Stated skein algebras of surfaces
Francesco Costantino Thang T. Q. Le
math.GT Primary 57N10, secondary 57M25













   






























#1
        y
                [0pt]
                        8pt#1
        
#1#1
#1#1#1


plain
theoremTheorem[section]
thmTheorem
lemma[theorem]Lemma
corollary[theorem]Corollary
proposition[theorem]Proposition
conjectureConjecture
questionQuestion
claim[theorem]Claim
definitionDefinition
definition
remark[theorem]Remark
example[theorem]Example






ℂ
ℕ
ℤ
ℝ
𝕋
ℚ


𝒜
ℐ
ℒ
ℳ
𝒪
𝒫
ℛ
𝒮
𝒯
𝒰
𝒵

Cob
HopfCoMod
Pt
øo
Pun
S
ℳ
𝒜ℳ

Mor
Mor2Mor
()


𝔏

𝒪_q^2(SL(2))

𝔟
𝔭
𝔱

⟨
⟩

Σ
rk
ℳ

λ̃
x̃
Q̃
Λ̃
𝒮
⊗
∘

Θ

ℰ
ℱ
𝕌
Mat
𝐤
𝐧
Δ
∘P
∘Q
𝒫
D̃
P̃
𝕍
∘
∘
Id
𝔖

𝔇
Δ

𝐂𝐡
∘Λ
K
𝒴
ev
𝒳
φ̃
θ̃
↪
∘Γ
∘φ
𝔛

in



∘Q
∘Λ
Im
Σ

𝒯
𝒱
𝒴
Q
ev
𝒳
𝒳
w

Col

∘Λ
∘A
↪
∘Γ
∘Q
∘Λ
Im
∘𝔐

∘Σ
φ
ϰ






∙=
ab
Σ
(Σ,)
∂Σ


Teichmüller 




Tr

Tr_



α
ε
 
    𝒯ΔΘθ𝒫Im𝔰θ𝔰𝔱̃𝔭𝔱𝔭𝔱CatRep𝒫ℳ𝒞÷divfinD̂Θ̂𝕋𝒞deg_lr𝔪ltldeglt≡fl[3][3]-8pt.8 cmleftve-8pt.8 cmleft_up-12pt1.2 cmarcpm-12pt1.2 cmarcall-12pt1.2 cmarcallflip-12pt1.2 cmreordpm-12pt1.2 cmreordmp-12pt1.2 cmarcmm-12pt1.2 cmarcpp-12pt1.2 cmempty0-8pt.8 cmloop1-8pt.8 cmleft_up_n-8pt.8 cmleft_upPP-8pt.8 cmleft_upNN-8pt.8 cmleft_down_n-8pt.8 cmleft_ve_d-8pt.8 cmleftvetwist-8pt.8 cmleft_ve_dl-8pt.8 cmempty-8pt.8 cmemptys-8pt.8 cmleft_down_p-8pt.8 cmleft_up_n[ -12pt1 cmpf1][ -12pt1 cmpf2][ -12pt1 cmpf3](  -10pt1 cmR3D) (  -10pt1 cmR3Dpositive) (  -10pt1 cmR3Dnegative) ( -10pt1 cmR3Dp)(  -10pt1 cmR3Dpnegative) (  -10pt1 cmR4a) (  -10pt1 cmR4b) (  -10pt1 cmpfR23a) (  -10pt1 cmpfR23b) (  -10pt1 cmpfR23c) (  -10pt1 cmpfR231) (  -10pt1 cmpfR232) (  -10pt1 cmpfR233) (  -10pt1 cmpfR234) (  -10pt1 cmpfR23d) (  -10pt1 cmpfR23d1) (  -10pt1 cmpfR23e) ρ̃-8pt.8 cmcross-8pt.8 cmresoP-8pt.8 cmresoN-8pt.8 cmkinkp-8pt.8 cmkinkn-8pt.8 cmkinkzero-8pt.8 cmreord1all-8pt.8 cmreord1allpp-8pt.8 cmreord1allnn-8pt.8 cmreord1pn-8pt.8 cmreord2pn-8pt.8 cmreord1allp-8pt.8 cmreordpn-8pt.8 cmleftmp-8pt.8 cmreord1-8pt.8 cmreord1-8pt.8 cmreord2-8pt.8 cmreord3-8pt.8 cmtrivloop-10pt1 cmcase1a-10pt1 cmcase1b-10pt1 cmcase2a-10pt1 cmcase2b-10pt1 cmcase3a-10pt1 cmcase3b-8pt.8 cmreord1z-8pt.8 cmreord1a-8pt.8 cmreord1aa-8pt.8 cmreord1b-8pt.8 cmreord1c-8pt.8 cmreord6z-8pt.8 cmreord6a-12pt1.3 cmcorner-12pt1.3 cmcorner2-12pt1.3 cmcorner3-12pt1.3 cmcornerp-12pt1.3 cmcorner2p-12pt1.3 cmcorner3p-12pt1.3 cmreord1gra𝖬𝗎𝗅𝗍𝗂𝖢𝗒𝖼𝖢𝗒𝖼𝖥𝗂𝗇𝖲𝖾𝗍𝖬𝗎𝗅𝗍𝗂𝖥𝗂𝗇𝖲𝖾𝗍𝖳𝗈𝗉𝖬𝗎𝗅𝗍𝗂𝖢𝗒𝖼𝖳𝗈𝗉𝖥𝗈𝗋𝖾𝗌𝗍𝖦𝗋𝖺𝗉𝗁𝖥𝗈𝗋𝖾𝗌𝗍𝖤𝖽𝗀𝖾𝗌𝖼𝗎𝗍(q^1/2)Francesco Costantino]Francesco CostantinoInstitut de Mathématiques de Toulouse, 118 route de Narbonne,
    F-31062 Toulouse, Francefrancesco.costantino@math.univ-toulouse.frThang  T. Q. Lê]Thang  T. Q. LêSchool of Mathematics, 686 Cherry Street,
     Georgia Tech, Atlanta, GA 30332, USAletu@math.gatech.edu Partially supported by ANR-11-LABX-0040-CIMI within the program ANR-11-IDEX-0002-02. 
    
    2010  Mathematics Classification: Primary 57N10. Secondary 57M25.
     Key words and phrases: Kauffman bracket skein module, TQFT.We study the algebraic and geometric properties of stated skein algebras of surfaces with punctured boundary. We prove that the skein algebra of the bigon is isomorphic to the quantum group 𝒪_q^2(SL(2))that provides a topological interpretation for its structure morphisms. 
    We also show that its stated skein algebra lifts in a suitable sense the Reshetikhin-Turaev functor and in particular we recover the dual R-matrix for 𝒪_q^2(SL(2)) in a topological way. 
    We deduce that the skein algebra of a surface with n boundary components is a comodule-algebra over 𝒪_q^2(SL(2))^⊗n and prove that cutting along an ideal arc corresponds to Hochshild cohomology of bicomodules. We give a topological interpretation of braided tensor product of stated skein algebras of surfaces as “glueing on a triangle”; then we recover topologically some bialgebras in the category of 𝒪_q^2(SL(2))-comodules, among which the “transmutation” of 𝒪_q^2(SL(2)).
    We also provide an operadic interpretation of stated skein algebras as an example of a “geometric non symmetric modular operad”.   
    In the last part of the paper we define a reduced version of stated skein algebras and prove that it allows to recover Bonahon-Wong's quantum trace map and interpret skein algebras in the classical limit when q→ 1 as regular functions over a suitable version of moduli spaces of twisted bundles.  Stated skein algebras of surfaces
        [
        December 30, 2023
    =================================
    
    ∂(,)𝒜ℬΣ𝔅pr𝒪∂_0Σ^rel_s^rel_s,0ℛbasic skein()()^(2)()^(2)()𝐦^()^()∂∂∂μ⃗ν⃗Gr𝐥↠St^↑()St^↑(')s^↑St_s𝒩(M,)∂ M_+___s()
    
    § INTRODUCTION
    
    This paper is devoted to study the notion of stated skein algebra of surfaces introduced by the second author in <cit.> in order to reinterpret in skein theoretical terms the construction of the quantum trace by Bonahon and Wong <cit.> as well as incorporating Muller's version of skein algebra <cit.>. 
    Although the definition of the stated skein module applies to 3-manifolds, this paper is entirely devoted to the case of surfaces: a forthcoming paper will describe how this fits in the framework of an extended topological field theory in dimensions 1,2,3. 
    Indeed the case of surfaces is sufficiently rich in algebraic and geometrical terms to deserve a separate treatment and we will now outline the results of this paper. 
     
     
     
    
     §.§ Skein algebras
     Let = [q^± 1/2] be the ring of Laurent polynomials in a variable q^1/2.
     Suppose  is the result of removing a finite number of points, called  punctures, from a compact oriented 2-dimensional manifold with possibly non-empty boundary. 
     The ordinary skein algebra (), introduced by Przytycki <cit.> and Turaev <cit.>, is defined to be the -module generated by isotopy classes of framed unoriented links in × (0,1)  modulo the Kauffman relations <cit.>
        =  q + q^-1
            =  (-q^2 -q^-2).
    
    The product of two links _1 and _2 is the result of stacking _1 above _2.  The skein algebra  has played an important role in low-dimensional topology and quantum topology and it serves as a bridge between classical topology and quantum topology. The skein module has connections to the SL_2()-character variety <cit.>, the quantum group of SL_2(), the Witten-Reshetikhin-Turaev topological quantum field theory <cit.>, the quantum Teichmüller spaces <cit.>, and the quantum cluster algebra theory <cit.>.
     
     In the definition of the skein algebra () the boundary  does not play any role, and we have () = (), where  is the interior of . In an attempt to introduce excision into the study of the skein algebra, the second author  <cit.> introduce the notion of  stated skein algebra, denoted in this paper by  , whose definition involves tangles properly embedded into × (0,1). These tangles can have end-points only  on  boundary edges of , which are open intervals connected components of the boundary.
     For details see Section <ref>.
     
     
    
    A key result about stated skein algebras is that they behave well under cutting along an ideal arc. Here an  ideal arc is a proper embedding c: (0,1) (so that its end points are the punctures). 
    Cutting  along c one gets a 2-manifold ' whose boundary contains two open intervals a and b so that one can recover  from ' by gluing a and b together, see Figure <ref>.
    
    cutCutting  along ideal arc c to get ', which might be disconnected3cm
    Then <cit.> (see 
    the splitting Theorem <ref> below)  says that  there is a natural injection of algebras 
    θ_c: ()↪('),  
    given by a simple state sum. The extension from () to  is unique (or canonical) if one wants the splitting theorem and a consistency requirement to hold.
     
     ()U_q^2(𝔰𝔩_2)
     The paper is a systematic study of the stated skein algebra . Let us now list the main results of the paper.
     
     
    
     §.§ Bigon and quantum SL_2() coordinate ring
     The quantized enveloping algebra  and its Hopf dual , known as the quantum coordinate ring of the Lie group SL_2(), play an important role in many branches of mathematics, see <cit.>. These algebras are usually defined by rather complicated presentations which are hard to comprehend.
     
    A first consequence of the splitting theorem is that the quantum coordinate ring  can be described by simple geometric terms, namely, it is naturally isomorphic to the stated skein algebra of the bigon , which is the standard disk without two points on its boundary, see Figure <ref>.
    
    bigon10Left: bigon. Right: splitting the bigon along the dashed ideal arc3cm
    By splitting the bigon along an ideal arc c (which is the dashed arc in Figure <ref>) we get a homomorphism Δ:=θ_c,
    
        Δ: →⊗,
    
    which turns out to be compatible with the product and makes  a bialgebra. Moreover, we will define using topological terms the counit, antipode and co-R-matrix which turns  into a “dual quasitriangular” (see  <cit.> Section 2.2) a.k.a. “cobraided” (see <cit.>, Section VIII.5) Hopf algebra, and will prove the following.
    
     The dual quasitriangular Hopf algebra  is isomorphic in a natural way to the quantum coordinate ring .
    
    
    
    This result allows to use skein theoretical techniques to study . We will show that many complicated algebraic objects and facts  concerning the quantum groups  and  have  simple transparent picture interpretations. For example, the above mentioned co-R-matrix has a very simple geometric picture description, see Theorem <ref>. 
    Another example is given by the reconstruction of Kashiwara's crystal basis, see Proposition <ref>. One can even “import” in  natural skein theoretical objects: in Subsection <ref> we define and provide some properties of the Jones-Wenzl idempotents in . 
    
    
    
     §.§ Lift of the Reshetikhin-Turaev invariant
    
    Suppose T is a tangle diagram in the bigon whose boundary ∂ T is in ∂ and the boundary points are labeled by signs ±. The Reshetikhin-Turaev  operator invariant theory <cit.> assigns to T a scalar Z(T)∈(q^1/2), see Section <ref>. On the other hand, such a labeled tangle T  defines an element in our skein algebra . We have the following result which shows that our “invariant”, which is T considered as an element of (), is a lift of the Reshetikhin-Turaev invariant.
     One has ϵ(T) = Z(T), where ϵ: →[q^± 1/2] is the counit.
      
    It would be interesting to understand this lift of the Reshetikhin-Turaev invariant in terms of categorification.
    
    
    
     §.§ Skein algebras as comodule over . Hochshild cohomology
     One important consequence of the identification of the bigon algebra with   is that for every boundary edge e of a surface , the skein algebra  has  a  right -comodule structure
    
        Δ_e : →.
    
    This map Δ_e is the splitting homomorphism (<ref>) applied to the an ideal arc parallel to e which cuts off an ideal bigon from  whose right edge is e, see Figure <ref>. 
    Similarly identifying the left edge of  to e  we get a left -comodule structure on .
    
    coproGeometric definition of the coaction: splitting the bigon along the dashed ideal arc3cm
    
    Using the comodule structure one can refine the splitting theorem by identifying the image of the splitting homomorphism, as follows. Let us cut  along an ideal arc c to get ' as in Figure <ref>. Then (') has a right -module structure coming from edge a and a left -module structure coming from edge b. Thus (') is a -bicomodule, and hence there is defined the Hochshild cohomology HH^0((')), for details see Section <ref>.
    
    
    Under the splitting homomorphism the skein algebra  embeds isomorphically into the Hochshild cohomology HH^0((')). In particular, when c cuts  into two surfaces _1 and _2, the splitting homomorphism  maps  isomorphically onto the cotensor product of (_1) and (_2).   
    
    
     §.§ Skein algebra  as module over 
     
    Since the Hopf algebra  is the Hopf dual of , then after tensoring with  each right -comodule is automatically a left -module. Thus each boundary edge e of  gives  a left -module structure. Note that finite-dimensional -modules are well-understood as they are quantum deformations of   modules over the Lie algebra 𝔰𝔩_2().
    
     Over the field (q^1/2), for every boundary edge the -module  is integrable, i.e. it is the direct sum of finite-dimensional irreducible -modules.
    
    Actually Theorem <ref> is much stronger: it provides an explicit decomposition and contains much more information about the decompositions as it deals also with the decomposition over Lusztig's integral version of .
    
    Using this result we also prove a dual version of Theorem <ref> which, with the notation of the theorem, shows that HH_0(⊗_('))=⊗_() (see Theorem <ref>). 
    
    
    
     §.§ Braided tensor product
     The co-R-matrix makes the category of -co­mo­du­les a braided category  and  in general given two algebras in that category (which are then -co­mo­dule algebras)  their tensor product can be endowed with the structure of an algebra  by using appropriately the braiding: this is  the  braided tensor product of the algebras, see <cit.>. 
     In Section <ref>, we generalise this notion to that of “self-braided tensor product” which applies to a comodule algebra having two commuting comodule structures.
    
    Suppose  is obtained by identifying two edges of a (possibly disconnected) surface  with two distinct edges of an ideal triangle as in Figure <ref>. 
    brtens-newGluing  to two distinct edges of an ideal triangle to get .3cm
    Then ()  has two natural commuting structures of -comodule algebra, and we have the following:
     As a -comodule algebra () is canonically isomorphic to the self-braided tensor product of (). In particular, if ='⊔” and the two edges belong to ' and ” respectively then () is canonically isomorphic to the braided tensor product of (_1) and  (_2).
    
    
    
    Through this theorem we easily compute the skein algebra of all “polygons”, “punctured bigons”, and “punctured monogons” in Subsection <ref>. It is remarkable that the skein algebras of the latter turn out to be bialgebra objects in the category of -comodules and that their structure morphism have natural topological interpretation. In particular the punctured monogon yields the “transmutation” of . 
    
    
    
     §.§ Modular operad
     
    
    The splitting homomorphism allows to put the theory of stated skein algebras of surfaces in the framework  of operad theory. We define  the notion of geometric non-symmetric modular operad in Section <ref> and prove the following.
    
     The stated skein algebra of surfaces gives rise to a non-symmetric modular operad in a category of bimodules over .
    
    
    
    To be more specific while leaving the details for Section <ref>, let us recall that, according to Markl (<cit.>) a “Non-symmetric modular operad in a monoidal category Cat” is a monoidal functor NSO:𝖬𝗎𝗅𝗍𝗂𝖢𝗒𝖼→ Cat, where 𝖬𝗎𝗅𝗍𝗂𝖢𝗒𝖼 is a suitable category of “multicyclic sets”.  
    In Section  <ref> we re-cast Markl's definition, by defining a category  whose objects are punctured surfaces  and whose morphisms are finite sets of ideal arcs (describing a way of cutting the surfaces). From this point of view, we then show in Theorem <ref> that stated skein algebras provide a symmetric monoidal functor from this category into a suitable category of modules and bimodules over copies of , thus providing a topological example of a NS modular operad. 
    
    
    
    ()ℐ^bad
    
     §.§ Reduced stated skein algebra, quantum torus, and quantum trace map
     The stated skein algebra  has a quotient = /, called the  reduced stated skein algebra, whose algebraic structure is much simpler as it can be embedded into the so called quantum tori. Here  is the ideal generated by elements, called bad arcs, described in Figure <ref> and is explained in Section <ref>.
    
    badarc0A bad arc.2cm
    
    
    We will show  that the ordinary skein algebra () still embeds into  and hence we can use  to study (). Most importantly, the splitting theorem still holds for .
     
