Legendrian curve L in S × R projects to an immersed curve in S, so it makes sense to speak about gradings of L. More intrinsically, we could replace T S in the above discussion by the rank two subbundle ξ = Ker(α) ⊂ T (S × R) cut out by the contact form, which is canonically identified with p * 2 T S. Given the above data we will sketch the definition of the partially wrapped Fukaya category F(S, N, θ, η, K) and the infinitesimally wrapped category F ∨ (S, N, θ, η, K). If every component of ∂S contains an element of N , then these two categories turn out to be isomorphic. An object of either category is given by an graded Legendrian curve L in S × R together with a local system of finite-dimensional K-vector spaces E on L. We require L to be compact and embedded in S ×R with ∂L ⊂ ∂S × R. In fact, for now we will also assume that the projection of L to S is also embedded and deal with the more complicated immersed case later. Additionally ∂L should be either disjoint from N × R or contained in it. In the former case, (L, E) belongs to F(S, N, θ, η, K), while in the latter it belongs to F ∨ (S, N, θ, η, K).
Before defining morphisms in the Fukaya category, we need a few more remarks about graded curves.

Grading
Analogously to how one can assign ±1 to a transverse intersection point of oriented submanifolds, one can assign an integer to a simple crossing of graded curves. Let L 0 = (I 0 , γ 0 ,γ 0 ) and L 1 = (I 1 , γ 1 ,γ 1 ) be graded immersed curves with transverse intersection at x = γ 0 (t 0 ) = γ 1 (t 1 ). Then define the intersection index where we use the fact that even though P(T x S) is not canonically identified with R, it does have a total order (since S is oriented) and action of Z. If p ∈ S such that there are unique t 0 ∈ I 0 and t 1 ∈ I 1 with p = p 1 (γ(t 0 )) = p 1 (γ(t 1 )) then we also write i p (L 0 , L 1 ) for i(L 0 , t 0 , L 1 , t 1 ).
When depicting graded curves in the plane we may as well assume that η is the horizontal foliation. A grading on an immersed curve γ : I → R 2 = C is then specified by a function φ : I → R with e πiφ(t) tangent to γ(t). To specify φ it suffices to label segments of γ where n = ⌊φ(t)⌋ is constant by that integer, see Figure 1.
all odd morphisms get their sign reversed. Thus, a morphism is formally an equivalence class of If either p 1 (L 0 ) is not transverse to p 1 (L 1 ) or both L 0 and L 1 have boundary, then it is necessary to perturb L 0 as graded Legendrian curve, which is equivalent to perturbing its projection to S by a Hamiltonian diffeomorphism. In particular this is always necessary when L 0 = L 1 . Up to quasiisomorphism, the resulting complex is independent of the choice of perturbation. Let us describe how to perturb near the boundary, first for objects in F ∨ . Endpoints of L 0 in N ×R should be moved by a small amount along ∂S in the direction of the natural induced orientation on the boundary. to be contractible (filtration preserving) is closed and hence holds in the limit.

Structure maps
The A ∞ structure maps of the Fukaya category are defined in terms of immersed polygons with boundary on the given Lagrangian curves. More precisely let L k , k = 0, . . . , n be graded Legendrian curves intersecting transversely and let x k ∈ p 2 (L k ) ∩ p 2 (L k+1 ), k = 0, . . . , n − 1 and x n ∈ p 2 (L 0 ) ∩ p 2 (L n ). Consider a smoothly immersed n + 1-gon φ : D → S, up to reparameterization, such that φ sends the k-th corner of D to x k and the side of D from the (k − 1)-st to the k-th corner to L k , see and if X k := (L k , E k ) are objects of the Fukaya category we put (3.2) m n : Hom(X n−1 , X n ) ⊗ · · · ⊗ Hom(X 0 , X 1 ) → Hom(X 0 , X n ), m n := where the sum is over all intersection points and immersed disks up to reparameterization as above.
For the above to be well-defined, we need to know that the set of disks with fixed L 0 , . . . , L n and x 0 , . . . , x n−1 is finite. For this, the assumption that all curves involved are Legendrian is essential.
Suppose first that the L i are not infinitely wrapped around cylindrical ends, hence compact. The area of an immersed disk φ : D → S as above is where z(L k , x k ) is the z-coordinate of L k over x k and we use the fact that θ = −dz along Legendrian curves in S × R. In particular, since z is bounded along all curves, the area of D is bounded by some constant depending only on the L k . But if there were infinitely many polygons, their areas would necessarily tend to infinity, regardless of whether the L k are Legendrian. In the case of infinite wrapping there can be infinitely many disks, but only finitely many for fixed choice of intersection points x 0 , . . . , x n−1 , so the structure maps are still well-defined.