     If ' is the result of cutting  along an ideal arc c, then the splitting homomorphism θ_c descends to an algebra embedding
     
        θ̅_c: (').
     
    The non-trivial fact here is that θ̅_c is injective.
      
     Except for a few simple surfaces, we can always cut  along ideal arcs so that the result is a collection of ideal triangles T_1, …, T_k. It follows that there is an embedding
     Θ: ⊗_i=1^k  (T_i). 
     
    The important thing with the reduced version is that for an ideal triangle T, unlike the full fledged (T), the reduced stated skein algebra (T) is a  quantum torus in three variables:
    
    
    The reduced stated skein algebra (T) of an ideal triangle has presentation
    
        (T) = ^± 1, β ^± 1, γ^± 1  / (βα = qβ, γβ = q βγ, γ=q γ).
    
    Moreover, the reduced stated skein algebra of the bigon is naturally isomorphic to the algebra [x^± 1] of Laurent polynomial in one variable, see Proposition <ref>.
    
    Consequently, the map Θ of (<ref>) embeds the reduced stated skein algebra  into a quantum torus in 3k variables.
    Geometrically the  variables ,β,γ in Theorem <ref> come from the corner arcs  of the ideal triangle. There is a similar quantum torus '(T) in 3 variables corresponding to the edges of T, and a simple change of variables gives us an embedding (T) '(T). Combining with Θ of (<ref>) we get an algebra embedding 
    
        _q: Θ⊗_i=1^k (T_i)  ⊗_i=1^k (T_i).
    
    
    There is a subalgebra  of ⊗_i=1^k (T_i), known as the Chekhov-Fock algebra associated to the triangulation. 
     The famous quantum trace map of Bonahon and Wong <cit.> is an algebra homomorphism _q: () →, where () is a coarser version of the stated skein algebra which surjects onto 
     (), see Section <ref>. 
    
    
     The image of _q is in . Thus _q restricts to an algebra embedding  _q:, and   the quantum trace map of Bonahon and Wong is the composition ()_q.
    
    
    The existence of the quantum trace map (for ()) was conjectured by Chekhov and Fock <cit.>, and was established  by Bonahon and Wong <cit.>. It is called the quantum trace map since when q=1 it becomes a formula expressing the trace of a curve under the holonomy representation of the hyperbolic metric in terms of the shear coordinates of the Teichmüller space.
    The second author <cit.> gave another proof of the existence of the quantum trace map based on the Muller skein algebra, which is actually a subspace of the state skein algebra . The above approach using the reduced stated  skein algebra  offers another proof, and also determines the kernel of the original quantum trace map _q.
    
    
    
    
    
     §.§ Classical limit
    
    The last section explores the natural question of “what is the classical limit of ?” In the case of the standard skein algebra () it is known <cit.> (see also <cit.>) that when the quantum parameter q is -1 and the ground ring is  then () is isomorphic as an algebra to the coordinate ring the SL_2(ℂ)-character variety of  and that in general the algebras at q and -q are isomorphic via the choice of a spin structure on  (<cit.>). Note that our stated skein algebra is not commutative when q=-1 though it is commutative when q=1. 
    
    We introduce the variety tw() of  “twisted  SL_2(ℂ)-bundles” over , which, roughly speaking, are flat SL_2(ℂ)-bundles over the unit tangent bundle U  of  with holonomy -Id around the fibers of π:U→ and are equipped with trivializations along the edges of , but which we reformulate in terms of groupoid representations. To deal with the non-oriented nature of the  arcs of the stated skein algebra we have to use a trick smoothing the arcs at their end-points so that one can compose arcs. 
    
     When q=1 and the ground ring is  the stated skein algebra  is naturally isomorphic to the coordinate ring of tw().
    
    
     In classical terms, the splitting theorem becomes an instance of a van-Kampen like theorem for groupoid representations.  
     
    
    Theorem <ref> highlights a relation between  and the coordinate ring of the character variety of . The study of quantizations of such rings has been performed with different techniques (based on Hopf algebras and lattice gauge theory) by  Alekseev-Grosse-Schomerus (<cit.>), Buffenoir-Roche <cit.>, Fock-Rosly (<cit.>) and, later, via skein theoretical approaches by Bullock-Frohman-Kania-Bartoszynska (<cit.>). 
    The relation of our work with these previous ones is still to be clarified, although it seems that one of the main differences between our approach and some of the above cited ones, is that we allow for “observables with boundary” and, as explained in the preceding paragraph, this endows the algebras we work with rich algebraic structures which in particular make computations much easier. 
    
    
    
    
     §.§ Related results
    
    
    While the authors were completing the present work,  D. Ben-Zvi, A. Brochier and D. Jordan <cit.> constructed a theory of quantum character variety for general Hopf algebras, based on completely  different techniques. Part of the results of this paper could probably be recasted in that theory, though we don't know the precise relation between the two theories. The substantial difference of the techniques used makes these works complementary. K. Habiro informed us that his  “quantum fundamental group theory” also 
    gives an alternative approach to the theory of quantum character variety.
    
    When the authors presented their works at conferences, Korinman informed us that he in joint work 
    A. Quesney obtained results similar to Theorem <ref> and Theorem <ref>, see their recent preprint <cit.>.
    
    
    
     Acknowledgements. The authors would like to thank F. Bonahon, R. Kashaev, A. Sikora,  V. Turaev, and D. Thurston for helpful discussions.  
    Both authors were partially supported by the CIMI (Centre International de Mathématiques et d'Informatique) Excellence program during the Thematic Semester on Invariants in low-dimensional geometry and topology held in spring 2017. The second author is partially support by an NSF grant.
    
    The authors presented the results of this paper in the form of talks or mini-courses at many conferences, including  “Thematic School on Quantum Topology and Geometry” (University of Toulouse, May 2017), “Algebraic Structures in Topology and Geometry” (Riederalp, Switzerland, January 2018), “TQFTs and categorification” (Cargese, March 2018), “Volume Conjecture in Tokyo” (University of Tokyo, August 2018), “Categorification and beyond” (Vienna, January 2019),  Topology mini-courses (university of Geneva, May 2019), “New developments in quantum topology” (Berkeley June 2019), “Expansions, Lie algebras and invariants” (Montreal July 2019) and would like to thank the organizers for the opportunities to present their work.
    
    
    
    
    
    § STATED SKEIN ALGEBRAS
    
    We will present the basics of the theory of stated skein algebras: definitions, bases of skein algebras, the splitting homomorphism,  filtrations and gradings. New results involve Proposition <ref> describing the inversion homomorphism and Proposition <ref> giving the exact value of the splitting homomorphism in the associated graded algebra.
    
    
     §.§ Notations
    
    Throughout the paper let
     be the set of integers,  be the set of non-negative integers,  be  the set of complex numbers.
    The ground ring   is a commutative ring with unit 1,  containing a distinguished invertible element q^1/2.
     For a finite set X we denote by |X| the number of elements of X.
     
     ∙ =
     We will write x  y if there is k∈ such that x = q^k y.
    
    𝔗
    
     §.§ Punctured bordered surface
     By a  punctured bordered surface we mean a  surface of the form = ∖, where  is 
     a compact oriented surface  with
    (possibly empty) boundary , and  is a finite non empty set such that every connected component of the boundary  has at least one point in .  We don't require  be to connected. It is easy to see that  is uniquely determined by . Throughout this section we fix a  punctured bordered surface .
    
    An  ideal arc on  is an immersion a: [0,1] → such that a(0),a(1)∈ and the restriction of a onto (0,1) is an embedding into . 
    Isotopies of ideal arcs are considered in the class of ideal arcs. 
    
    
    A connected component of  is called a  boundary edge of  (or simply an edge), which is an ideal arc.
     
    
    
    
    
    
     The fact that each connected component of ∂ is an open interval is not a serious restriction as for the purpose of the constructions of this paper a point-less boundary component is treated as a puncture; so that in the end the only excluded surfaces are closed ones without punctures.
    
    
     §.§ Ordinary skein algebra
      Let M=× (0,1). For a point (z,t)∈× (0,1) its  height is t. A vector at
     (z,t) is called  vertical if it is a positive vector of z × (0,1).  A framing of a 
     1-dimensional submanifold  of M=× (0,1) is 
     a continuous choice of a vector transverse to  at each point of .
    
    
     A  framed link in × (0,1) is a closed 1-dimensional unoriented submanifold   equipped with a framing.  The empty set, by convention, is considered a framed link. 
     
     A link diagram on  determines an isotopy class of framed links in × (0,1), where the framing is vertical everywhere. Every isotopy class of framed links in × (0,1) is presented by a  link diagram.
    
    
    
    The  skein module(), first introduced  by Przytycki <cit.> and Turaev <cit.>, is defined to be the -module generated by the isotopy classes of framed unoriented links in × (0,1) modulo the Kauffman relations
    
        =  q + q^-1
            =  (-q^2 -q^-2)
    
    We use the following convention about pictures in these identities, as well as in other identities in this paper. Each  shaded part is a  part of , with a link diagram on it.
    Relation (<ref>) says that if  link diagrams D_1, D_2, and D_3 are identical everywhere except for a small disk in which D_1,D_2, D_3 are like in respectively the first, the second, and the third shaded areas, then [D_1] = q[D_2] + q^-1 [D_3] in the skein module (). Here [D_i] is the isotopy class of links determined by D_i. The other relation is interpreted similarly.
    
    For two framed links _1 and _2 the product _1 _2  is defined as the result of stacking _1 above _2. That is, first isotope _1 and _2 so that _1 ⊂× (1/2,1) and _2 ⊂× (0, 1/2). Then _1 _2= _1 ∪_2.
     It is easy to see that this gives rise to a well defined product and hence an -algebra structure on ().
     
    
    
    
    
     §.§ Tangles and order
     In order to include the boundary of  into the picture, we will replace framed links by more general objects called -tangles. Recall that M= × (0,1) and its boundary is = × (0,1).
    
    In this paper, a -tangle is
    an unoriented,
     framed, compact,  properly embedded 1-dimensional submanifold ⊂ M=× (0,1) such that:
     
      *   at every point of ∂=∩ the framing is  vertical, and
    
      *   for any boundary edge b, the points of ∂_b():=∂∩ (b × (0,1))  have distinct heights.
     
    
    
    For a -tangle  define a partial order on ∂ by: x>y if there is a boundary edge b such that x, y ∈ b × (0,1) and
     x has greater height. If x>y and there is no z such that x>z>y, then we say
    x and y are  consecutive. 
    
     Isotopies of  -tangles are considered  in the class of -tangles. It should be noted that isotopies of -tangles preserve the height order.
    The empty set, by convention, is a -tangle which is isotopic only to itself. 
    
    
    As usual, -tangles are depicted by their diagrams on , as follows.
    Every -tangle is isotopic to one with vertical framing.
    Suppose a vertically framed -tangle  is in general position with respect to
    the standard projection π: × (0,1) →, i.e.  the restriction π |_:→ is an immersion with transverse  double points as the only possible singularities and there are no double points on the boundary of . 
    Then D=π (), together with 
    
      *   the over/underpassing information at every double point, and 
    
      *   the linear order on π() ∩ b for each boundary edge b induced from the height order
     
    is called a -tangle  diagram, or simply a tangle diagram on .
     Isotopies of  -tangle diagrams are  ambient isotopies in .
    
    Clearly the -tangle diagram of a -tangle  determines the isotopy class of .
     When there is no confusion, we identify a -tangle diagram with its isotopy class of -tangles.
    
    𝔬
     Let  be an  orientation of ,   which on a boundary edge may or may not be equal to the orientation inherited from . A -tangle diagram D is -ordered if for each boundary edge b the order of ∂ D on b is  increasing when one goes along b in  the direction of . It is clear that every -tangle, after an isotopy, can be presented by an  -ordered -tangle diagram.
     If  is  the orientation coming from , the -order is called the  positive order.
    
    
    
    𝔬
    
     §.§ Stated skein algebra
    
    
    A  state on a finite set X is a map s: X →{±}. We write # s= |X|.
    A  stated -tangle   (respectively  a stated -tangle diagram) is a -tangle (respectively  a  -tangle diagram)  equipped with a  state on its set of boundary points.
    
    
    The   stated skein algebra() is  the -module freely spanned
     by isotopy classes of stated -tangles modulo the  defining relations, which are the old  skein relation (<ref>) and the trivial loop relation (<ref>), and the new boundary relations (<ref>) and (<ref>):
    
        =  q + q^-1
            =  (-q^2 -q^-2)
             =  q^-1/2 ,      =0,     = 0   
            =  q^2  +  q^-1/2
    
    Here each   shaded part is a  part of , with a stated -tangle diagram on it.
    Each arrowed line is part of a boundary edge, and the order on that part is indicated by the arrow and the points on that part are consecutive in the height order. The order of other end points away from the picture can be arbitrary and are not determined by the arrows of the picture. On the right hand side of the first identity of (<ref>), the arrow does not play any role; it is there only because the left hand side has an arrow.
    
    
    
    Again for two -tangles _1 and _2 the product _1 _2  is defined as the result of stacking _1 above _2. The product makes () an -algebra. In <cit.> it is proved that if  is a domain then   does not have non-trivial zero-divisors, a fact known earlier for the case when  has no boundary <cit.>.
    
    
    If   _1 and _2 are two punctured bordered  surfaces, then
    there is a natural isomorphism
    
        (_1 ⊔_2) ≅ (_1) _ (_2).
    
    
    
    
    Since the interior  of  does not have boundary, we have () = ().^+()
    The subalgebra  spanned by -tangles whose  states  are all + is naturally isomorphic to the skein algebra defined by Muller <cit.>, see <cit.>. If in the definition of () we use only two relations (<ref>) and (<ref>), we get a coarser version () which was defined by Bonahon and Wong <cit.>.  Relations (<ref>) already appeared in <cit.>. Relation (<ref>) appeared  in <cit.> where the stated skein algebra was introduced.
    ⋆⟶-8pt.8 cmleftvep
    
     §.§ Consequences of defining relations
     
    
    
    
    Define C^ν_ν' for ν, ν'∈{±} by
    
    C^+_+ = C^-_- = 0,    C^+_-= q^-1/2,    C^-_+ = -q^-5/2.
    
    In the next lemma we have the values of all the trivial arcs.
    
    In () one has
    
        -q^-3    =    =  -q^3
            =  C^ν _ν'
             =     =  - q^3 C^ν' _ν
    
    
    The next lemma describes how a skein behaves when the order of two consecutive boundary points is switched.
    
    
    
     (a) One has
    
        =  q^-1  ,       =  q^-1  ,    = q  
        
        q^3/2  - q^-3/2   = (q^2 -q^-2) .
    
    
    (b) Consequently, if q=1 or q=-1, then for all ν, ν'∈{±},
     =
     q .
      Because of relation  (<ref>), in general  () is not commutative when q=-1. This should be contrasted with the case of the usual skein algebra   (), which is
      commutative and is canonically equal to the SL_2() character variety of π_1() if = and q=-1 (assuming  is connected), see <cit.>.
    𝕀idχℒχ̃
    
     §.§ Reflection anti-involution
    Suppose =[q^± 1/2]. There exists a unique -linear map χ: () →(), such that
    
    
      * χ(q^1/2)= q^-1/2, 
    
      * χ is an anti-automorphism, i.e. for any x,y ∈(),
    
        (x+y)= (x) + (y),   (xy) = (y) (x),
    
      *  if  is a stated -tangle diagram then χ() is the result of switching all the crossings of  and reversing the linear order on each boundary edge.
    
    Clearly ^2=𝕀. We call  the reflection anti-involution.
    
    inv𝕀ide_re_l
    
     §.§ Inversion along an edge
      Let  e be a boundary edge of a punctured bordered surface  and  f:{±}→ be a function such that f(+) f(-)= - q^-3.
    
    There exists a unique -linear homomorphism _e,f: () →() such that if  is a stated ∂-tangle diagram with a state s and with positive order on e,
    then 
    _e,f()=  ( ∏_x ∈ (∩ e) f(s(x))  )  ',
     where ' is the same  except that the height order of ' on e is given by the negative direction of e and the state of ' on e is obtained from that of  by switching ν∈{±} to -ν at every boundary points in ∩ e.
    
    If e' is another boundary edge and f'(+) f'(-)= -q^-3, then 
    _e,f∘_e',f'  = _e'.f'∘_e,f.
      Let T be the set of isotopy classes of stated, positively ordered -tangle diagrams. Since T spans (), the  uniqueness of _e,f is clear. 
    
    Let  be the -module freely span by T, and _e,f: '→() be the 
     -linear map defined  by (<ref>). To show that  _e,f descends to a map _e,f: ()→() one needs to prove _e,f is invariant under isotopy in M:=× (0,1) and under the moves generated by the defining relations (<ref>)-(<ref>). More precisely, we have to show that  _e,f() = _e,f() for any ,'∈ T   whenever 
    
      (i) and ' are isotopic as -tangles, or
    
      (ii) and ' are respectively the left hand side and the right hand side of the defining relations (<ref>)-(<ref>).
    
    
    It is known that  and ' are isotopic as -tangles if and only if they are related by a sequence of the 3 framed Reidemeister moves of <cit.>. 
    The invariance under the 3 framed Reidemeister moves follows from the invariance under the defining relations (<ref>) and (<ref>), see <cit.>. Clearly _e,f is invariant under the moves generated by the  defining relations (<ref>) and (<ref>). There remains relations (<ref>) and (<ref>).
    
    Let us consider (<ref>).  Using the definition, f(+)f(-)=q^-3, and then (<ref>), we have:
    
    
    
    
        _e,f()=f(+)f(-) =-q^-3 =-q^-3(-q^3)C_-^+ = q^-1/2,
    
    which proves the first identity of (<ref>). The other two identities of (<ref>) are trivial.
    
    Let us consider (<ref>). By definition and Lemma <ref>,
    _e,f()=f(+)f(-)=(-q^-3)  ( q  )= -q^-2.
    
        _e,f(q^2+q^-1/2)     =q^2(-q^-3)+q^-1/2
           =-q^-1( q^-3+ q^-3/2(q^2-q^-2))+q^-1/2
           =-q^-4+q^-9/2,
    
    and the right hand sides of (<ref>) and (<ref>) are equal due to (<ref>).
    
    Identity (<ref>) follows immediately from the definitions.
    
    
    
    
    
    
    
    
    There are two important cases for us. Define
    _e:= _e,C,   and _e:= _e,C̅,
    
    where  
    C(+)= C̅ (-) = -q^-5/2,    C(-) = C̅(+) = q^-1/2.
    
    
    Note that C(ν) = C^-ν_ν and C̅(ν) = C(-ν) for  ν∈{±}. 
    For a stated tangle diagram  with a state s on the boundary edge e define 
     
     C_e() = ∏_x∈∩  e C(s(x))= ∏_x∈∩  e C^-s(x)_s(x).
      
    If  has positive order on e then, with '  defined as in Proposition <ref>, one has
    _e() = C_e() ',
     The definition (<ref>) works only for stated -tangle diagrams with positive order on e. If the order is not positive, the formula will be different.
    
    In general _e,f is not an algebra homomorphism.
    
    
     §.§ Basis of stated skein module
     A -tangle diagram D is  simple if it has neither double point nor
    trivial component. Here a closed component of D is  trivial if it bounds a disk in ,
     and an arc component of  is  trivial if it can be homotoped relative to its boundary  to a subset of a boundary edge.
     By convention, the empty set is considered as a simple stated -tangle diagram with 0 components.
    
    
    Define an order on {±} so that the sign - is less than the sign +. If X is a partially ordered set, then a state s: X →{±} is  increasing if s is an increasing function, i.e. f(x)  ≤ f(y) whenever x ≤ y.
    
    
    
    ndB̃
     Choose an orientation    of
    .
    Let  B(;) be the set of
     of all isotopy classes of increasingly stated, -ordered simple -tangle diagrams. From the defining relations  it is easy to show that the set B(;) spans  over . 
    
    
    
    
     Suppose  is a punctured bordered surface and  is an orientation of .
    Then B(;) is an -basis of ().
      dec Theorem <ref> means that the coefficients given in the defining relations (<ref>) and (<ref>) are consistent in the sense that they do not lead to any more relations among the set B(;).
      B
    
    The subset (;)⊂ B(;) consisting of ∈ B(;) having no arcs is a basis of the ordinary skein algebra (). Similarly, the subset B^+(;)⊂ B(;) consisting of ∈ B(;) having only positive states is a basis of the Muller  skein algebra , see <cit.>. Hence we have the following.
     Both the ordinary skein algebra () and the Muller skein algebra  are subalgebras of the stated skein algebra .  
    𝔄
    
     §.§ Filtration and grading
     Suppose a is either an ideal arc or a simple closed curve on  and  is a simple -tangle diagram on . The geometric intersection index I(a,) is
    
        I(a,) = min |a ∩'|,
    
    where the minimum is over all the simple -tangle diagrams ' isotopic to .
    
    For a collection ={ a_1, …, a_k}, where each a_i is either an ideal arc or a simple closed curve, and n∈ let F^_n() be the -submodule of  spanned by all stated simple -diagrams  such that ∑_i=1^k I(a_i, ) ≤ n. It is easy to see that the collection {F^_n()}_n∈ forms a filtration of  compatible with the algebra structure, i.e. with F_n= F^_n() one has
    
        F_n⊂ F_n+1,  ⋃_n∈ F_n = ,  F_n F_n'⊂ F_n+n'.
    
    One can define the associated graded algebra 
    ^(()):
     
        ^(())= ⊕_n=0^∞^_n(()) with ^_n(())= F_n/F_n-1 ∀ n≥ 1  and _0= F_0.
     
    
    This type of filtration has been used extensively in the theory of the ordinary skein algebra, see e.g. <cit.>. 
    
     The following is a consequence of Theorem <ref>:
      Let   be an orientation of the boundary of a punctured bordered surface , and ={a_1,…, a_k} be a collection of boundary edges of .
    
     (a) The set {∈ B(;)|∑_i=1^k I(, a_i)≤ n}
    is an -basis of F^_n(()).
    
    (b) The set {∈ B(;) |∑_i=1^k I(, a_i)= n} is an -basis of ^_n(()).
    _∂lt
    For what concerns the grading, for each non-negative integer m and a boundary edge e let  G^e_m be the -subspace of () spanned by stated -tangle diagrams   with 
     δ_e():= ∑_u ∈ (∩ e) s(u) =m, where s is the state and 
      we identify + with +1 and - with -1.  
     
    From the defining relations it is clear that ()= ⊕_m  ∈ G^e_m and G^e_m G^e_m'⊂ G^e_m+m'. In other words,  () is a graded algebra with the grading { G^e_m}_m∈.
    
    Also the following is a consequence of Theorem <ref>:
    
    Let  be a punctured bordered surface and  be an orientation of .
     The set {∈ B(;) |δ_e()=m} is an -basis of G^e_m(()).
    
    
    
     
     
     
    If ' is another orientation of the boundary , the change from basis B(;) to B(;') might be complicated. For the associated space ^(), the change of bases is simpler. 
    
    Recall that ' means = q^m ' for some m∈.
    
     Suppose  is stated tangle diagram on  and I(,e)=k where e is a boundary edge. Let's alter  to get ' by changing the height order on e and the states on e such that δ_e() = δ_e('). Then  one has
     '      in ^e_k().  
    
    One can get ' from  by a sequence of moves, each is either (i) an exchange of the heights of two consecutive vertices on e, or (ii) an exchange of states of two consecutive vertices on b. We can assume that ' is the result of doing a move of type (i) or type (ii).
    
    In case of move (i), the identities (<ref>) and (<ref>) prove (<ref>).
    
    In case of move (ii), the identity (<ref>) prove (<ref>).
    
    
     §.§ Splitting/Gluing punctured bordered surfaces
    
    Suppose a and b are distinct boundary edges of a punctured bordered surface ' which may not be connected. Let = '/(a=b) be the result of gluing a and b together in such a way that the  orientation is compatible. The canonical projection : ' →  induces a projection : M=' × (0,1) → M=× (0,1).
    Let c= (a)=(b). It is an interior ideal arc  of .
    
    Conversely if c is an ideal arc in the interior of , then there exists ', a,b as above such that = '/(a=b), with c being the common image of a and b. We say that ' is the result of splitting  along c.
    
    
    
     A -tangle  ⊂ M=× (0,1),  is said to be  vertically transverse to c if
     
      *  is transverse  to c × (0,1),
     
      *  the points in ∂_c := ∩ (c × (0,1)) have distinct heights, and have vertical framing.
     
     Suppose  is a stated -tangle vertically transverse  to c. Then :=^-1() is a '-tangle
     which is stated at every boundary point except for newly created boundary points, which are points in  ^-1 (∂_c ).
       A lift of  is a stated '-tangle β which is  equipped with states on ^-1 (∂_c ) such that if x,y∈^-1 (∂_c ) with (x)=(y) then x and y have the same state. If |∂_c |=k, then   has 2^k lifts.
    
    
     Suppose c is an ideal arc in the interior of a  punctured bordered surface  and ' is the result of splitting  along c.
    
    
    (a)
    There is a unique -algebra homomorphism
    θ_c: () →('), called the splitting homomorphism along c,
    such that if  is a stated -tangle  vertically transverse  to c, then
    θ _c()=∑β,  where the sum is over all lifts β of .
    
    
    (b) In addition, θ_c is injective.
    
    (c) If c_1 and c_2 are two non-intersecting ideal arcs in the interior of , then
    
        θ_c_1∘θ_c_2 = θ_c_2∘θ_c_1.
     The coefficients of the right hand sides of the defining relations (<ref>) and (<ref>) were chosen so that one has the consistency (see Remark <ref>) and the splitting theorem. It can be shown that if one requires the consistency and the splitting theorem, then the coefficients are unique, up symmetries of a group isomorphic to /2 ×/2.
    
    
     §.§ Splitting homomorphism and filtration
     Fix an orientation  of the boundary edges of ∂. Let  ' be the result of splitting  along an ideal arc c, with c being lifted to boundary edges a and b of '. 
    Choose an orientation of c and lift this orientation to a and b which, together with , gives an orientation ' for '.
    Assume D is a stated simple -ordered -tangle diagram which is  taut with respect to c, i.e. |D∩ c|=I(D,c). 
    For each each i=0,1,…, m:=|D ∩ c| let (,s_i) be the '-tangle diagram where = ^-1(D),  and the states on both a and b are '-increasing and having exactly i  minus signs. Then each (, s_i) is in the  basis  of the free -module ^{a,b}_2m((')) described in Proposition <ref>.
    
    For non-negative integers n,i the quantum binomial coefficient is defined by
    
        ni _q = ∏_j=n-i+1^n (1-q^j)/∏_j=1^i (1-q^j) .
     In ^{a,b}_2m((')) one has
    θ_c(D) = ∑_i=0^m mi_q^4 (, s_i).
    -13pt1.3 cmreord1gram-13pt1.3 cmreord1grbm-13pt1.3 cmreord2grb-13pt1.3 cmreord2grbm-13pt1.3 cmreord9-13pt1.3 cmreord9m For s: D ∩ c →{±} let (,s) be the stated '-ordered '-tangle diagram with state s on a and b.
    By definition,
    θ(D) = ∑_s: D ∩ c →{±}  (, s)   ∈(').
     
    Taking into account the filtration, from relations (<ref>) and (<ref>), we see that in ^{a,b}_2m((')),
      = q^2 ( ),    = q^2  ( ).
    
    It follows that ^{a,b}_2m((')) we have
     = q^4  (  ).
    
    Suppose s: D ∩ c →{±} has i minus values. 
    For k=1,…, i let x_k be the number of plus states (of s) lying below the k-th minus state. By doing many switches, each changing  a pair of  consecutive (-, +) to (+,-), we can transform s into s_i. The number of switches is x_1 + … + x_i. Hence from (<ref>) we see that 
    
        (,s) = q^4(x_1+ … x_i) (, s_i).
    
    Taking the sum over all s: D ∩ c →{±} with  i minus values, we get
     θ(D) = ∑_i=0^m ( ∑_0≤ x_1 ≤…≤ x_i ≤ m-i q^4( x_1 + … + x_i))   (, s_i)   in ^{a,b}_2m((')).
    
    By induction on i one can easily prove that
    ∑_0≤ x_1 ≤…≤ x_i ≤ n q^4( x_1 + … + x_i) =n+ii_q^4,
    
    from which and (<ref>) we get (<ref>).
    
    
     §.§ The category of punctured bordered surfaces and the functor .
    
    
    
    A  morphism from  one bordered punctured surface  to another one ' is an isotopy  class of orientation-preserving embeddings from  to '. Here we  assume that the embeddings map a boundary edge of  into (but not necessarily onto) a boundary edge of '.
    
    Very often we identify an embedding f: ' with its isotopy class. 
    
    
    inv𝕀ide_re_l
    
    
    Suppose f: →' is an embedding representing a morphism from  to '. Define an -linear homomorphism f_*: ()→(') such that if  is a stated tangle diagram on  with positive order then f_*() is given by the  stated tangle diagram f(), also with positive order.
     It is clear that f_* is an -linear homomorphism, and it does  not change under isotopy of f. 
    
    In general f_* is not an -algebra homomorphism. However, if every edge of ' contains the image of at most one edge of , then f_* is an -algebra homomorphism.
    
    
    
     A bigon  is the standard closed disk in the plane with two points on its boundary removed. Suppose a⊂ is an arc whose two end points are in distinct boundary edges e_1 and e_2, where 
     Let e be a boundary edge of  and '= ∖{v}, where v ∈ e. The embedding ι: ' induces an -linear homomorphism ι_*: (') → which is surjective but not injective in general.
     
     
     Suppose e'⊂ e is one of the two boundary edges of ' which is part of e. There is a diffeomorphism g: →' ∖{e'} which is unique up to isotopy. Thus, we have a morphism f: →', which is the composition
     
        g ⟶' ∖{e'}'.
    
    The morphism f induces an injective (but not surjective) algebra morphism f_*: ('). 
    μ⃗ν⃗η⃗
    
    § HOPF ALGEBRA STRUCTURE OF THE BIGON AND 
    
    
    We will define using geometric terms a  dual quasitriangular (a.k.a. cobraided) Hopf algebra structure on the stated skein algebra () of the bigon  and then show that it is naturally isomorphic to the dual quasitriangular Hopf algebra . We also show simple pictures of the canonical basis of , and discuss the Jones-Wenzl idempotents in (). In this section =[q^± 1/2] unless otherwise stated.
    
    
    -13pt1.3 cmmonogon-13pt1.3 cmmonogona
    
     §.§ Monogon and Bigon
     Let D be the standard disk
    
        D = { (x,y) ∈^2 | x^2 + y^2 ≤ 1}
    
    and v_1=(0,-1) and v_2 = (0,1) are two points on the circle ∂ D. The punctured bordered surface = D∖{v_1} is called the  monogon, and 
     = D∖{v_1, v_2} is called the  bigon. Let , be the two  boundary edges of  as depicted in Figure <ref>. For =(μ_1, …, μ_k) and =(η_1,…,η_k) in {±}^k let _∈() be the element presented by k parallel arcs as in Figure  <ref>, with (η_1,…,η_k) being the states on  in increasing order and (μ_1, …, μ_k) being the states on  in increasing order.
    
    bigon0Monogon, bigon with its edges e_l and e_r, elements _+- and  _. Note that for _ the height order is indicated by the arrows.2.6cm
    We have ()=. 
    Moreover, one has
    
    
    
     =  
    Here the circle enclosing x and two vertical lines attaching to it stand for a stated -tangle diagram. The proof follows by using the skein relation (<ref>), then the loop relation (<ref>), and finally the arc relation (<ref>) to reduce x to a scalar.
    
    
    
    
    We study the algebra () in this section.
    rot
     Let : → be the rotation (of the plane containing ) by 180^o about the center of , which is a self-diffeomorphism  of  and induces an -algebra involution
     _*: () →().
     
    
     §.§ Coproduct
    
    Suppose e is a boundary edge of a punctured bordered surface  . Let ' be the result of cutting out of  a bigon  whose right edge  is identified with e.
    Since ' is canonically isomorphic to  in the category of punctured bordered surfaces, we will identify () with (').
     The splitting homomorphism gives  an injective algebra homomorphism 
     
        (')  (⊔) ≡() ⊗ _().
     Since we identify () with ('), this map becomes an -algebra homomorphism 
    _e: ()   () ⊗ _().split1The coaction _e.2.5cm
    
    Suppose x∈ B(, ) is a basis element, where  is a given orientation of ∂. Assume the state of x on e is , then we have (see Figure <ref>):
    _e(x) = ∑_∈ S_x∩ e x_⊗ _,  
    where S_x ∩ e is the set of all states of x∩ e and x_ is x with the state on e switched to . 
    
    
    
      
      
      
      
      
    In particular, when = and e=, we get an -algebra homomorphism Δ= Δ_,
    Δ: () →() ⊗_ R ().
    
    
    Theorem <ref>(c) implies that  Δ is coassociative, i.e.
     
    (Δ⊗𝕀) Δ = (𝕀⊗Δ ) Δ.
    
    
    Applying (<ref>) to x=_νμ with ν,μ∈{±}, we get
    Δ(_νμ )= ∑ _η∈{±}_νη⊗_ημ.
    
    
     §.§ Presentation of ()
     A presentation of the algebra () was given in <cit.>. We give here a presentation of () in a form which is suitable for us. Recall that C(η)= C^η̅_η for η∈{±} were defined by (<ref>). We form the following matrix
    
      C:= [ C^+_+ C^+_-; C^-_+ C^-_- ] = [       0  q^-1/2; -q^-5/2       0 ] -18pt1.5 cmantipode-18pt1.5 cmap2
    The -algebra () is generated by {α_ν,μ|ν,μ∈{±}} with the following relations: 
    
        C    = A^t C A   
        
        C   = A C A^t,
    
    where A^t is the transpose of A and
    
        A := [ _++ _+-; _-+  _– ].
     Explicitly, the relations (<ref>) and  (<ref>) are respectively
    
        C^ν_μ· 1    = ∑_η∈{±} C(η̅) _ην_η̅μ= C^+_-α_+να_-μ+C^-_+α_-να_+μ    ∀ν,μ∈{±}
        
        C^ν_μ· 1   =  ∑_η∈{±} C(η̅) _νη_μη̅= C^+_-α_ν+α_μ-+C^-_+α_ν-α_μ+    ∀ν,μ∈{±}.
    
    
    
    Let e be the only boundary edge of the monogon . Because ()= and ⊗()= (), the -algebra map _e: () →() ⊗_() is an -algebra map _e: →(). As any -linear map, we must have _e(c)= c · 1, where 1 is the unit of (). Apply Δ_e to the simple arc in the monogon whose endpoints are stated by ν and μ and we get a 
     proof of (<ref>) as follows:
    
        C^ν_μ= _e⟶∑_η, η'∈{±}=  ∑_η∈{±} C(η̅) _ην_η̅μ.
    
    Equation (<ref>) is obtained from Equation (<ref>) by applying the map _* of (<ref>).
    
    Using  Theorem <ref> one sees that the set
      B=  {α_++^hα_-+^kα_–^l    | h,k,l∈}∪{α_++^hα_+-^kα_–^l | h,k,l∈ℕ, k ≥ 1}
    is an -basis of (). In particular, () is generated by _νμ with ν,μ∈{±}.
    
    
    Using these relations it is easy to check that any monomial in the _νμ can be expressed as an -linear combinations of   B. The proposition follows.
    