Relation to ribbon graph approach
In [HKK17] the category F was defined using a choice of arc system, i.e. decomposition of the surface into polygons, and was also shown to be Morita equivalent to a certain homotopy colimit over a ribbon graph as a special case of the "Lagrangian skeleton" approach to Fukaya categories proposed by Kontsevich [Kon09]. For the approach using arc systems it is also convenient to allow S to have corners and replace set N ⊂ ∂S of marked points by marked intervals connecting corners of S. Morphisms were defined in [HKK17] as paths along the boundary instead of explicitly perturbing the arcs and taking intersection points. There are also some minor differences in convention between [HKK17] and the present paper, regarding the grading, direction of wrapping, and signs.
An arc system provides a generator of F given by the direct sum of all arcs (with arbitrary grading and trivial rank one local system). i.e. to find the image under the Yoneda-embedding, amounts to intersecting L with all the arcs in the arc system and counting disks which have one side on L and the remaining sides on the arcs.
We will not use the equivalence of the two approaches directly, except in the case of the disk and annulus (see Section 4).

Immersed curves and Maurer-Cartan elements
If we allow immersed curves, in particular ones bounding immersed 1-gons ("teardrops"), we get a curved A ∞ -category. According to the general philosophy, the true objects depend on an additional choice of Maurer-Cartan element.
Let L ⊂ S × R be an embedded graded Legendrian curve with boundary in ∂S ⊂ N × R.
The projection p 1 (L) ⊂ S is not required to be embedded, but should have only transverse selfcrossings. Also fix a local system of vector spaces E on L. The additional choice alluded to above is a δ ∈ Hom 1 ((L, E), (L, E)) >0 satisfying the A ∞ Maurer-Cartan equation. In order the define or compute Hom((L, E), (L, E)), we need to perturb L slightly to some L ′ so that p 1 (L ′ ) is transverse to p 1 (L). Fortunately, the positive part Hom((L, E), (L, E)) >0 has an alternative definition which does not require us to perturb L. This is already evident from Chekanov's definition in [Che02] which does not require a choice of perturbation. Namely, define Hom((L, E), (L, E)) >0 as a direct sum over self-intersection points of p 1 (L) with summands as in Subsection 3.1.2, going from the upper branch over the self-intersection to the lower branch, and define the structure of a (non-unital) curved A ∞ -algebra on this filtered graded vector space as in Subsection 3.1.3 by counting immersed disks with boundary on L and corners at self-intersection points. This is called the augmentation category in [NRS + ]. It is clear from the definition that there is a natural functor C(L) 1 → F ∨ . As the notation suggests, there is a bigger category C(L) where E is allowed to have arbitrary rank and be Z-graded, so C(L) is independent of the grading on L. A category which turns out to be equivalent to C(L) was defined in [STZ14], for L ⊂ R 3 , in terms of constructible sheaves with singular support on the front projection of L.