    
     §.§ Counit
     
     The embedding ι: gives rise to an -linear map ι_*: () →()=. Define ϵ: () → as the composition ϵ= ι_* ∘_e_r, 
    
        ϵ: () _e_r⟶() ι_*⟶()=,
    
    where _e_r is defined in Section <ref>.
    Explicitly, if  is a  stated ∂-tangle diagram as in Figure <ref>, then
    ϵ() = C_e_r()  ',
    
    where ' is described in Figure <ref> and  C_e_r() is defined by (<ref>).
    
    counitHow to obtain ' and ” from  in the definition of counit and antipode. Height order  is  indicated by the arrows on the boundary edges. Then ' is the same , but considered as a tangle diagram in  with  its states on the edge e_r switched from ν to ν̅=-ν. And ” is obtained from  by a rotation of 180^o, and switching all the states ν to ν̅.
     4cm
    
    Using (<ref>) and the values of C(η), one can check that 
    ϵ(_νμ) = δ_νμ: =  1      if ν=μ
    
    0    if ν≠μ.
    -18pt1.5 cmcounit3-18pt1.5 cmcounit4-18pt1.5 cmcounit5-18pt1.5 cmcounitx-18pt1.5 cmcounity The algebra () is a bialgebra with  counit ϵ and coproduct .
    
    
    
     We already saw that  is an algebra homomorphism and is associative.
    It remains to show that ϵ is an algebra homomorphism, and 
    
    (ϵ⊗𝕀 ) ∘Δ(x) =x =  (𝕀⊗ϵ  ) ∘Δ(x).
    
    Let x, y∈() be presented by  tangle diagrams schematically depicted as in Figure <ref>. 
    counit2Elements x,y∈(). Each horizontal strand stands for several horizontal lines which are tangled in the two small disks.1.6cm
    
    The following shows that ϵ(xy)= ϵ(x)ϵ(y), i.e. ϵ is an algebra homomorphism:
    
        ϵ(xy)=  ϵ(  ) = 
        C_(x)  C_(y)  
        = C_(x)  C_(y)    
        = ϵ(y) ϵ(x).
    
    Here we use equality (<ref>) in the third identity.
    
    
    As both  and ϵ are algebra homomorphisms, one only needs to check (<ref>) for the generators x=_νμ with ν,μ∈{±}. Using (<ref>) and (<ref>), we have
    
        (ϵ⊗𝕀 ) ∘Δ(_νμ) =  (ϵ⊗𝕀 )  ∑_η_νη⊗_ημ = _νμ,
    
    which proves the first identity of (<ref>). The other identity is proved similarly.
    
    
     §.§ Antipode
    
     Define 
    S: () →(), byS:= _* ∘ ( _∘ ( _)^-1), where  and  are defined in Subsection <ref>.
    Explicitly, if  is a  stated ∂-tangle diagram as in Figure <ref>, then
      S() = C_() / C_() ”,
    
    where ” is described in Figure <ref>. In particular, we have
    
        S(_νμ) = C( μ)/C( ν)_μ̅ν̅.
    
    Explicitly,
     
    S(_++) = _–,  S(_–)= _++,  S(_+-) = -q^2_+-,  S(_-+) = -q^-2_-+.
    
    The map S is an antipode of the bialgebra (), making () a Hopf algebra.
    
    From the definition (<ref>) one  sees that S is an anti-homomorphism, i.e.
    
        S(xy)= S(y) S(x).
    
    It remains to check the following property of an antipode:
    ∑  S(x') x” = ϵ(x) 1 = ∑  x' S(x”) ,
    
    where we use the Sweedler's notation for the coproduct x = ∑ x' ⊗ x”.
    Since S is an anti-homomorphism and ϵ is an algebra homomorphism, it is enough to check (<ref>) for generators x= _νμ. In that case, using  (<ref>) we have
    (x)=  (_νμ)=∑_η_νη⊗_ημ, and
    
        ∑ S(x') x”   = ∑_η∈{±}  S(_νη)  _ημ
            = ∑_η∈{±}C(η)/C(ν)_η̅ν̅_ημ  by  (<ref>)
           = C^ν̅_ μ· 1 /C(ν)  by  (<ref>)
           = δ_νμ· 1 = ϵ(_νμ) · 1   by definition of C(ν) and  (<ref>),
    
    which proves the first identity of (<ref>). The second identity of (<ref>) is proved similarly.
    
    
     §.§ Quantum algebra 
    
    Let us recall the definition of the  quantum coordinate ring  of SL_2(), which is the Hopf dual of the quantum group
    . See <cit.>.
    
    The Hopf algebra  is the -algebra generated by a,b,c,d with 
    relations 
    
        ca    =q^2ac,  db=q^2bd,  ba=q^2ab, dc=q^2cd, 
        
         bc    =cb,   ad-q^-2bc=1    and  da - q^2cb=1 .
    
    Its coproduct structure is given by
     
        Δ(a)=a⊗ a+b⊗ c,    Δ(b)=a⊗ b+b⊗ d,
            Δ(c)=c⊗ a+d⊗ c,     Δ(d)=c⊗ b+d⊗ d.
     
    Its counit is defined as ϵ(a)=ϵ(d)=1,ϵ(b)=ϵ(c)=0
     and its antipode is defined by S(a)=d,S(d)=a,S(b)=-q^2 b,S(c)=-q^-2 c.
    
    There exists a Hopf algebra isomorphism ϕ: () →
    given on the generators by 
    ϕ(α_+,+)=a,    ϕ(α_+,-)=b,    ϕ(α_-,+)=c,    ϕ(α_-,-)=d.
    
    Furthermore, 
    
    under the identification of () with  via the isomorphism ϕ, the involution _*: ()→(),  given by the rotation of 180^∘ around the center of the bigon (see (<ref>)), becomes the -algebra involution r:→ given by
     r(a)=a,r(b)=c,r(c)=b, r(d)=d. Moreover, r is a co-algebra antimorphism, i.e. 
     
        (r⊗ r) ∘Δ^op=Δ∘ r.
    By Lemma <ref>, the -algebra () is generated by _±± with relations (<ref>) and (<ref>). Under the assignment ϕ given 
    on generators _±±
    by (<ref>), 
    the matrix relations (<ref>) and (<ref>) become respectively
    
        ca   = q^2 ac ,  db= q^2 bd,  ad- q^-2 cb=1,  da -q^2 bc=1 
        
        ba   = q^2 ab,  dc= q^2 cd,  ad- q^-2 bc=1, da - q^2 cb=1
    
    Each of these identities is a consequence of the relations in (<ref>) and (<ref>). Conversely, all the relations in (<ref>) and (<ref>), except for bc=cb, are among the identities  (<ref>) and (<ref>). The remaining relation bc=cb is obtained by taking the difference between the last identity of (<ref>) and the last identity of (<ref>). Hence ϕ is an -algebra isomorphism.
    
    
    
    To check that ϕ is a Hopf algebra isomorphism it is sufficient to check this on the level of generators where it is straightforward. 
    
    The last statement is a direct verification.
    
    
     §.§ Geometric depiction of co-R-matrix, a lift of the co-R-matrix
     The Hopf algebra  is “dual quasitriangular” (see  <cit.> Section 2.2) or “cobraided” (see e.g. <cit.>, Section VIII.5) , i.e. it has a co-R-matrix with the help of which one can make the category of -modules a braided category. Formally, a  co-R-matrix  is a bilinear form 
    
        ρ : ⊗→
     such that there exists another bilinear form ρ̅: 
    ⊗→ (the “inverse" of ρ)
    satisfying for any x,y,z∈ U,
    
        ∑ρ (x'  y') ρ̅(x” y”)   = ∑ρ̅(x'  y') ρ (x” y”)= ϵ (x) ϵ(y) 
        ∑ρ (x” y”) y'x'     = ∑ρ (x' y') x” y”
        ρ (xy  z)    = ∑ρ (x'  z') ρ  (y” z”) ϵ(x”) ϵ (y')  
        
        ρ (x  y z)    = ∑ρ  (x'  z') ρ (x” y”) ϵ(z”) ϵ (y')
    
    Here we use Sweedler's notation for the coproduct.
    Relations (<ref>) and (<ref>) show that ρ is totally determined by  its values at a set of generators of the algebra , and the values ρ at a set of generators are given by (see <cit.>)
    ρ[  a a b  b  a b b  a;  c c d  d c  d d  c;  a c b  d a  d b  c;  c a d  b c  b d  a ] = 
    [        q        0        0        0;        0        q        0        0;        0        0     q^-1 q - q^-3;        0        0        0     q^-1 ]-9pt.9 cmx-9pt.9 cmy-9pt.9 cmxy-9pt.9 cmxybar-9pt.9 cmyxbar-10pt1 cmxyz-10pt1 cmxyzp-10pt1 cmxyzpp-10pt1 cmbc-10pt1 cmbcp-10pt1 cmbcpp∂ Under the identification of () and  via the isomorphism ϕ,
     the co-R-matrix ρ and its inverse ρ̅  have the following geometric description
     
        ρ ()     = ϵ(  )  
        ρ̅ ()     = ϵ(  ).
    Here a circle enclosing x and two lines adjacent to the circle stand for a stated -tangle diagram, also denoted by x. The left hand side of (<ref>) stands for ρ(x y). 
     Let ρ' be the map defined by the r.h.s. of (<ref>): we will show that ρ'=ρ. For this it is enough to show that ρ' satisfies (<ref>), (<ref>), and the initial values identity (<ref>), all with ρ replaced by ρ'. 
    We have, where a line labeled by, say x, stands for the stated -tangle diagram x,
    
        ρ'(xy  z)= ϵ()
    
    Split the bigon by the vertical middle ideal arc, then use the fact that ϵ(u) =∑ϵ(u') ϵ(u”) (in any Hopf algebra) where Δ(u)= ∑ u'  u”, we have
    
        = ∑ϵ() ϵ()   
           = ∑ρ  (x'  z') ρ  (y” z”) ϵ(x”) ϵ (y').
    
    This proves (<ref>) for ρ'. The proof of (<ref>) is similar.
    
    To check (<ref>) we have to check 16 identities, all of which are easy. For example, the most difficult one is the identity of the (3,4) entries:
    
        ρ'(b c)    = ϵ( ) = q  ϵ( ) + q^-1 ϵ( )  
           = q - q^-3    by (<ref>) and (<ref>).
    
     This proves (<ref>) for  the (3,4) entries. Identity (<ref>) for  other entries are similar.
     The bilinear form ρ':  ⊗→ defined by
    ρ'(x ⊗ y) := ρ̅(y ⊗ x) =  ϵ(  ) 
    gives a new co-R-matrix for , which is the mirror reflection of ρ.
    
     §.§ The Jones-Wenzl idempotents as elements of the bigon algebra
    
    In this subsection we will work over the ring ^loc obtained by localizing  over the multiplicative set generated by {[n]=q^2n-q^-2n/q^2-q^-2, n≥ 1}.
    Recall that the n^th Temperely-Lieb algebra T_n (see e.g. <cit.>) is the ^loc-algebra generated by non-stated simple (n,n)-tangle diagrams in  modulo isotopy (rel to the boundary) and relation (<ref>). The product is obtained by concatenating horizontally. 
    
    
    A ^loc-basis of T_n is given by simple (n,n)-tangle diagrams without closed components; 
    define ϵ:T_n→^loc be the dual of the element 1 with respect to this basis. 
    The n^th Jones-Wenzl idempotent is an element JW_n∈ T_n defined by recursion as explained in Figure <ref>.
    
    
    The following is the key property of the JW_n, see e.g. <cit.>.
     One has ϵ(JW_n)= 1. 
    For all x∈ T_n, 
    it holds JW_n  x=x JW_n=ϵ(x)JW_n. In particular JW_n^2=JW_n.
    
    For a simple (n,n)-tangle diagram x∈ T_n and μ⃗_⃗l⃗,μ⃗_⃗r⃗∈{±}^n let x(μ⃗_⃗l⃗,μ⃗_⃗r⃗) be the stated -tangle diagram which is x with states μ⃗_⃗l⃗ on e_l and states μ⃗_⃗r⃗ on e_r, and the height order on each of e_l and e_r is from bottom to top. By linearity, for  y∈ T_n, we define  y(μ⃗_⃗l⃗,μ⃗_⃗r⃗). This is well-defined since (<ref>) is part of the defining relations of stated skein algebra. Thus, if y is trivial (n,n)-tangle diagram, then y(, ) is the element _ described in Figure <ref>.
     If all the components of  are the same and all the components of  are the same, then 
    JW_n(, )= _.
     Indeed JW_n is equal to trivial (n,n)-tangle diagram plus the linear combination of diagrams each contains an arc whose endpoints are both in e_l or in e_r; such an arc is 0 by (<ref>).
     For ∈{±}^n let o()  be obtained by reordering increasingly the states of  and no(μ⃗) is the minimal number of exchanges needed to do so. Then
    
        JW(μ⃗_⃗l⃗,μ⃗_⃗r⃗)    =q^2 no(μ⃗_⃗l⃗)+2 no(μ⃗_⃗r⃗) JW(o(μ⃗_⃗l⃗),o(μ⃗_⃗r⃗)) 
        Δ(JW(μ⃗_⃗l⃗,μ⃗_⃗r⃗))   =∑_j=0^n nj_q^4 JW(μ⃗_⃗l⃗,η⃗_⃗j⃗)⊗ JW(η⃗_⃗j⃗,μ⃗_⃗r⃗),
    
    where η⃗_⃗j⃗ is the increasing state containing j signs + and n-j signs -.
    
    Observe that if one exchanges a sign - and a + which are not in the increasing order along e_r then by (<ref>) one gets q^2 times the reordered term and q^1/2 times a term killed by JW_n. Since a similar argument (using Lemma <ref>) shows that each reordering along e_l multiplies JW by q^2,  the first statement follows. 
    The second statement is a consequence of the fact that JW_n^2=JW in T_n  and Proposition <ref>. 
    B_can
    
     §.§ Kashiwara's basis for 
    
     We will see that the celebrated Kashiwara canonical basis of , see <cit.>, is the geometrically defined basis B(,_+) of Theorem <ref>, up to powers of q. Here _+ is the positive orientation of the boundary  of the bigon .
    
    bigon8Elements β_ (left) and β'_ (right) with =(-,+,+) and =(–+); the difference is the direction of the left edge.2cm
    
    First recall B(,_+). For sequences ,∈{±}^n
     let β_, be the stated -tangle diagram consisting of n horizontal arcs and stated by  on the left edge and  on the right edge, with height order given by _+; see 
     Figure <ref>. 
    Then 
     
        B(,_+)= {β_,|,  are increasing}.
    
    Let β'_, be the same β_, with reverse height order on the left edge, see Figure <ref>. Define
    
        := {β'_,|β_,∈ B(;_+) }
        ={_,| is decreasing,  is increasing.}
         .
    
     Here _, is defined in Figure <ref>.
     
    From Lemma <ref> which deals with height exchange, one can easily show that
    β'_, = q^h()β_,,  h():= 1/2(n_+ n_- + n_+ + n_- -n_+^2 - n_-^2  )
    
    where n_+ (respectively n_-) is the number of + (respectively -) in the sequence . It follows that  is also an -basis of ().
    
    
    Via the isomorphism of Theorem <ref>, the basis  coincides with the canonical basis defined by Kashiwara  <cit.>. Both bases  and B(;_+) are positive with respect to the product and to the coproduct, i.e. for B= or B(;_+) and α,β∈ B one has
    
        αβ∈ℕ[q^±1]· B
    
        Δ(α)∈ℕ[q^±1]· B⊗ B.
    
    The first statement is an observation directly following Theorem <ref>: in <cit.> the basis is {c^la^mb^n, l,m,n≥ 0}⊔{c^ld^mb^n, l,n≥ 0, m>0}.
    
    As B(;_+) is equal to  up to powers of q, one needs only to prove the second statement for .
    
    For positivity of multiplication, it is sufficient to check it on pairs of generators: there are then 16 cases. 
    All of them are straightforward; we provide some instances among the most complicated cases  where the right hand sides are in  ℕ[q^±1]·:
     
        α_+-·α_-+=α_-+·α_+-, α_+-·α_–=q^-2α_–·α_+-, α_–·α_-+=q^2α_-+·α_–
    
        α_+-·α_++=q^2α_++·α_+-, α_++·α_-+=q^-2α_-+·α_++, α_++·α_-+=q^-2α_-+·α_++.
    
    
    Once positivity is known for multiplication, the statement for comultiplication can be checked on generators where it is straightforward. 
     A direct proof of positivity using pictures is also easy and left to the reader.
    
    
    § COMODULE STRUCTURES, CO-TENSOR PRODUCTS AND BRAIDED TENSOR PRODUCTS
    Δ_e _eΔ-bal
    
    
    
    
    In this section we show that given any edge of  the skein algebra  has a natural structure of -comodule algebra. We show how to decompose this comodule into finite dimensional comodules. We then identify the image of the splitting homomorphism using the Hochshild cohomology, and give a dual result using Hochshild homology. When  is the result of gluing two surfaces _1 and _2 to two edges of an ideal triangle, we show that the skein algebra  is canonically isomorphic to the braided tensor product of (_1) and (_2). In this section =[q^± 1/2].
    
    
    
    
     §.§ Comodule
     Suppose  e is a boundary edge of a  punctured bordered surface . Recall that by cutting out of  a bigon  whose right edge is e and canonically identifying ∖ int() with , we get an -linear map
    
        : () →() ⊗(),
    
    see Figure <ref>.
    Similarly cutting out of  a bigon  whose left edge is e and canonically identifying ∖ int() with , we get an -linear algebra homomorphism
    
        : ()→(⊔) ≡() ().
     (a) The map _e: () →() ⊗() gives () a right comodule-algebra structure over the Hopf algebra (). Similarly  gives gives () a left comodule-algebra structure over the Hopf algebra (). 
     
     (b) It holds =∘ (_()⊗_*)∘ where (x⊗ y)=y⊗ x and _*:()→() is the algebra involution defined by (<ref>).
    
    (c)  If e_1,e_2 are two distinct boundary edges, the coactions on the two  edges commute, i.e.  
        (Δ_e_2⊗𝕀)∘Δ_e_1=(⊗𝕀)∘(Δ_e_1⊗𝕀)∘Δ_e_2.
     (a) The associativity of _e follows from the
    the commutativity  of the splitting maps of theorem <ref>(c). Applying (𝕀ϵ) to Equation (<ref>) and using the value of ϵ(_) from (<ref>), we get that 
    
        (𝕀ϵ) _e(x) = x_=x,   ∀ x ∈ B(,).
    
    Hence _e gives () the structure of a right ()-comodule.
    
    Recall that () is a comodule-algebra over the bialgebra (), see e.g. <cit.>, if and only if the map : () →() ⊗() is an algebra homomorphism. The last fact follows easily from the definition of _e.
    
    (b) Observe that (_*⊗_*)∘Δ^op=Δ∘_*. 
    
    (c) is clear from the definition.
    
    By identifying  with  using Theorem <ref>, the above proposition also provides  with the structure of a -comodule. 
    More in general, we will use the following terminology:
    
    A punctured bordered surface  has indexed boundary if its boundary edges are partitioned into two ordered sets (the left and right ones, with indices L and R respectively): e^L_1,… e^L_n, e^R_1,… e^R_m.
    
    
    If  has indexed boundary then ()  is naturally endowed with a structure of 
        (^⊗ n,^⊗ m)- bicomodule
     by the left coaction Δ^L:()→^⊗ n⊗() defined by 
    
        Δ^L:=
        (_^⊗ n-1⊗e^L_n)∘ (_^⊗ n-2⊗e^L_n-1)∘⋯∘ (_⊗e^L_2)∘e^L_1
     and the right coaction 
    
        Δ^R:=(Δ_e^R_1⊗_^⊗ m-1 )∘ (Δ_e^R_2⊗_^⊗ m-2 )∘⋯∘ (Δ_e^R_m-1⊗_ )∘Δ_e^R_1.
     
    
    Furthermore notice that () is not only a bicomodule but a bicomodule-algebra as each of the above maps Δ_e^R_i or e^L_j are also morphisms of algebras. 
    
    
    
    
     §.§ Quantum group 
    
    Recall that the quantized enveloping algebra  is the Hopf algebra generated over the field ℚ(q^1/2) by K^±1,E,F with relations 
        KE=q^4 EK,   KF=q^-4FK,   [E,F]=K-K^-1/q^2-q^-2.
    
    The coproduct and the antipode are given by 
    
        Δ(K)=K⊗ K,  Δ(E)=1⊗ E+E⊗ K,   Δ(F)=K^-1⊗ F+F⊗ 1
        
        S(K)=K^-1, S(E)=-EK^-1, S(F)=-KF.
    
      
    We emphasize that  is defined over the field (q^1/2). There is an integral version U^L_q^2(𝔰𝔩_2), defined by Lusztig <cit.>, which is the -subalgebra of  generated by K^±1 and the divided powers  E^(n):=E^n/[n]!,F^(n):=F^n/[n]!. Here [n]! = ∏_i=1^n (q^2i - q^-2i) /(q^2 - q^-2).
    
    
    
    One has a non degenerate Hopf pairing  
    ·, ·: _→.
    
    This is a Hopf duality since it satisfies (with Sweedler's coproduct notation)
     x, y_1 y_2  = ∑ x', y_1  x”, y_2 ,     x_1 x_2, y = ∑ x_1, y'  x_2, y”
    The values of the form on generators are given by
    
        ⟨ K, [ a b; c d ]⟩= [  q^2    0;    0 q^-2 ],  ⟨ E, [ a b; c d ]⟩= [ 0 1; 0 0 ],  ⟨ F, [ a b; c d ]⟩= [ 0 0; 1 0 ].
     The form (<ref>) on U^L_q^2(𝔰𝔩_2) is integral, i.e. it restricts to a map
    
        U^L_q^2(𝔰𝔩_2)_→= [q^±1/2]  .
     It is enough to check that ⟨ E^(n),x⟩, ⟨ F^(n),x⟩∈ for n≥ 1 and x∈. 
    Since Δ(E^(n))=∑_i=0^n q^2i(n-i)E^(i)⊗ E^(n-i)K^i it is sufficient to check the statement for the evaluations of E^(i) and K^j on a,b,c,d where this is a straightforward computation. Similarly for F^(n). 
    Recall that the rotation by 180^∘ of the bigon induces the involution r:→, see Theorem <ref>. Let r^*:→ be the adjoint of the map r. We will show that r^* is equal to the map ρ of Lusztig's book <cit.>, which is used in the study of canonical bases of quantum groups. 
    
    
    
    Let -Mod (respectively Mod-) be the monoidal category of left (respectively right) -modules.
    
    
    
    (a) The map  r^* is an algebra antimorphism involution and a coalgebra morphism, i.e. for x,y∈  one has 
        (r^*)^2(x)=x     r^*(xy)=r^*(y)r^*(x),   Δ(r^*(x))=(r^*⊗ r^*)∘Δ(x).
    
    
    Explicitly, the value of r^* on the generators is 
     r^*(E)=q^2KF,      r^*(K)=K,     r^*(F)=q^-2EK^-1.
    
    
    (b) The map r^* induces monoidal functors 
        LR:-Mod→ Mod-,  and RL:Mod-→-Mod
     which are inverse to each other as follows: for each left (resp. right) module M
     the module LR(M) (resp. RL(M)) is the right (resp. left) module whose underlying vector space is M and on which the action of x∈ 
     is given by, with on α∈ M,
      
        α· x:=r^*(x)·α   (resp.   x·α:= α· r^*(x) ).
     Formula (<ref>) shows that r^* is equal to ρ of <cit.>.
    
    (a) Since r is a algebra involution and a co-algebra antimorphism by  Theorem <ref> and the Hopf pairing is non-degenerate,  its dual r^* is an algebra antimorphism involution and a coalgebra morphism.
    
    
    
    It is sufficient to compute r^* on the generators where one can verify the values provided in the statement.
    For instance:
        ⟨ r^*(E),a⟩=⟨ E,a⟩=0=⟨ q^2 KF,a⟩=⟨ q^2K⊗ F,a⊗ a+c⊗ b⟩
    
        ⟨ r^*(E),b⟩=⟨ E,c⟩=0=⟨ q^2 KF,b⟩=⟨ q^2K⊗ F,a⊗ b+b⊗ d⟩
    
        ⟨ r^*(E),c⟩=⟨ E,b⟩=1=⟨ q^2 KF,c⟩=⟨ q^2K⊗ F,c⊗ a+d⊗ c⟩
    
        ⟨ r^*(E),d⟩=⟨ E,d⟩=0=⟨ q^2 KF,d⟩=⟨ q^2K⊗ F,c⊗ b+d⊗ d⟩
    
    
    The verification for the pairings with a,b,c,d for r^*(F) and r^*(K) are similar. 
    
    
    (b) LR and RL are functors as r^*:→ is an algebra antimorphism; they are inverse to each other as r^* is an involution. Monoidality is a consequence of the fact that r^* is a coalgebra morphism. 
    
    
     §.§ Module structure of ().
     As usual, the Hopf duality implies that 
    every right (resp. left) -comodule V  has a natural structure of a left (resp. right) -module, via the following construction. For a ∈ and v ∈ V, one has
      a · v:= ∑ a, b'  v',   where  Δ_r(v) = ∑ v'  b'.
    
    
    To be precise, we have to replace V by V_, since  is defined over .
    
     
     _
     
     
    In particular, for an edge e of  the right comodule structure : → gives  a left module structure over , and we want to understand this module structure.
    
     
     Fix an orientation  of the boundary . Recall that B(;) is an -basis of (). For each edge e let B_e,d(;) ⊂ B(;) be the set of all ∈ B(;) such that |∩ e|=d and all the states on ∩ e are signs +. Let
    B_e(;) = ∪ _d=0^∞ B_e,d(;). 
    For α∈ B_e,d(;) and η⃗∈{±}^d, let α(η⃗) be the same α except for the states of e∩α which are given by s(x_i)= η_i, where x_1, …, x_d are the points of ∩ e listed in decreasing order. In particular let α_j:=α(+,+,⋯ ,+,-,-, …,-) where the number of - is j. For example, α=α_0.
     
    The left action of  on , dual to ,  is: 
    
        K_left(α_j)=q^2(d-2j)α_j
    
        E_left(α_0)=0,  and E_left(α_j)=[j]_q^2α_j-1 F^e_d-1
    
        F_left(α_d)=0,  and F_left(α_j)=[d-j]_q^2α_j+1  F^e_d-1
    
    where  F^e_d-1= F^e_d-1() is the -span of elements β∈ B(;) with |β∩ e| < d. 
    The right action, dual to , is given by the left action of r^*(K),r^*(E),r^*(F) (see Lemma <ref>). 
    
    
    
    By (<ref>),  Δ(α(η⃗))=∑_ϵ⃗∈{±}^dα(ϵ⃗) ⊗α_ϵ⃗η⃗ where  α_ϵ⃗η⃗ is defined in Figure <ref>. If we define inductively Δ^[d] = (Δ𝕀^ (d-2) )∘Δ^[d-1] for d≥ 3, with Δ^[2] = Δ, then
    
        Δ^[d] (K)   = K^ d,
        Δ^[d] (E)   = ∑_j=1^d 1^ (j-1) E  K^  (d-j),  
        Δ^[d] (F)   = ∑_j=1^d (K^-1) ^ j F  1^ d-j-1.
    
    Applying these to  compute the Hopf pairing of K,E,F with α_ϵ⃗η⃗ we get
    
        K_left(α(η⃗))=q^2∑η_iα(η⃗),
    
        E_left(α(η⃗))=∑_j=1^d (δ_η_j,-) q^2∑_k=j+1^d η_kα(η_1,⋯ ,η_j-1, +, η_j+1,⋯, η_d),
    
        F_left(α(η⃗))=∑_j=1^d (δ_η_j,+) q^-2∑_k=1^j-1η_kα(η_1,⋯ ,η_j-1,-,η_j+1,⋯ ,η_d).
    
    Now the main claim is a direct computation using relation (<ref>).
    
    
    
    
    
    
    
    Let   have indexed boundary ∂={e^L_1,… e^L_m, e^R_1,… e^R_n} as explained in Subsection <ref>. The Hopf duality gives  an  algebra bimodule structure over (^⊗ n,^⊗ m). (Notice the inversion between left and right when passing to modules). 
    
    For each m⃗∈ℕ^m and n⃗∈ℕ^n, let B_m⃗,n⃗(;) be defined as:
    
        B_m⃗,n⃗(;)=(⋂_i=1^m B_e_i^L,m_i(;))∩(⋂_j=1^n B_e_j^R,n_j(;)).
    
    Let also, for each j⃗≤m⃗ and h⃗≤n⃗ (component-wise) and each α∈ B_m⃗,n⃗(;) let α_j⃗,h⃗∈ B(;) be the skein identical to α but for its state which is increasing and contains j⃗ (resp. h⃗) signs - on the left (resp. right) edges. 
    
    
     Suppose that  has indexed boundary ∂={e^L_1,… e^L_n, e^R_1,… e^R_m}.
    
    (a) For each m⃗∈ℕ^m and n⃗∈ℕ^n and each ∈ B_m⃗,n⃗(;), 
     the (^⊗ n,^⊗ m)-bimodule generated by α (namely ^⊗ n·α·^⊗ m) is irreducible and isomorphic to V^L_n_1⊗⋯⊗ V^L_n_n⊗ V^R_m_1⊗⋯⊗ V^R_m_m, where V^L_k (resp. V^R_k) is the irreducible left (resp. right) module on  with highest weight k. 
    
    (b) As (^⊗ n,^⊗ m)-bimodules, we have
     = ⊕^⊗ n·α·^⊗ m.
    
    where the sum is taken over all m⃗∈ℕ^m,n⃗∈ℕ^n, and all α∈ B_m⃗,n⃗(;).
    In particular, the bimodule () is a direct sum of finite dimensional bimodules over (^⊗ n,^⊗ m).
    
    (c) Furthermore the bimodule structure restricts to that of a (U^L_q^2(𝔰𝔩_2)^⊗ n,U^L_q^2(𝔰𝔩_2)^⊗ m)-bimodule (where U^L_q^2(𝔰𝔩_2) is the integral version of ) and a decomposition similar to the above one holds:
    () = ⊕_j⃗≤m⃗,h⃗≤n⃗( U^L_q^2(𝔰𝔩_2)^⊗ n·α_j⃗,h⃗· U^L_q^2(𝔰𝔩_2)^⊗ m)
    
    where the direct sum is taken over all m⃗∈ℕ^m,n⃗∈ℕ^n and all α∈ B_m⃗,n⃗(;), and the _j⃗,h⃗ symbol stands for the non direct sum.
     
     
    
    (a) Fix m_i, n_j and α as in the statement and let JW(α) be the skein obtained by inserting a JW_m_i near e^L_i and a JW_n_j near e^R_j for all i,j. By Example <ref> it is clear that =JW() and by Lemma <ref> that it is a highest weight vector of weight q^2m_i for the action of the i^th-copy of  for each i≤ m; similarly it is a highest weight vector of weight q^2n_j for the right action of the j^th-copy of .  Furthermore, by Lemma <ref> and the fact that the m^th-Jones Wenzl projector kills the self-returns, the orbit of  is exactly the span of the vectors JW(α_j⃗,h⃗) with j⃗≤m⃗ and h⃗≤n⃗. 
    
    (b)
    It is straightforward from a) and from Theorem <ref>.
    
    (c) If a left (resp. right) -module weight M (over ) has a basis formed by weight vectors over which the action of E^(r), F^(r),r≥ 1 has coefficients in , then M restricts to a U^L_q^2(𝔰𝔩_2)-module; 
     we claim that  the basis B(;) of () has this property. Indeed since the structure of module is induced by the right (resp. left) comodule structure on each edge and the Hopf pairing between  and , and since the comodule structure is integral in the basis B(;), it is sufficient to observe that the Hopf pairing between  and =() extends to a -bilinear Hopf pairing between U^L_q^2(𝔰𝔩_2) in the basis B(,): this is the content of point (1) of Lemma <ref>. 
    
    To prove that the direct sum decomposition still holds,
    let 
     
        B^<_m⃗,n⃗(;):={α∈ B(;)|# (α∩ e^L_i)≤ m_i ,# (α∩ e^R_j)≤ n_j, ∀ i≤ m, ∀ j≤ n }∖ B_m⃗,n⃗(;).
     
     
    To prove the claim, we will show that for each α∈ B_m⃗,n⃗(;), the following holds:
    
        (· B^<_m⃗,n⃗(;))⋂(_j⃗≤m⃗,h⃗≤n⃗( U^L_q^2(𝔰𝔩_2)^⊗ n·α_j⃗,h⃗· U^L_q^2(𝔰𝔩_2)^⊗ m))={0}.
    
    
    
    We start by remarking that if α∈ B_m⃗,n⃗(;), then α=JW(α) and so, over , its orbit is a direct summand of () and thus it has trivial intersection with the -span of B^<_m⃗,n⃗(;): 
    
        (· B^<_m⃗,n⃗(;))⋂( ^⊗ n·α·^⊗ m)={0}.
    
    
    
    Furthermore, by the point (b), given j⃗≤m⃗,h⃗≤n⃗,  there exist c(j⃗,h⃗)∈∖ 0 and ℓ(j⃗,h⃗)∈· B^<_m⃗,n⃗(;) such that c(j⃗,h⃗) α_j⃗,h⃗+ℓ(j⃗,h⃗) is in the orbit of α: 
    
        c(j⃗,h⃗) α_j⃗,h⃗+ℓ(j⃗,h⃗)∈ U^L_q^2(𝔰𝔩_2)^⊗ n·α· U^L_q^2(𝔰𝔩_2)^⊗ m.
     
    
    
    Now suppose that for some l_i∈ U^L_q^2(𝔰𝔩_2)^⊗ m and r_i∈  U^L_q^2(𝔰𝔩_2)^⊗ n and some j⃗_i,h⃗_i it holds 
        ∑_i l_i·α_j⃗_i,h⃗_i· r_i∈· B^<_m⃗,n⃗(;)∖{0}.
    
    Then, multiplying by ∏_i c(j⃗_i,h⃗_i) (which gives a non-zero vector as () is free as a -module) we also get that 
    ∑_i l_i ·α· r_i ∈· B^<_m⃗,n⃗(;)∖{0},
    which as already argued is impossible. 
    
    
    Let  be the bigon whose edges e_l and e_r are declared to be respectively of type L and R. Then by Theorem <ref>, () is the right and left module -module : the left action is induced by the right comodule structure coming from e_r^R and the right action from e_l^L. 
    If we let ^R be the bigon where both e_l and e_r are declared to be of type R (right), then (^R) is a left ()^⊗ 2-module; the action of x⊗ y on a skein b∈(^R) is given by 
    
        (x⊗ y) · b=x· b· r^*(y)
     where r^*(y) is the algebra antimorphism provided in Lemma <ref> and the left and right actions are those on () described above. 
    
    
     §.§ Co-tensor product
     Suppose  U is a coalgebra over a ground ring . Assume 
    M is a left U-comodule with coaction  Δ_M:M→ U⊗_ M, and N a right U-comodule with coaction Δ_N:N→ N⊗_R U. 
    Then the cotensor product N  _U M  is
    
        N_U M:={v∈ N⊗ M | (Δ_N⊗𝕀_M)(v)=(𝕀_N⊗Δ_M)(v)}.
    
    
    Cotensor product is a special case of the following notion of Hochshild cohomology. 
    Assume V is a -module with a left U-coaction and a right U-coaction:
    
        _r: V → V  U,    _lΔ: V → U  V .
    
    The 0-th Hochshild cohomology of V is defined by
    
        HH^0(V)= { x∈ V |_r(x) =  (_l(x)),
    
    where : V  U→ U  V is the flip (x y) = y  x.
    