Example: Trefoil
To illustrate the definition we consider the simple but non-trivial example of the (right-handed) trefoil knot L, see Figure 5. Equip L with a rank one local system E with monodromy λ ∈ K × which is trivialized away from some point on the right tear-drop in Figure 5. A basis of Hom((L, E), (L, E)) >0 is given by the self-intersection points u, v, x, y, z where |u| = |v| = 2 and |x| = |y| = |z| = 1. Looking for possible immersed disks one finds the following non-zero terms of the A ∞ -structure:

Relation to the Chekanov-Eliashberg DGA
We begin with a general algebraic construction. Let A be a curved A ∞ -algebra (curved A ∞ -category with a single object) such that A = A >0 . Assume that A is finite-dimensional, then m k = 0 for ⊗k is a differential graded algebra.
If we apply the above to A := Hom((L, E), (L, E)) >0 we obtain (an algebra isomorphic to) the Chekanov-Eliashberg algebra of L, more precisely the refined version of [EES05], with formal parameters t i specialized to the monodromy of E. This is clear at least over Z/2, since both definitions involve counting the same disks. Presumably, the signs can be made to agree as well and the statement is true in arbitrary characteristic.

Smoothing intersections
In this subsection we discuss the relation between smoothing of intersection points and Maurer- The graded Legendrian curves L + , L − , L s are well-defined up to isotopy in a neighborhood of (x, z).
See Figure 6. The reader may wonder whether L s really lifts to a Legendrian curve. When reconnecting the two strands in S there could a difference in z-value of the endpoints lifts, however this difference must be small since L has a self-intersection not just in S but in S × R. This jump in z-value can be fixed by adding a bump in one direction or another to one of the strands.
L + L − L s Figure 6: Resolving a self-intersection. Suppose a local system of vector spaces E is given on L − or equivalently L + and let E i be the fiber of E over t i , i = 1, 2. For any choice of isomorphism g : E 0 → E 1 we also get a local system E g on L s by identifying E 0 and E 1 via g for the left branch of L s and via −g for the right branch of L s . both components of L meeting at p are not closed. In this case we can assume that the local system E on L − is trivial and g = δ p = 1 is the identity matrix. We claim that the morphisms α 1 + α 2 and β 1 − β 2 (see Figure 7) are inverses isomorphisms (to zeroth order) between (L − , E, δ) and (L s , E g ).
If two of the four paths starting at p eventually meet, then there is one additional crossing of the projections of L − and L s to S. There are six different cases to consider, see the top two rows in Figure 10. If all four of the paths starting at p eventually meet, then there are two additional crossings of the projections of L − and L s to S. There are three different cases to consider, see the To complete the proof we apply Proposition 2.2 with X = (L − , E, δ p ), Y = (L s , E g ), and f 0 = α 1 + α 2 . In general the proposition would not give a unique γ ∈ MC(Y ), however in our case we have Hom(X, X) >0 = Hom(Y, Y ) >0 and the map φ constructed in the proof of the proposition is an isomorphism, not just a homotopy equivalence.

Boundary point
Suppose L = (I, γ,γ) is a graded Legendrian curve where γ : I → S × R is an embedding except for a transverse self-intersection point at (x, z) := γ(t 0 ) = γ(t 1 ) with x ∈ N ⊂ ∂S on the boundary. We order t 0 , t 1 so that in the clockwise order the strand of L belonging to t 0 comes before the strand belonging to t 1 , i.e. the t 0 strand is the upper one in Figure 11. Assume furthermore that the gradingγ is such that i(L, t 0 , L, t 1 ) = 1

Legendrian skein algebras
This section contains the heart of the paper. We introduce the skein algebra in Subsection 4.1 using the relation given in the introduction. Some comments about the front projection are also found Figure 10: Showing that α 1 + α 2 is closed. There are similar pictures for β 1 − β 2 .
here. In Subsection 4.2 we define the homomorphism Φ from the skein algebra to the Hall algebra of the Fukaya category. The main point is to show that the relation (S1) holds, which uses the result of the previous section. Subsection 4.3 discusses Legendrian tangles in preparation for the final two subsections, where we specialize to the case of a disk and an annulus, respectively, in which case we can say more about Φ. Remark 4.1. While Legendrian skein modules have not previously been explicitly considered in the literature, the defining relations are well-known to experts in Legendrian knot theory and appear (in slightly different but equivalent form) for example in work of Rutherford [Rut06]. Non-Legendrian skein modules, on the other hand, have been studied extensively. Let us mention here just one particularly intriguing results due to Turaev [Tur91], that the HOMFLY-PT skein algebra of a product threefold S × [0, 1] is a quantization of the Goldman Lie algebra, which acts on moduli spaces of local systems.