    With M and N as above, define a left U-coaction and a right U coaction on N _R M by
    
        _r    : N _R M → N _R M _R U,   _r(n ⊗ m) = ∑ n'  m  u'  if _N(n) = ∑ n'  u' 
        _l   : N _R M →  U _R N _R M,   _l (n ⊗ m) = ∑ u” n  m” if _M(m) = ∑ u” m”.
    
    
    Then the cotensor product N_U M= HH^0(N⊗_ M).
    
    
    
    
    
     §.§ Splitting as co-tensor product and Hochshild cohomology
     _c_c _l_r
    
    Suppose c_1, c_2 are distinct boundary edges of a punctured bordered surface ' and = '/(c_1=c_2), with c⊂ being the common image of c_1 and  c_2.  The splitting homomorphism gives an embedding 
    θ_c: () (') and  we will make precise the image of θ_c. 
    
     
     
     
     
     split2(a) The middle shaded part is the bigon, while the left and the right shaded parts are part of '. Gluing c_1=e_l gives the right coaction  and gluing =c_2 gives the left coaction .
     (b) Element x _∈('). The horizontal lines are part of x. Note the order of indices.2.6cm 
    Suppose c_1, c_2 are distinct boundary edges of a punctured bordered surface ' and = '/(c_1=c_2). The splitting homomorphism  
    θ_c: () (') .
    
    maps  isomorphically onto the Hochshild cohomology HH^0((')), which is a ()-bimodule via the left coaction := _c_2 and the right coaction := _c_1 (see Figure <ref>(a)).
    
    
    In particular, if c_1 is a boundary edge of '_1 and c_2 is a boundary edge of '_2 which is disjoint from '_1, and =('_1 ⊔'_2)/(c_1=c_2),  then θ_c maps () isomorphically onto
    the cotensor product of ('_1) and ('_2) over ().
    coefτ⃗ Let us identify  with its image under θ_c.
     From the splitting formula (<ref>) it is easy to see that ()⊂ HH^0((')). Let us prove the converse inclusion.
    Assume  0≠ v ∈ HH^0((')).  By definition, this means
    (v) - ((v))=0.
    
    
    
    Choose an orientation  of  and an orientation of c. Then  and the orientation of c_1 and c_2 induced from c give an orientation ' of '. 
    Recall that  B(';') is a free -basis of ('). Let B̃(';') be the set of all
    isotopy classes x of '-ordered '-tangle diagrams which are increasingly stated on every boundary edge except for c_1 and c_2. 
    If  is a state of x ∩ c_1 and  is a state of x∩ c_2, let x _ be the  stated '-ordered '-tangle
     diagram whose states on x ∩ c_1 and x ∩ c_2 are respectively  and . See Figure <ref>(b). 
    If  and  are increasing, then x _∈ B(', ') is a basis element. For each i=1,2 let S_x∩ c_i and S_x ∩ c_i^↑ be respectively the set of all states and the set of all increasing states of x ∩ c_i. Then
    
        B(',') = { x_| x ∈B̃(', '), ∈ S_x ∩ c_1^↑, ∈ S_x ∩ c_2^↑}.
    
    Using the above  -basis B(';') of ('), we can present v∈(') in the form
     v=∑_x∈ X(v) ∑_∈ S_x ∩ c_1^↑ ∑_∈ S_x ∩ c_2^↑ (v, x_)   x_,
    
    where X⊂B̃(';') is a minimal finite set, so that for each x∈ X, there are , such that  the coefficient  (v, x_) is non-zero.
    
    Let m(v)=max{ |x ∩ c_1|, |x∩ c_2|,  x∈ X(v)}. We show by induction on m(v) that v ∈().
    
    For i=1,2 let X_i(v)= { x∈ X(v) , |x ∩  c_i| = m}.   If x∈ X_1(v)∩ X_2(v), then |x∩ c_1|=|x ∩ c_2|= m(v), and there is an element x̅∈(,) such that x has coefficient non-zero in the result of splitting x̅ along c. From the definition of the splitting map we have
    (θ(x̅), x_)= 1, 
    where = (+)^m is the state consisting of m plus signs. Let 
    
        v' = v - ∑ _x∈ X_1(v) ∩ X_2(v)(v, x_ )  θ(x̅).
     
    If m(v') < m(v) then  we are done by induction. Assume that m(v')=m(v). One of X_1(v'), X_2(v') is not empty, and
    without loss of generality assume X_2(v') ≠∅. Formula (<ref>) for v' has the form
     
    v'=∑_x∈ X(v') ∑_∈ S_x ∩ c_1^↑ ∑_∈ S_x ∩ c_2^↑ (v', x_)   x_,
    
    and because of (<ref>) we can assume that there is no x_ττ on the right hand side of (<ref>).
    
    Let p^c_2_m:  (') →(') be
     the projection onto the homogeneous part G^c_2_m((')),  and p^_m: () →() be the projection onto the homogeneous part G^_m(()), see the end of Section <ref>. Explicitly, for x∈ X(v') we have
    
    p^c_2_m (x_)=  0      if ≠
    
    x_   if  = τ,
       p^_m (_)=  0      if ≠τ
    _   if  = .
    
    
    
    From Formula (<ref>) for the coaction, we have, for x∈ X(v') and (,) ≠ (,), 
    
        x _   ⟶∑ _∈  S_x ∩ c_1  x _ _p^c_2_m  p^_m⟶ 0 
        x _   ⟶∑ _∈  S_x ∩ c_2 _   x _⟶∑ _   x _ _p^c_2_m  p^_m⟶   0      if  x ∉X_2(v') 
        
        x_ _     if  x ∈ X_2(v').
    
    It follows that
    
        0= (p^c_2_m  p^_m )( ( (v')) - (v'))=  ∑_x∈ X_2(v')∑_∈ S_x ∩ c_1^↑ ∑_∈ S_x ∩ c_2^↑ (v', x_)   x_ _.
    
    As the right hand side is a linear combination of elements of a basis, all the coefficients 
    v there are 0. This means X_2(v')=∅, a contradiction. Thus m(v') < m(v) and we are done.
    
    Theorem <ref> holds also if we change the base ring to ℂ by evaluating q to a non-zero complex number. 
    (') _
    Using the above result together with Theorem <ref> we can deduce a similar result for -modules.
    Recall that for a bi-module V over a -algebra U the 0-homology group is
    
        HH_0(V) = V/- span  a · v - v · a | a ∈ U, v ∈ V.
    
    Suppose c_1, c_2 are distinct boundary edges of a punctured bordered surface ' and = '/(c_1=c_2), with c being the common image of c_1 and c_2. Then the composition
    θ_c⟶(') _R → HH_0((') _R )
    
    is an isomorphism of -vector spaces. Here  is a -bimodule via the dual actions of Δ_c_1 and c_2.
    
    
    
    In particular, if '= _1 ⊔_2 with c_1 ⊂_1 and c_2 ⊂_2, then the map in (<ref>) is an isomorphism between   and  ((_1) _R ) _ ((_2) _R ). 
    
    
    The decomposition of (') _R given by part (b) of Theorem <ref> shows that it is sufficient to prove that if V^R_m_2 (resp. V^L_m_1) is the irreducible m_2+1-dimensional (resp. m_1+1-dimensional) right (resp. left) -module, then the composition of natural map
      HH^0(V^L_m_1⊗ V^R_m_2) ↪ V^L_m_1⊗ V^R_m_2 
     HH_0( V^L_m_1⊗ V^R_m_2)
    
    is an isomorphism of vector spaces. We will see that this follows from the fact that every finite-dimensional -module V is equivalent to its dual V^*.
    
    First observe that since the pairing between  and  is non degenerate, HH^0(V^L_m_1⊗ V^R_m_2) can be equivalently defined as 
    
        HH^0(V^L_m_1⊗ V^R_m_2)={v⊗ w∈ V^L_m_1⊗ V^R_m_2| x· v⊗ w=v⊗ w· x, ∀ x∈}.
    
    Then using the isomorphism between V^R_m_2 and (V^L_m_2)^* we have :
    
        HH^0(V^L_m_1⊗ V^R_m_2)=Hom_(V^L_m_2,V^L_m_1)=δ_m_1,m_2
    
    by Schur's lemma.
    
    Now let's take the dual of the above equation and get: 
      (HH_0( V^L_m_1⊗ V^R_m_2))^*↪ 
      (V^R_m_2)^*⊗ (V^L_m_1)^* (HH^0(V^L_m_1⊗ V^R_m_2))^*
     
     where the first arrow maps an element of (HH_0( V^L_m_1⊗ V^R_m_2))^* to some f∈ (V^L_m_1⊗ V^R_m_2)^* such that f(x · v⊗ w)=f(v⊗ w· x) for all x∈ and v⊗ w∈ V^L_m_1⊗ V^R_m_2.  Using again the isomorphism between (V^R_m_2)^* and V^L_m_2 we have that the image of (HH_0( V^L_m_1⊗ V^R_m_2))^* in V^L_m_2⊗ (V^L_m_1)^* is Hom_(V^L_m_1,V^L_m_2)=δ_m_1,m_2 by Schur's lemma. 
     
     To conclude, observe that if m_1=m_2 then the image of the inclusion HH^0(V^L_m_1⊗ V^R_m_1) ↪ V^L_m_1⊗ V^R_m_1≃ V^L_m_1⊗ (V^L_m_1)^*=Hom(V^L_m_1,V^L_m_1) is given by the multiples of the identity map. But the kernel of the projection V^L_m_1⊗ V^R_m_1 HH_0(V^L_m_1⊗ V^R_m_1) is the sub vector space of Hom(V^L_m_1,V^L_m_1) spanned by the matrices of the form xM-Mx where x represents the action of an element of  and M∈ Hom(V^L_m_1,V^L_m_1); thus it is contained the set of matrices with zero trace and so the projection of HH^0(V^L_m_1⊗ V^R_m_1) in HH_0(V^L_m_1⊗ V^R_m_1) is nonzero. 
    
    
    
    By the splitting theorem and Proposition <ref>, () is both a submodule and a quotient module of (').
    
    Clearly, if in Theorem <ref>c_1=e_i^R and c_2=e_j^L belong to two distinct connected components of ', then one can restate the HH_0 simply as a tensor product over a copy of  acting on the left on the skein algebra of one component and on the right on the other.
    
    In particular, if  is obtained by glueing a bigon  along its right edge to a left edge of ' then ()=(')⊗_() is isomorphic to (') as it can be seen  directly by Theorem <ref>.
     
    If ^R is the bigon whose edges are both declared to be of type R (right), then (^R) is a left module over ^⊗ 2 (see Example <ref>). Then glueing ^R to ' along one edge of type L, shows that 
    
        ()=(')⊗ _(^R).
    
    The resulting surface  is still homeomorphic to ' but the edge on which the glueing has been performed has been transformed from an edge of type L to one of type R. This corresponds to applying Lemma <ref> to the module structure coming from that edge. 
    
    If c'_1,c'_2 are two other edges of ∂' (and then of ∂), (<ref>) is an isomorphism of -modules for the structure associated to c'_1 and c'_2. Furthermore the theorem can be applied independently to glue also c'_1 and c'_2 and the final isomorphism between ('/(c_1=c_2,c'_1=c'_2))⊗ and  HH^0((')⊗) (with respect to the ^⊗ 2-bimodule structure) does not depend on the order in which the glueing was performed.
     _1Δ()∗
    
     §.§ Braided tensor product
     
    
    
    Let U be a dual quasitriangular Hopf algebra. 
    Assume A is an algebra admitting two right U-comodule-algebra structures Δ_1:A→ A⊗ U and Δ_2:A→ A⊗ U which commute, i.e.
     (Δ_1⊗_U)∘Δ_2 =(_A ⊗ )∘(Δ_2⊗_U)∘Δ_1 
    
    where :U⊗ U→ U⊗ U is the flip operator. Denote the common operator of (<ref>) by Δ_12. 
    
     
    Observe that since Δ_1 and Δ_2 commute,  A can be endowed with a right U-comodule structure :=Δ_1 Δ_2: A → A  U defined by
    (x) = ∑ x'  u_1 u_2  if Δ_12(x) = ∑ x' u_1  u_2.
    
    
    
    
    However : A → A  U is not an algebra homomorphism, i.e. A is not a right U-comodule algebra with respect to , even though it is a right U-comodule algebra with respect to each of  Δ_1 and Δ_2. So we define a new product. For x,y∈ A let
    
     x y = ∑ x' y' ρ(u  v)  if Δ_2 (x) =∑ x'  u,  
    Δ_1(y) =∑ y'  v.
    
     It is easy to check that  gives A a new associative product, and we call A with this new product  the self braided product of Δ_1 and Δ_2 and denote it A.
     
     
    
     
    With respect to the product  and the right U-comodule given by =Δ_1 Δ_2, the algebra A is a right U-comodule-algebra. 
     We have to show that for x,y∈ A one has 
     (x y)= (x)  (y).  
    Let us now  write Δ_12(x) and Δ_12(y) as
    
        Δ_12(x)= ∑ x'  u_1  u ∈ A  U  U 
        Δ_12(y)= ∑ y'  v   v_2 ∈ A  U  U.
    
     Using  the commutativity of Δ_1 and Δ_2 and a simple calculation, we obtain
    
        (x y)    = ∑ρ(u” v”)   x' y'   u_1   v'u'    v_2      
        (x)  (y)     = ∑ρ(u' v')   x' y'   u_1   u” v”     v_2    ,
    
    where Δ(u)= ∑ u'  u”, Δ(v) = v'  v” are coproducts in U.
    The right hand sides of (<ref>) and (<ref>) are equal thanks to (<ref>). This proves (<ref>). For those who are familiar with graphical calculations in Hopf algebras,
    we provide a graphical proof in Figure <ref>.
    [Braided tensor product]
    The first  example of the above structure is the well-known  braided tensor product of two right comodule-algebras, which we describe in details for the reader's convenience.
    Suppose A_1, A_2 are right comodule-algebras over a dual quasitriangular  Hopf algebra U. The tensor product 
    A=A_1 _ A_2 has two commuting right U-comodule structures. Namely  Δ_1=(_A_1⊗ )∘ (Δ_A_1⊗_A_2) and Δ_2=_A_1 ⊗Δ_A_2. 
    
    
    
    
    By the above construction, A has  the structure of a right U-comodule algebra, with the coaction = Δ_1 Δ_2 and the product . Explicitly, for x∈ A_1 and y∈ A_2, the coaction is
    
        (x y)= ∑ (x'  y')  uv  if Δ_A_1(x)= ∑ x'  u, Δ_A_2(y)= ∑ y'  v.
    
    
    Let us now describe the product . 
    Identify A_1 with A_1 {1} and A_2 with {1} A_2 (as subsets of A_1_A_2).  Then the new product (<ref>) is given by
    
    x y =  xy        if  x, y ∈ A_1  or  x, y ∈ A_2 
    
    x  y    if  x ∈ A_1 , y ∈ A_2 
    ∑ρ(u v)(y'   x')
       if  x ∈ A_2, y ∈ A_1, 
    
    where Δ_A_2(x)= ∑ x' u,Δ_A_1(y)= ∑ y',   and ρ is the co-R-matrix.
    
    
    The algebra A_1 _A_2 with this new product is called the  braided tensor product of A_1 and A_2, and is denoted by A_1 _U A_2.
    For details see <cit.>.
    
    [Transmutation]  
    Assume that U is a dual quasitriangular Hopf algebra and let A=U.  Then Δ_2:= Δ: A → A U gives A a right U-comodule algebra structure. To get another right U-comodule structure one converts the standard left comodule structure to a right one by
    Δ_1(x) := ∑ x'  S(u)  if Δ(x) = ∑ u  x'.
    
    
    However Δ_1 is not compatible with the algebra structure of A. One can twist the product of A using the co-R-matrix ρ of U to make both Δ_1 and Δ_2 right comodule algebras as follows.
     Define a new product on A using the common value of (<ref>), i.e.
       x ⊙ y = ∑ρ (x' y') x” y”, if Δ(x) =∑ x'  x”, Δ(y) = ∑ y'  y”.
     
    It is easy to check that this gives A a new product, with which both Δ_1 and Δ_2 give A  right U-comodule algebra structures. Besides, Δ_1 and Δ_2 commute.
    
    
    Our construction now gives A a right U-comodule algebra structure whose coaction =Δ_1Δ_2 and whose product  are given by
    
        (x)    = ∑ x” S(x') x”'  if  (Δ_U)∘Δ(x) = ∑ x' x” x”'
        
        
        x  y   = ∑   x” y”ρ( S(x' x”')  S( y')  ) if Δ(y) = ∑ y' y”.
     
    It turns out that the coaction (<ref>) is exactly the  right coadjoint action, see <cit.> and the product  is 
     exactly the  covariantized product of <cit.>. The algebra A, with this 
    new product  and the original coproduct Δ,  is known as the  transmutation of A, and is a  braided group in the braided category of U-comodules, see <cit.>.
    
    
    
     §.§ Attaching an ideal triangle is a braided tensor product
    
    Suppose e, e_1, e_2 are oriented edges of an ideal triangle _3  as depicted in Figure <ref>.
    brtensor0Left: Ideal triangle _3. Middle: Glueing  and _3 by a_1=e_1 and a_2=e_2 to get . Right: tangle diagram x∈() and its image f(x)∈()2.8cm
    
    
    
    Let  be a (possibly disconnected) punctured bordered surface, with two boundary edges a_1,a_2 ⊂∂. Define = (⊔_3)/(e_1=a_1, e_2= a_2), see Figure <ref>. (A special case is when =_1⊔_2 and a_1⊂',a_2⊂”.)
    For i=1,2 the algebra  has a right comodule algebra structure Δ_i:= Δ_a_i: () →. 
    The two coactions Δ_1 and Δ_2 commute, see Lemma <ref>. Hence we can define the self braided tensor product () of Δ_1 and Δ_2, which gives  a new right comodule algebra structure over . On the other hand, Δ_e: () →() gives () a right comodule algebra structure over .
    