Skein relations
We explain briefly how to relate our skein relations to the ones in [Rut06] which give the graded ruling polynomial as it is usually defined in the Legendrian knot theory literature. Given a graded Legendrian link L in S × R without boundary define its writhe as the following signed number of crossings of the projection to S: n This is easily seen to be an isotopy invariant by checking Reidemeister moves. If we replace L by q 1 2 w(L) L then the skein relation (S1) becomes, after multiplication with q ± 1 2 , (S1')

Front projection
So far we have been depicting Legendrian links using the Lagrangian projection, i.e. the projection to the (x, y)-plane, from which the Legendrian curve can be recovered by integration. Generally, the front projection, which is the projection to the (x, z)-plane, turns out to be more useful since the Legendrian curve can be recovered simply by looking at the slope: y = dz/dx, at least where dx = 0 on the curve. For a generic Legendrian curve the front projection has three types of singularities, left cusps (≺), right cusps (≻), and transverse crossings (×), away from which the projection is the graph of a smooth function z = f (x). In particular, unlike for non-Legendrian knots, it is not necessary to indicate which strand passes over the other, though we will often do so for convenience.
Here is what the skein relations look like under front projection. (S1)

From Skein to Hall
Fix data S, N , θ, η, and K := F q as usual. The goal of this subsection is to define a homomorphism of Q-algebras We begin by defining the image of a graded Legendrian link L ⊂ S × R in the Hall algebra.
Recall that to L we attach the A ∞ -category C(L) 1 of rank one local systems on L together with Maurer-Cartan element. This category has finitely many objects and a weighted counting measure for which we give a more explicit formula below. We define Φ(L) by pushing µ C(L) 1 forward along the functor Before stating the following lemma, we assign an integer e(L) to a generic graded Lagrangian link L = (I, γ,γ). For each self-crossing x ∈ Cr(L) of p 1 (L) we have an intersection index i x := i(L, t 0 , L, t 1 ) ∈ Z where p 1 (γ(t 0 )) = p 1 (γ(t 1 )) = x and p 2 (γ(t 0 )) > p 2 (γ(t 1 )). Define e(L) as the number of x ∈ Cr(L) with i x (L) ≤ 0 and even, minus the number of x ∈ Cr(L) with i x (L) ≤ 0 and odd.
Lemma 4.2. Let L be a graded Legendrian link, then the formula holds, where E ranges over all isomorphism classes of rank one local systems on L.
Proof. The weighted counting measure µ C on an A ∞ -category C can be written as µ C = A ! (1) where A : C → 0 is the functor to the final category with only the zero object. If we factor the functor C 1 (L) → 0 through G : C(L) 1 → C(L) 1,0 , where C(L) 1,0 has morphisms Hom ≥0 /Hom >0 as in Subsection 2.2, we get Isomorphism classes of objects in C(L) 1,0 correspond to isomorphism classes of rank 1 local systems on L, so the weighted counting measure has the simple form Combining this with the formula for G ! from Proposition 2.6 and noting that essentially by definition, we obtain (4.2).
Our main result is the following. of Q-algebras.
The proof will be completed in the remainder of this subsection.

Skein relations in the Hall algebra
(S1). Possibly performing a rotation by a right angle, we may assume that m ≤ n. Let us first look at the case where m = n. In the notation of Subsection 3.3, we have an immersed graded Legendrian curve L with single self-intersection over p ∈ S and its three resolutions L + , L − , and L s , which are, from left to right, the three links which appear in the relation (S1). Given a rank one local system E on L + let E 0 (resp. E 1 ) be the fiber of E over the upper branch (resp. lower branch) crossing over p. Also, for a Maurer-Cartan element δ ∈ MC(L + , E) let δ p ∈ Hom(E 0 , E 1 ) be the component of δ belonging to p. The main idea is to distinguish the case δ p = 0 and δ p = 0.
We have a corresponding sum in degree zero. We need to show that the second summand is (q − 1)Φ(L s ).
Rank one local systems over K are classified by first cohomology with coefficients in K × . There are surjective pullback maps g i : H 1 (L; K × ) → H 1 (L i ; K × ) where i ∈ {+, −, s}. The map g i is (q − 1) ǫ + -to-1, where ǫ i = 1 if the two branches of L i near the intersection point belong to the same component, and ǫ i = 0 if they belong to distinct components. Given E ∈ H 1 (L + , K × ) and δ ∈ MC(L + , E) with δ p = 0 we can use δ p to identify the fibers of E over p to get a local system over L, which in turn pulls back to a local system over L s and a corresponding δ ∈ MC(L s , E) by Proposition 3.2. Taking into account the different sizes of the H 1 (L i ; K × ) we get which proves the claim since e(L + ) = e(L s ) and The proofs of (S1b) and (S2b) are similar to (S1) and (S2), respectively.

Tangles
Formally, a graded Legendrian tangle is a graded Legendrian link We will refer to them simply as tangles, and generally use the front projection to depict them. The front projection is also implicit in the language used below: The positive x-axis points to the right, the positive z-axis points up. Denote the left (resp. right) boundary of a tangle L by ∂ 0 L ⊂ R (resp. ∂ 1 L ⊂ R). These are subsets of R with grading, i.e. a labeling of the points by integers. Besides vertical composition -stacking -of tangles, there is a horizontal composition, well-defined up to isotopy, for tangles L 0 , L 1 with matching ends, i.e ∂ 1 L 0 isotopic to ∂ 0 L 1 .
Remark 4.4. One way of summarizing the situation is to say that there is a braided monoidal category whose objects are finite graded subsets of R up to isotopy and morphisms from X to Y are graded Legendrian tangles L up to isotopy with ∂ 0 L = X and ∂ 1 L = Y . Composition in this category is horizontal composition of tangles, while the monoidal product is given by vertical composition. This monoidal category has an equivalent description in terms of an object of D b (k) with a pair of complete flags in the derived sense. For more details see [Hai].
All tangles are obtained from the following elementary tangles under vertical and horizontal composition. Second, a permutation braid is a tangle without cusps and with any pair of strands crossing at most once. The first condition ensures that the tangle defines a permutation σ : ∂ 0 L → ∂ 1 L, and the ungraded braid is determined up to Legendrian isotopy by σ. One way to construct the braid corresponding to σ ∈ S n is to draw straight lines in the front projection. From this it is clear that the number of crossings of the braid is equal to inv(σ), the number of inversions of the permutation. Proof. If L is an arbitrary tangle then we need to show that it may be written, as an element of the skein, as a linear combination of tangles of the form described in the statement of the proposition.
We show this by induction on the number |∂L| of endpoints of L.
As a first step we show that L is a linear combination of tangles which have, from left to right, a sequence of right cusp tangles, a braid (tangle without cusps), and a sequence of left cusp tangles.
Let us take care of the left cusps first. The strategy, following Rutherford [Rut06], is to focus on the right-most left cusp, which is part of some maximal left cusp tangle C. If there are no more right cusps or crossings to the right of C we can continue by induction with the part of the tangle to the left of C, which has less boundary points. Otherwise, look at the basic tangle immediately to the right of C. A case-by-case analysis shows that using Reidemeister and skein moves we can always reduce to tangles with fewer crossings to the right of the right-most left cusp. We refer to [Rut06] for details. Apply the same strategy to right cusps.
The second step is to reduce from braids to permutation braids. Use induction on the number of crossings of the braid, so we need to analyze the following situation. Suppose P is a permutation braid and we compose it on the left with a basic tangle B with a single crossing of the i-th and (i + 1)-st strands. There are two cases depending on B. If the strands of P starting at the i-th and and (i + 1)-st boundary point on the left of P do not cross in P , then the composition BP is a permutation braid. Otherwise, if the strands do cross in P , we can apply third Reidemeister moves to P to move this crossing to the left, so P = BP ′ and we want to simplify BBP ′ . We apply the skein relation (S1) to BB which allows us to write BBP ′ as a linear combination of BP ′ , P ′ , and a tangle CC ′ P ′ with a right cusp in C and a left cusp in C ′ , where some terms may not be present depending on the grading. The tangles BP ′ = P and P ′ are permutation braids and induction takes care of C ′ P ′ which has two boundary points less.