    
    The right -comodule algebra () is naturally isomorphic to the self braided tensor product (), defined with the co-R-matrix ρ' of (<ref>).
    
    In particular when =_1⊔_2 and a_1⊂_1,a_2⊂_2, the right comodule algebra () over  is naturally isomorphic to the braided tensor product (_1) (_2).
    -13pt1.3 cmbrtensor1-15pt1.5 cmbrtensor2-13pt1.3 cmbrtensor3-8pt.8 cmbrtfour-13pt1.3 cmbrtensor1-new-15pt1.5 cmbrtensor2-new-13pt1.3 cmbrtensor3-new-8pt.8 cmbrtensor4-new  Let _+ be the positive orientation of the boundaries of  and .  In the proof all tangle diagrams will have positive height order.
    
    For a stated -tangle diagram x let f(x) be the stated ∂-tangle diagram obtained from x by extending the strands ending on a_1⊔ a_2 until they end on e, with order on e given by its positive direction,
    see  Figure <ref>. We require that f(x) has no crossing inside _3 and this makes f(x) unique up to isotopy. Since f clearly preserves the defining relations of a stated skein algebra, we can extend it to an -linear map f:()→().
    
    
    Recall that () is the same  with a new product  given by (<ref>).
    
    The map f: () →() is an algebra homomorphism.
     
    Let x,y be stated -tangle diagrams.
    
    brtensorxy-newxy,f(xy), and f(x) f(y)2.5cm
    We present xy, f(xy), f(x)f(y) schematically as in Figure <ref>. By splitting along the dashed line in the picture of f(x)f(y) and using the counit property which says u = ∑ u' (u”), we get, with Δ_2(x) = ∑ x' u and Δ_1(y) = ∑ y' v,
    
        f(x) f(y)
           = ∑  ( ) 
        
           = ∑   f ( x'  y')ρ' (u   v),
             using co-R-matrix ρ' of  (<ref>)
           = f(x y)      by   (<ref>).
    
    Thus f is an algebra homomorphism.
    
    
    It remains to show that f is an -linear isomorphism.
    For l∈ let F_l(())= F_l^a_1, a_2(()) and F_l(())= F_l^a_1, a_2(()) be the filtrations defined in Subsection <ref>. In other words, F_l(()) is
    the -submodule spanned by stated tangle diagrams  such that I(, a_1)+ I(,a_2) ≤ l, and similarly  for F_l(()).
    Denote by _* the corresponding graded -modules.
    It is clear that f preserves the filtrations F_l. It is enough to show that (f) is a bijection. 
    
    
    Let B_m,n be the set of isotopy classes of simple -tangle diagrams  such that I(, a_1)=m, I(, a_2)=n and  is increasingly stated on each boundary edge, except for a_1 and a_2 where it is not stated.
     Then 
    
        _l (())    = ⊕ _ m+n=l,  x∈ B_m,n V(x)
        _l (())    = ⊕ _ m+n=l,  x∈ B_m,n W(x).
    
    Here V(x) is the -submodule of _l (()) spanned by ∈ B(;_+) such that =x if we forget the states on a_1∪ a_2, and W(x) is the -submodule of _l (()) spanned by
    z∈ B(;_+) such that z∩= x. It is enough to show that (f) is an isomorphism from V(x) to W(x) for x∈ B_m,n. 
    
    Note that both V(x) and W(x) are free -modules, and both have rank
    (m+1)(n+1).
    Indeed there are m+1 increasing states on x∩ a_1 and n+1 increasing states on x∩ a_2 and these can be chosen independently, thus _(V(x))=(m+1)(n+1). For what concerns W(x), observe that if  z∈ B(;_+) such that z∩= x then z∩_3 consists of  k arcs (for some k∈ [0,min(m,n)]) connecting e_1 and e_2, m-k arcs connecting e_1 and e, and finally n-k arcs connecting e_2 and e; furthermore z∩ e is increasingly stated so that there are exactly (m+n-2k+1) such z.
     Thus we have:
    
        _(W(x))=
        ∑_k=0^min(m,n)(m+n-2k+1)=(m+1)(n+1).
    
    The reordering relation (<ref>) implies the relation in Figure <ref>, which converts arcs connecting e_1 and e_2 to arcs with one end in e.  This shows that (f): V(x) → W(x) is surjective.
    brten6-new1.7cm
    
     Since both V(x) and W(x) are free -modules having the same rank, we conclude that (f): V(x) → W(x) is an isomorphism. This completes the proof of the theorem.
    
    
     §.§ Examples: Polygons, punctured bigons, and punctured monogons
    [Polygon] 
    The  polygon _nis the standard  disc with n punctures on its boundary removed.
    Note that the  triangle _3 is the result of attaching an ideal triangle to two bigons. By Theorem <ref> we have 
    (_3) ≅_,
    
    where each copy of  is a right -comodule algebra via the coproduct. Consequently, (_3) ≅_ as -modules, and its algebra structure is described by (<ref>). From here one can get a presentation of (_3). In <cit.>, a presentation of (_3) was obtained by brute force calculation.
    
    The n-gon _n is the result of attaching an ideal triangle to the disjoint union of _n-1 and the bigon. By induction, we obtain
     One has
    
        (_n)≅⋯,
    
     where there are (n-1) copies of .
     swro[Punctured bigons] Let _n be the bigon  with n interior punctures removed. 
    For example 
    _0=. Like in the case of , we will show that (_n) has a natural structure of a Hopf alagebra where all the operations can be defined geometrically. Recall that we  denote e_l and e_r the left and right boundary edges of .
    bigonnThe embedding of the 4-punctured bigon _4 into _8 and the splitting homomorphism. The composition gives the coproduct.2cm
    
      
    Let ι:_n↪_2n be the inclusion identifying _n with the complement of n closed disjoint arcs, each  connecting two punctures of  the 2n-punctures of _2n; See Figure <ref>. Let  Δ:(_n)→(_n)⊗(_n) be the map induced by ι and then splitting along the vertical arc connecting the two boundary ideal vertices of _2n and identifying the two halves with _n. 
    
    Let also ϵ:(_n)→ be the map obtained by including _n in _0= and then applying ϵ :()→. 
    
    Finally  define S:(_n)→(_n) as the -linear map whose value on a stated tangle  α in _n × (-1,1) is obtained by first switching all the states η to -η and all the framing vectors v to -v, then rotating α by 180^∘ around the axis passing through the two boundary ideal vertices and finally multiplying the result by (√(- 1)   q)^(δ_e_l(α)-δ_e_r(α)) (where δ_e(α) was defined in Subsection <ref> as the sum of the states on α∩ e). It is easily checked that all the defining relations (<ref>)-(<ref>) are preserved so that S is well-defined. 
    
    
    
     For each n≥0 the skein algebra (_n), endowed with Δ,ϵ,S, is a Hopf algebra.
    
    
     From the definition of the splitting homomorphism it is clear that Δ is an algebra homomorphism. The
    coassociativity of Δ is a direct consequence of the fact that applying twice ι induces the same morphism as the identification of _n with the complement of n disjoint arcs in _3n each containing 3 punctures as depicted here:
    
        -1cm2cmbigonnn
    
    The map ϵ is a morphism of algebras by its definition; to verify that (ϵ⊗)∘Δ= observe that if in Figure <ref> we fill the left punctures then we obtain the initial _n.
    
    That S is an antimorphism is a consequence of the fact the revolution by 180^∘ about the 
    the axis connecting the two boundary vertices reverses the height order in _n × (-1,1).
    
    We are left to prove  that  
     (S⊗) ∘Δ=(⊗ S) ∘Δ=ϵ, 
     and since S is an antimorphism, it is enough to check this identity on a set of generators. From relation (<ref>) we get that the set of  horizontal arcs, with all possible states, generates (_n).
    If _ημ is a horizontal arc with state η on the left and state μ on the right, then the definition gives
    
        S(α_ηη)=α_η η   and   S(α_ηη)=-q^2ηα_ηη,
     so that on _0=, S coincides with the antipode defined in (<ref>). Now Identity (<ref>) for _ημ follows from the same identity for the antipode in ()=.
     (a) Since _n is the result of attaching 2n ideal triangles to n+1 bigons, Theorem <ref> can be used to show that as -algebras (_n) ≅^ (n+1) where the tensor product is over  and the algebra structure  of ^ (n+1) is the unique one determined by:
    
    (i) the subset A_i= 1^ (i-1)__ 1^ (n+1-i)⊂^ (n+1) is isomorphic to   as  -algebras for each i=1,…, n+1, and 
    
    (ii) for a∈ A_i and b∈ A_j with i< j, one has ab= a b and
    
        ba  = ∑ρ̅'(b' a') ρ'(b”' a”')  a” b”
    
    where ρ' is the co-R-matrix defined by (<ref>) and  ρ̅' is its inverse.
    
    (b) For the  case  n=1, Proposition  <ref> and a presentation of (_1) were also independently obtained in <cit.> via a direct calculation. 
    
    [Punctured monogons] Let _n  be the monogon  with  n punctures in its interior removed, see  Figure <ref>. Let  :(_n)→(_n)⊗() be the right ()-comodule algebra structure induced by the only boundary edge of _n.
     
     
    Similarly (_n) has two commuting right comodule-algebra structures over () induced by e_l and e_r, let Δ_1 be the one induced e_l and Δ_2 be the one induced by e_r. 
    
    
    
    
    Like in the bigon case, let ι:_n↪_2n be the inclusion identifying _n with the complement of n  disjoint arcs, each  connecting two punctures of _2n. Observe that _2n is the result of attaching an ideal triangle to two copies of _n, see Figure <ref>.
    By Theorem <ref> we have isomorphism of algebras (_2n)=(_n)(_n).
     Let  Δ:(_n)↪(_2n)=(_n)(_n) be the map induced by ι_* and this isomorphism. 
    
    (a) For each n≥ 0 the algebra (_n) endowed with the map Δ and the map ϵ:(_n)→(_0)= induced by inclusion is a bialgebra object in the category of -comodules (i.e. its product, coproduct, unit and counit are morphisms of -comodules). 
    
    (b) 
    The  comodule-algebra (_n) is isomorphic to the self braided tensor product (_n-1).
    In particular (_1) is isomorphic as a Hopf algebra to BSL_q(2),  the “transmutation” of , or “braided version” or “covariant version” of  (see <cit.> Examples 4.3.4 and 10.3.3). 
    
    (a) The inclusion ι induces an injective  algebra  homomorphism  ι_*:(_n)↪(_2n)=(_n)(_n). As in the case of bigons, the coassociativity follows from the fact that applying twice ι induces the same morphism as the identification of _n with the complement of n disjoint arcs in _3n each containing 3 punctures as depicted here:
    
        -1cm
            < g r a p h i c s >
        ↪-1cm
            < g r a p h i c s >
        ↪-1cm
            < g r a p h i c s >
        .
    
    The map ϵ is a morphism of comodule-algebras by its definition; to verify that (ϵ⊗)∘Δ= observe that if in Figure <ref> we fill the left punctures then we obtain the initial _n. 
    The last fact to verify is that Δ is a morphism of comodules, i.e. denoting :(_n)→(_n)⊗() the right comodule structure, that 
        (Δ⊗)∘=(⊗⊗ m)(⊗⊗)∘(⊗)∘Δ,
     where m:()⊗()→() is the multiplication and  is the flip. 
    Since Δ is a morphism of algebras, it is sufficient to check this on generators of _n. For this we refer to Figure <ref>, where we depict the case n=1 but the proof is similar for other n. Letting α(x,y)∈ B(_n,) be the generator represented by a horizontal arc whose states are y,x∈{±} depicted in the figure, the equality in the figure shows that we have: 
        Δ(α(x,y))=q^1/2α(+,y)α(x,-)-q^5/2α(-,y)α(x,+).
     Also, from the definition of  one computes (α(x,y))=∑_ϵ,ηα(η,ϵ)⊗ (α_η xα_η y) where α_xy denote the standard generators of ().
    Now the verification is a straightforward computation.
    A· -S
    (b) As _n is obtained by attaching an ideal triangle to _n-1, the claim follows from Theorem <ref>.
    The isomorphism of (_1) with BSL_q(2) then follows from Example <ref>. Let us make explicit the isomorphism.
    
    Let T:()→(_1) be the -linear isomorphism obtained on a skein a by embedding _*(_e_l^-1(a)) (see Subsection <ref>) 
    in the monogon through the embedding depicted in Figure <ref> (a) and finally extending the strands of _*(_e_l^-1(a)) until they hit the boundary of _1 (i.e. by applying the map f defined in the proof of Theorem <ref>). We claim that pulling back through T the coaction of (_1) we get the right adjoint coaction on =() and pulling back the product  * on (_1) we get the product  on BSL_q(2) defined in Example <ref>:
    
        Δ^coad=(T^-1⊗ Id)∘∘ T      and    =T^-1∘(· *·)∘ (T⊗ T).
    
    Using the definitions of T, of the antipode on ()(<ref>) and of , one verifies directly that (T^-1⊗ Id)∘∘ T(x)=x”⊗ S(x')x”' for any x∈=() (see Figure <ref> (b)). 
    Then since * and  are compatible it is sufficient to check that the pullback of * equals  on the generators of ; this is a straightforward computation. 
    A graphical explanation is as follows. Observe that if x=T(a),y=T(b), then x*y=(T(a*b))_(1)ϵ((T(a*b))_(2)) (this holds in general comodule algebras), so that we have:
    
    
        < g r a p h i c s >
    1cm=
        < g r a p h i c s >
    1cm·ϵ(
        < g r a p h i c s >
    1cm)
    
    and that the right hand side equals 
        T(a_(2)· b_(2)) ρ' (S(a_(1)) a_(3) S(b_(1)))ϵ(b_(3))=T(a_(2)· b_(2)) ρ' (S(a_(1)) a_(3) S(b_(1)))
     where ρ' was defined in (<ref>). This proves that =T^-1∘(· *·)∘ (T⊗ T).
    
    
    § A LIFT OF THE RESHETIKHIN-TURAEV OPERATOR INVARIANT
    
    In this section we show that a Reshetikhin-Turaev operator invariant of tangles can be lifted to an invariant with values in . In this section =[q^± 1/2].
    
    Ṽ∂_r∂ _l
    
     §.§ Category of non-directed ribbon graphs
     We will present the category of  non-directed ribbon graphs<cit.>, also known as  framed tangles<cit.>, in the form convenient for us.
    
     The bigon is canonically isomorphic (in the category of punctured bordered surface) to the square S= [0,1] × (0,1).
    Under the isomorphism  and   are mapped respectively to {0}× (0,1) and {1}× (0,1), and abusing notation we also denote {0}× (0,1) and {1}× (0,1) respectively  and . We identify S with S ×{0} in M:=S × (-1,1). We have ∂ M = ∂ S × (-1,1) = (∪) × (-1,1).
    
    tangleLeft: Square S=[0,1] × (0,1), with edges  and e_r. Middle: tensor product β⊗β'. Right: composition β∘β', which can be defined only when |β|=|β'| .2.5cm
    
    Recall that in the definition of a ∂ M-tangle we require the boundary points over any boundary edge have distinct heights (see Subsection <ref>).
    If we change this requirement to: all boundary points are in ∂ S (in particular they all have the same height) we get the notion of a ∂ S-tangle. Formally, a ∂ S-tangle is a framed compact 1-dimensional unoriented manifold β properly embedded in M=S × (-1,1) such that ∂β has height 0, i.e.  ∂β⊂∂ S =∪, and the framing at every boundary point of β is vertical. Let β= β∩ and β = β∩.
    Two ∂ S-tangles are ∂ S-isotopic if they are isotopic in the class of ∂ S-tangles. If |β|=k and |β|=l, then our notion of a  ∂ S-tangle is the notion of  a non-directed ribbon (k,l)-graph without coupons in <cit.>.
    
    After an isotopy we can bring β to a generic position (with respect to the projection from S × (-1,1) onto S) and make the framing vertical everywhere. The projection of β together with the over/under information at every crossing, is called a ∂ S-tangle diagram of β. The isotopy class of β is totally determined by any of its diagrams. 
    
    The  non-directed ribbon graph category is the category whose set of objects is  and a morphism from k to l is an isotopy class of ∂ S-tangleβ such that |β|=k and |β|= l, with the usual composition (see Figure <ref>). If the tangles are oriented, then one would get the usual ribbon tangle category.
    
    
    If β, β' are two ∂ S-tangles, define their tensor product β⊗β' as the result of putting β above β' as in Figure <ref>.
    Under the tensor product and the composition, morphisms of the non-directed ribbon tangle category are generated by the five  elementary ∂ S-tangles depicted in Figure <ref>. 
    
    eletanglesFive elementary tangles1.5cm-7pt.7 cmcap-7pt.7 cmcup-7pt.7 cmXp-7pt.7 cmXn
    
    
     From the ribbon category of finite-dimensional modules over the quantum group  we get the Reshetikhin-Turaev operator invariant of ∂ S-tangles, see <cit.>.
      Let us describe this operator invariant in a special case. Let V be the free -module with basis g_+, g_-. The above mentioned operator invariant is the unique 
     functor Z from the non-directed ribbon tangle category to the category of -modules preserving the tensor product such that Z(n) = V^ n and the values of the elementary tangles are given by
    
        Z ()    : V^⊗ 2→ R,    g_+  g_-  →  q^-1/2,  g_-  g_+ →  -q^-5/2,  g_+  g_+ → 0,   g_-  g_- → 0  
        
        Z ()    : R→ V^⊗ 2,    1 →  -q^5/2  (g_+  g_- )+ q^1/2   (g_-  g_+)  
        
        Z ()   : V^⊗ 2→ V^⊗ 2,   Z () = q 𝕀 + q^-1(  Z () ∘ Z ()),  
        
        Z ()   : V^⊗ 2→ V^⊗ 2,   Z () = q^-1 𝕀 + q (  Z () ∘ Z ()),
    
    see <cit.>.  Here our g_± are related to the basis vectors  g_1/2 in <cit.> by
    
        g_+= -√(-1)   q^-3/2 g_1/2,    
        g_-= g_-1/2.
    ν⃗μ⃗
    
    Thus if β is a ∂ S-tangle with |β|=l and |β|=k then Z(β) is an -linear map  V^ k→ V^ l which depends only on the isotopy class of β. 
     