Disk
In this subsection we consider the special case where S is a disk with n + 1 = |N | marked points on the boundary. For concreteness we take S to be the closed unit disk in C and N = {p 0 , . . . , p n } the set of (n + 1)-st roots of unity, where p k := exp(2πik/(n + 1)). Since S is simply connected, all grading structures are equivalent. For brevity write Skein := Skein(S, N, θ, η).
The Fukaya category F = F(S, N, θ, η, K) ∼ = F ∨ (S, N, θ, η, K) is equivalent to the bounded derived category D b (Rep(A n )) of representations of any A n quiver, see for example [HKK17]. We will not use this equivalence directly here, but for computing Hall(F) we use two facts about F. The first is the classification of indecomposable objects and the second is a particular slicing.
Isomorphism classes of indecomposable objects in F are indexed by elements of the set where the object E i,j ∈ Ob(F) corresponding to (i, j) ∈ I has an underlying curve whose projection to S is an arc (say, straight line) connecting p i and p j with grading such that 2) φ ∈ [0, 1) if and only if −(n + 1)/2 ≤ i + j < (n + 1)/2.
Here φ ∈ [0, 1) corresponds to the grading where the arc is labeled by "0" with our usual conventions.
The precise Legendrian lift of the straight line in S to S × R will be irrelevant. Also, all K-linear rank one local systems on E i,j are of course isomorphic.
The category F admits a slicing (F φ ) φ∈R where F φ has indecomposable objects E i,j with i+j n+1 = φ. Each F φ is semisimple category with simple objects represented by parallel disjoint arcs. This slicing comes, up to reparametrization of R, from the stability condition for the quadratic differential exp(z n+1 )dz 2 , see [HKK17]. The stability condition is also characterized, up to a C × factor, by the fact that it is preserved by Aut(F) up to the action of C × , i.e. lies at the orbifold point in the space of stability conditions. Theorem 4.6. For the disk with n + 1 marked points on the boundary, Φ is an isomorphism.
Thus, the graded Legendrian skein algebra of the disk with n + 1 marked points on the boundary, specialized at a prime power q, is isomorphic to the Hall algebra of the bounded derived category of representations of the A n quiver over F q .
Lemma 4.7. Skein is generated, as an algebra, by the straight the line segments E i,j , (i, j) ∈ I.
Proof. When using the front projection it will be convenient to adopt a half-plane model where S is the closed right half-plane {Re(z) ≥ 0} ⊂ C and p k = − √ −1k, k = 0, . . . , n. Thus, under front projection, a strand ends at p k if the slope at the boundary is −p k . In depictions of Legendrian links one can simply draw the strands as horizontal near the boundary (omitting a tiny "bend" at the end) and label them by k if they end at p k . From this discussion it is also clear that, after breaking the cyclic symmetry of the disk with marked points, there is a map from the skein of tangles L with ∂ 1 L = ∅ and endpoints in ∂ 0 L labeled by {0, . . . , n} to the skein of the disk.
We will need to modify links by moving endpoints past each other in the front projection. If they are labeled by the same integer, that is end at the same p k , then the boundary skein relation (S1b) holds. If i > j then which is just Legendrian isotopy and does not use any skein relations, and if i < j then by (4.5) and (S1). (Warning: The integer labels i, j here refer to the endpoints p i , p j and not the grading as in (S1b).) Let A ⊆ Skein be the subalgebra generated by straight line segments and let B k ⊂ Skein be the submodule generated by links with ≤ k endpoints. We will prove by induction that B k ⊂ A for all k, which implies the statement of the lemma since k B k = Skein.
By Proposition 4.5 the submodule B k is generated by links L with |∂L| ≤ k and no left cusps under front projection. Suppose L is such a link and decompose it into basic tangles. If left-most basic tangle of L has a crossing, we can use (4.5), (4.4), or the boundary skein relation (S1b) to write L in terms of links with fewer crossings and possibly less endpoints. Thus, by induction, it remains to deal with the case where the left-most basic tangle of L has a right cusp. Let p i (resp. p j ) be the endpoint of the lower (resp. upper) strand starting at that cusp. If i = j we can remove the left cusp using the boundary skein relation (S2b) to get a link with fewer endpoints. If there are no strands below the cusp, then L is a product of the link with just the cusp, which is isotopic to a straight line segment, and a link with fewer endpoints. If there is a strand below the cusp ending at p k then there are the following cases to consider.
• If k < i or k > j then we can move the cusp past the strand below it by an isotopy.
• If k = i or k = j then we can move the cusp past the strand below it possibly modulo links with less endpoints by the boundary skein relation (S1b).
• If i < k < j then we can move the cusp past the strand below it modulo links with a left cusps which connects p r , p s with |r − s| < |i − j|.
This is easier to see in the Lagrangian projection. Thus by induction we eventually reduce to one of the base cases above.
Lemma 4.8. Skein is spanned, as a module, by products of the form Proof. This follows from the previous lemma and its proof. It is convenient to return to the disk model and the Lagrangian projection where the E i,j are straight lines. The boundary skein relation (S1b) ensures that we can change the order of two straight line segments modulo terms where the total lengths of the segments is strictly smaller.
Proof of Theorem 4.6. We will define a linear map and show that it is inverse to Φ. Let E ∈ Ob(F), then it has a decomposition The summands in this decomposition and their order are unique up to permutation of those E i k ,j k with the same i k + j k . Let We first check ΦΨ = id. Let

Annulus
In this subsection we look at the case S = R/Z × [−1, 1], N = ∅, θ = −ydx, and η = ∂/∂x. Let where The image of Φ can be described more explicitly and we will do so below after making appropriate definitions. We will prove Theorem 4.9 by finding an explicit basis of the skein algebra. A basis of the ordinary (non-Legendrian) skein algebra was found by Turaev [Tur88,Tur91]. The Legendrian case is a bit more subtle, for example the algebra is noncommutative.
Generators of the skein algebra are closed curves which wind around the annulus some number of times. To introduce some notation, let C k be the Legendrian curve as in Figure 13 which winds around the annulus k > 0 times and with grading function φ with range in (−1/2, 1/2). To see that the curve depicted in Figure 13 really is the Lagrangian projection of a Legendrian curve, at least up to planar isotopy, we need to show feasibility of a system of inequalities [Che02]. Let A i , i = 1, . . . , k − 1 be the area of the i-th region (from the top) cut out by the projection of C k and h i > 0, i = 1, . . . , k − 1 the difference in z-coordinates of the strands over the self-crossings points, then by the Legendrian condition  where m ≥ 0, n i ∈ Z, n 1 ≥ n 2 ≥ . . . ≥ n m , k i > 0, and k i ≥ k i+1 if n i = n i+1 .
The proof of this proposition has two parts. The first is to show that the elements (4.7) span the skein, which uses Proposition 4.5 for tangles, and the second is show linear independence, for which we will use the homomorphism Φ to the Hall algebra. Lemma 4.11. Closures of tangles L with ∂ 0 L = ∂ 1 L sorted span the skein of the annulus.

Spanning the skein
Proof. Given an arbitrary tangle L we need to show that its closure is a linear combination of closures of tangles with sorted boundary. We will prove this by induction on the disorder δ = δ(∂ 0 L). In the base case δ = 0 there is nothing to show. Assume δ > 0, then the there is a pair of neighboring points x 1 < x 2 in ∂ 0 L with m := deg(x 1 ) > deg(x 2 ) =: n. Let 1 b (resp. 1 t ) be the tangle with horizontal strands corresponding to points in ∂ 0 L which are below x 1 (resp. above x 2 ). In the skein of tangles we have L = q −(−1) m−n L(1 b ⊗ (σ m,n σ n,m ) ⊗ 1 t ) + δ m,n+1 (q − 1)L(1 b ⊗ (ρ n λ n ) ⊗ 1 t ) by (S1). The closure of the first tangle is equal to the closure of (1 b ⊗ σ n,m ⊗ 1 t )L(1 b ⊗ σ m,n ⊗ 1 t ) and the closure of the second tangle is equal to the closure of (1 b ⊗ λ n ⊗ 1 t )L(1 b ⊗ ρ n ⊗ 1 t ), both of which have boundary with strictly smaller disorder. Proof. Let L be a tangle of the form described in Proposition 4.5, with a sequence of right cusp tangles, a permutation braid, and a sequence of left cusp tangles. Suppose L has a right cusp, then looking at the endpoints of the two strands starting at that cusp we see that ∂ 0 L is not sorted, similarly for left cusps and ∂ 1 L. Thus, if L has sorted boundary, then it is just a permutation braid. Moreover, by definition of a permutation braid any pair of strands crosses at most once, so if ∂ 0 L = ∂ 1 L then no pair of strands with different grading can cross, thus L is a vertical composition of permutation braids each which is made up of strands of the same degree, and so that degree increases when going upwards in the front projection. Proposition 4.5 and Lemma 4.11 thus imply the claim.
It remains to show that the closure of a permutation braid with strands of the same degree is a linear combination of braids as in (4.7) with n i = 0. A somewhat indirect argument which utilizes the existing literature goes as follows. Fix n and consider those braids in the skein of tangles which are have n strands, all of which are in degree zero. Among these the defining relations of the Iwahori-Hecke algebra H n hold: The quadratic relation (S1) and the braid relation aka is the companion matrix. (The companion matrix is characterized as k × k-matrix, up to similarity, by having minimal polynomial x k + a k−1 x k−1 + . . . + a 1 x + a 0 .) Proof. The morphism space Hom((C k , E), (C k , E)) >0 , where E is any local system on C k , is concentrated in degree one with basis the k − 1 self-intersection points of C k , so MC(C k , E) = Hom((C k , E), (C k , E)) >0 , for which we get a basis by trivializing E away from the point at the very bottom of C k . To find the representation of K[x ± ] corresponding to a given choice of monodromy and Maurer-Cartan element we need to intersect with the arc L = {0} × [−1, 1], which gives a complex concentrated in a single degree with basis y 1 , . . . , y n , and look for triangles with edges on a perturbed copy of L, L, and C k , see Figure 14. We get CMAT(a 0 , a 1 , . . . , a k−1 ) where −a 0 is the monodromy of E and a i , i = 1, . . . , k − 1, is the coefficient of the Maurer-Cartan element at the i-th self-intersection point from the bottom.
In order to prove injectivity of Φ we will consider the full subcategory Mod un (K[x ± ]) of Mod fd (K[x ± ]) consisting of those modules where x acts by a unipotent endomorphism U , or equivalently the spectrum of U is concentrated at 1 ∈ K. To simplify the notation write Mod un := Mod un (K[x ± ]) and  where m ≥ 0, n i ∈ Z, n 1 ≥ n 2 ≥ . . . ≥ n m , k i > 0, and k i ≥ k i+1 if n i = n i+1 .
To relate this to the elements in (4.7) indexed by the same set, we proceed as follows. Let where k i , n i are as in Proposition 4.10 and Lemma 4.14.
Proof. First of all, J ! (Φ(C k )) = (q − 1) −1 I k by Lemma 4.13 and the fact that the companion matrix is unipotent only for a single choice of a i 's, in which case we get a single Jordan block.
It is not true for general A, B ∈ D b (Mod fd ) that either, so both sides of (4.10) vanish. In the second case, Ext 1 (B, A) = 0, the product in the Hall algebra has only one term coming from the direct sum so (4.10) holds by a similar reasoning.
Proof of Theorem 4.9 and Proposition 4.10. By the previous lemma the elements (4.7) map, up to scalar factor, to a basis, but also span the skein, so must themselves form basis. Furthermore, J ! • Φ is an isomorphism, thus Φ injective.