    For =(ν_1,…,ν_l)∈{±}^l and =(μ_1,…,μ_k) ∈{±}^k we can define the matrix entry _ Z(β) _∈ R such that
     
    Z(β) (g_μ_1… g_μ_k )= ∑_∈{±}^l(  _ Z(β) _)   g_ν_1… g_ν_l.
     In fact V and all its tensor powers are  modules over the quantum group , and all the operators Z(β) are -morphisms. But we don't need the structure of -modules here. 
    When k=l=0, we have Z(β)∈ R, which is equal to the Kauffman bracket polynomial of β.
    
    
    
     §.§ From ∂ M-tangles to ∂ S-tangles
     Suppose γ is a ∂ M-tangle. We can  ∂ M-isotope γ so that its diagram D   has the height order on  and  determined by the arrows in Figure <ref>. This diagram determines a unique class of ∂ S-tangle, denoted by γ̅. Note that the arrows of ,  are irrelevant for γ̅. It is easy to see that the map γ→γ̅ is a bijection from the set of ∂ M-isotopy classes of ∂ M-tangles to the set of ∂ S-isotopy classes of ∂ S-tangles.
    
    tangle2Direction of boundary edges, used to determine the height order2cm
    
    Suppose |γ∩|=l and |γ∩|=k, and  =(ν_1,…,ν_l)∈{±}^l and =(μ_1,…,μ_k) ∈{±}^k. Let _γ _ be the stated ∂ M-tangle  whose underlying tangle is γ and whose states on γ∩ (respectively on γ∩) from top to bottom by the height order are ν_1,…, ν_l (respectively μ_1,…, μ_k).
    
    Assume the above notation. Consider _γ _ as an element of (). Then
    ϵ ( _γ _)  = _ Z(γ̅) _. 
    
    Thus we see that the tangle invariant of  _γ _ with values in () is stronger than the Reshetikhin-Turaev operator invariant. 
    
     Suppose γ_1, γ_2 are ∂ M-tangles. 
    Since γ_1 γ_2= γ̅_1 ⊗γ̅_2, if (<ref>) is true for γ = γ_1 and γ=γ_2, it is true for γ= γ_1 γ_2.
    
    η⃗
    
    Now suppose γ_1, γ_2 are obtained by splitting a ∂ M-tangle β  along an ideal edge. By the splitting formula (<ref>) and the definition of , 
    
        Δ(  _β _) = ∑_ _ (γ_1)_  _ (γ_2) _.
    
    Applying ϵ𝕀 to the above, we get
    
        _β _ = ∑_ϵ(  _ (γ_1)_)   _ (γ_2) _.
    
    Applying ϵ to the above, we get
    
        ϵ(  _β _ ) = ∑_ϵ(  _ (γ_1)_)  ϵ( _ (γ_2) _),
     
    which shows that if (<ref>) holds for γ=γ_1 and γ=γ_2 then it holds for β=γ_1 ∘γ_2.
    
    Thus it is enough to check (<ref>) for the elementary tangles, for which  (<ref>) follows from the explicit formulas (<ref>)–(<ref>).
    
    
     §.§ A 1+1-TQFT
    𝖲𝖾𝗍𝖠𝖡𝗂𝗆𝖴_𝗊^2(𝔰𝔩_2)-𝖿𝗂𝗇𝖡𝗂𝗆
    
    Let _1,1 be the symmetric monoidal category whose:
    
     Objects are numbered disjoint unions of open unoriented segments.
     Morphisms are diffeomorphism classes of punctured bordered surfaces  with indexed boundary. Explicitly if ∂=e_1^L,⋯ ,e^L_m, e^R_1,⋯, e^R_n then ∈ Mor(e_1^L⊔⋯⊔ e^L_m,e^R_1⊔⋯⊔ e^R_n) and the composition of morphisms is given by the glueing of marked surfaces explained above (associativity of compositions is ensured by the fact that we consider diffeomorphism classes of surfaces). In particular the identity morphism of e_1⊔⋯⊔ e_n is a disjoint union of n copies of .
     Tensor product is the disjoint union, where the components of (e_1⊔⋯⊔ e_n)⊔ (e'_1⊔⋯⊔ e'_m) are ordered as e_1⊔⋯ e_n⊔ e'_1=e_n+1⊔ e'_m=e_m+n.
    
    In order to define the target category of our TQFT functor, let us fix some notation. 
    Given a finite set C, we will then denote by ^⊗ C the algebra obtained as the tensor product ⊗_c∈ C where each copy of  in the tensor product is indexed by a distinct element of C.
    
    
    Let  be the category whose objects are pairs (C,[M]) where C is a finite set, M is a right module over ^⊗ C which is a direct sum of finite dimensional modules and [M] is its isomorphism class. A morphism from (C,[M]) to (C',[M']) in  is the isomorphism class of a bimodule B over (^⊗ C,^⊗ C') which is a direct sum of finite dimensional bimodules and such that [M⊗_^⊗ C B]=[M']. The composition of [B]:(C,[M])→ (C',[M']) and [B']:(C',[M'])→ (C”,[M”]) is [B⊗_^⊗ C'B'] (the composition is associative as we consider bimodules up to isomorphisms).  The monoidal structure on  is given by (C,[M])⊗ (C',[M']):=(C∪ C',[M⊗_ M']) and its symmetry is given by exchanging (C,[M]) and (C',[M']). 
    
    
    Then let :_1,1→ be defined as 
        (e_1⊔⋯⊔ e_n)=(C={e_1,… e_n},[⊗_()^⊗ C])
     and for a punctured bordered surface  whose boundary is a union ofC={e_1^L,… ,e^L_n} and C'={e_1^R,… ,e^R_m} let () be the isomorphism class of the (^⊗ C,^⊗ C')-bimodule ⊗_().
    
    The functor  is a symmetric monoidal functor into .
    
    By point b) of Theorem <ref> it holds ⊗_( )=⊕_i≥ 0 V^L_i⊗ V^R_i where V^L_i (resp. V^R_i) is the irreducible i+1-dimensional left (resp. right) module over . 
    Then, arguing exactly as in the proof of Theorem <ref> one sees that for each j≥ 0 it holds 
        [V^R_j⊗_(⊗_())]=[V^R_j].
      Then ⊗_() represents the identity morphism ({e},[M])→ ({e},[M]) (for any edge e) if restricted to finite dimensional right -modules (which are all direct sums of V^R_j's). 
    Let ' and ”  be two bordered punctured surfaces with boundaries indexed so that ∂^L'
    ={e_1,… ,e_n}=∂^R” and let  be the surface obtained by glueing ' and ” by identifying the corresponding edges of ∂^L ' and ∂^R” via an orientation reversing diffeomorphism. Then (') (resp. (”)) is a right (resp. left) module over ^∂^L' (resp. over ^∂^R”=∂^L'). To conclude, a repeated application of Theorem <ref> shows that the following holds up to isomorphism:
    
        ()=(')⊗_^⊗∂^L'().
    
     The previous construction can be improved by passing to the setting of 2-categories in order to consider objects no longer up to isomorphisms. This requires to consider marked surfaces and bimodules and will be dealt with in another work.  
     Bθ̅∂Σ
    
    § A NON-SYMMETRIC MODULAR OPERAD
    
    
    
    In this section we show that stated skein algebras provide an example of “non symmetric geometric modular operad”. Such objects were defined by Markl (<cit.>) as a generalisation of “modular operads” initially defined by Geztler and Kapranov (<cit.>). Given a monoidal category C, Markl defined a NS modular operad in C as a monoidal functor NSO:𝖬𝗎𝗅𝗍𝗂𝖢𝗒𝖼→ C where 𝖬𝗎𝗅𝗍𝗂𝖢𝗒𝖼 is a suitable category of 𝖬𝗎𝗅𝗍𝗂𝖢𝗒𝖼“multicyclic sets”. 
    In this section we rephrase Markl's definition in the case of a suitable category of punctured bordered surfaces ; then we define a NS geometric modular operad as a monoidal functor NSO:→ C. Finally we re-interpret skein algebras as an example of an NS geometric modular operad with values in  (see Definition <ref>). 
    
    
    
     §.§ The category of topological multicyclic sets .
     
    In this section all surfaces will be oriented and all homeomorphisms will preserve the orientation. 
    
    A cutting system in a bordered punctured surface  is a finite linearly ordered set α of pairwise disjoint ideal oriented arcs α_1,⋯ ,α_k⊂ (see Subsection <ref>); a homeomorphism of cutting systems α and β in  is a homeomorphism ϕ:→ such that ϕ(α)=β so that it preserves the ordering and the orientations of the arcs. 
    Cutting along all the arcs of a cutting system α produces a bordered punctured surface _α() whose homeomorphism class depends only on the homeomorphism class of α. We will say that a cutting system α is disconnecting  if each arc in α disconnects . 
    
    If the connected components of  are linearly ordered then one can order the connected components of _α() as follows. Since _α()=_α_k( _α_k-1( ⋯_α_1 ())), it is sufficient to define how to do it for  of the cut along a single ideal arc α. 
    If α does not disconnect, then there is a natural bijection between the components of  and _α() which induces the ordering on those of the latter surface. If α disconnects , since both α and  are oriented there is a well defined notion of the connected component of _α()“lying at the left” and “at the right of α” we then order them so that left precedes right and they are in the same position in the global ordering of the components of  as the component they come from.
    
    
    Let  be the category whose objects are homeomorphism classes of punctured bordered surface whose connected components are linearly ordered, and where a morphism ' → is a homeomorphism class of a cutting system α in  such that _α() is homeomorphic to '.
    The category  is the subcategory whose objects are disjoint unions of polygons (see Example <ref>) and whose morphisms are those represented by disconnecting cutting systems. 
    
    If ϕ:'→_α() and ψ:→_β(”) are homeomorphisms, then the composition of the morphisms associated to α  and β is the homeomorphism class of ψ(α)⊔β⊂” where the numbering of the arcs of ψ(α) is lower than those of β. 
    The identity morphism is represented by the class of the empty cutting system and it is straightforward to check that the composition is associative, so that the above are indeed categories. 
    
    Both  and  are symmetric monoidal categories. Indeed the tensor product of '='_1⊔⋯⊔'_k and =_1⊔⋯⊔_h (where '_i,_j are connected for all i,j and the linear order of the components is increasing from left to right) is defined as 
        '⊗:='_1 ⊔⋯'_k⊔_1⊔⋯⊔_h.
    
    On the level of morphisms, if α⊂_1 and β⊂_2 are two cutting systems then α⊗β=α⊔β where the linear order of the arcs of α is lower than that of the arcs in β. 
    The symmetry is given by exchanging the components, so with the above notations s('⊗)=⊗' and s(α⊗β)=β⊗α. 
    
    
     The following definition is a reformulation of Markl's <cit.> (Definition 4.1) in the context of punctured bordered surfaces:
    
    Let C be a symmetric monoidal category.
    A NS (non symmetric) geometric modular operad in C is a symmetric monoidal functor 
        O:→ C.
    
    A NS cyclic operad in C is a symmetric monoidal functor O:→ C.
    
    
     §.§ NS geometric modular operads from skein algebras
    
    Recall that if  is the bigon with one edge of type “left” and one of type “right”, then ()= as a (,)-bimodule.  Let also ^R be the bigon whose edges are declared to be both of type R (right edges) then (^R) is the left module over ^⊗ 2 whose underlying space is  and on which the action of x⊗ y∈^⊗ 2 is given by x⊗ y· b=x· b· r^*(y) (see Example <ref>). 
    
    
    
    
    There is a geometric NS-modular operad NSO in  defined on an object  of  as 
        NSO()=((),[⊗_()])
     where  is the surface whose edges are all indexed to be of type L (left) and where we see ⊗_() as a right module over ^⊗() as explained in Subsection <ref>.
    
    
    If ϕ:'→ is a morphism associated to a cutting system α, then let 
        NSO(ϕ)=[⊗()^⊗()⊗(^R)^⊗α]
    
    where (^R)^⊗α is the skein algebra of a disjoint union of one copy of ^r per arc α_i∈α whose boundary edges correspond to the edges of ∂' lying respectively at the left and at the right of α_i. 
    
    First of all observe that the functor is well defined as all surfaces are seen up to orientation preserving diffeomorphism and all modules and bimodules in  are seen up to isomorphism. 
    Then we observe that the skein algebra of a disjoint union of n bigons 
        ⊗_(⊔_j=1^n _j)=⊗_()^⊗ n=⊗_^⊗ n
     is the identity of ({1,2,⋯ ,n},[M]) (where i is the left edge of the i^th-bigon) for any right module M which is a direct sum of finite dimensional modules over ^⊗ n. Indeed M is a direct sum of modules of the form W⊗ V^R_j where V^R_j is the j+1-dimensional irrep of  and W is a right module over ^⊗ n-1 which is itself a tensor product of finite dimensional modules. As proved in Theorem <ref>()==⊕_i V^L_i⊗ V^R_i so that by the same arguments as in the proof of Theorem <ref> it holds:
    
        [W⊗ V^R_j⊗_U_q^2(𝔰𝔩_2)_n( ⊕_i V^L_i⊗ V^R_i)]=⊕_i[W⊗(V^R_j⊗_ V^L_i)⊗ V^R_i ]=[W⊗ V^R_j ].
    
    This shows that tensoring over  with a single copy of ⊗_() provides the identity morphism; by Remark <ref> repeating this along all the boundary edges one gets that tensoring with ⊗_^⊗ n is the identity of ({1,2,⋯ ,n},[M]) for any M decomposing into a direct sum of finite dimensional modules.
    
    Now we prove that if i,j are two distinct boundary edges of a (possibly disconnected) surface ', then
    
        [(')⊗_^{i,j}(^R)]=[()]
    
    where (^R) is seen as left module over ^{i,j} and  is the surface obtained by glueing the edges i,j by an orientation reversing homeomorphism. 
    Indeed by Remark <ref> and Example <ref> to glue ^R along i and j, one can first glue ' and ^R along i thus obtaining the surface ' whose edge i has been changed to type R (see Example <ref>) and then operating a self-glueing along i and j on this surface. 
    By Theorem <ref> the overall result is ⊗_().
    Then if α is a cutting system given by c arcs, by  Remark <ref> applying c times (<ref>) we get that tensoring with ⊗_(^R)^⊗α is performing the glueing inverting the cut associated to the cutting system α.   
    
    
    
    § REDUCED SKEIN ALGEBRA
     We show that the stated skein algebra  has a nice quotient , called the  reduced stated skein algebra, which can be embedded in a quantum torus. This quotient is still big enough to contain the ordinary skein algebra and the Muller skein algebra. Unlike the case of the full fledged version , when  is an ideal triangle, the reduced version  is a quantum torus. The construction of the quantum trace map follows immediately from the splitting theorem for the reduced stated skein algebra. 
    
    Throughout we fix a  punctured bordered  surface = ∖ and we will denote =().
    
    
    
     §.§ Definition
    
    
    A non-trivial arc ⊂, which is the closed interval [0,1] properly embedded in  not homotopic relative its endpoints to a subset of the boundary , is called a  corner arc 
     if  it is as that depicted in Figure <ref>(a), i.e. it cuts off from  a triangle with one ideal vertex. Such an ideal vertex is said to be surrounded by the corner arc . 
     
     A  bad arc is a stated corner arc whose states 
      are as in the figure Figure <ref>, i.e. they are - followed by + if we go along the arc counterclockwise around a surrounded vertex.
    badarc3(a) a bad arc (b) the splitting of a bad arc2.5cm 
    The reduced stated skein algebra  is defined to be  the quotient of () by the 2-sided  ideal  generated by bad arcs. 
    
    
    
    
    
    
    
     
     θ̅
    
     §.§ Basis
    
    Let  _+ be the orientation of ∂ induced by that of , i.e. every boundary edge has positive orientation.
    Then B:=B(;_+) is an -basis of .  Let =()⊂ B  be the subset consisting of all elements in B which contain no bad arc.
     
      The set  is a free -basis of the -module .
       Let A⊂ be the -span of  and A'⊂ be the -span of B ∖. One has = A ⊕ A'. Let us prove that the ideal  is equal to A'. 
    
     Proof that A'⊂. Let γ∈ (B ∖), i.e. γ contains a bad arc. We have to show that γ∈. If an arc in γ (at some corner) is bad, then the positive orientation and  increasing  states imply that all the arcs closer to the vertex of that corner are bad, see Figure <ref>. 
    badarc2If the outer arc is bad, then all inner arcs are bad, too.1.5cm
    
    
    
    Thus we assume that γ has a bad arc which is an inner most arc, see Figure <ref>(b).
    
    move1(a) The product β, here  (in red) is a bad arc, (b) an element γ∈ B which has a bad  innermost arc (in red). 3cm
    
    We have the relations in Figure <ref>, which are part of Lemma <ref>.
    rel5Moving endpoint with negative state (left) and positive state (right)2.3cm
    The first relation allows us to move the end of the red arc with state - (in γ) up until we get the diagram in Figure <ref>(a), which is of the form β, where  is a bad arc. The result is that γβ. Thus, γ∈.
    
     Proof that ⊂ A'. We have to show that  β, β∈ A' for any bad arc  and any β∈ B. 
    
    The product β: In this case, β is presented as in Figure <ref>(a). We already saw that βγ, where γ is as in Figure <ref>(b). Since γ∈ A', we see that β∈ A'.
    
    move2(a) Product β, where  is a bad arc (in red), (b) the diagram γ2.5cm
    
    The product β: In this case, β is presented as in Figure <ref>(a). Using the 2nd relation in Figure <ref>, we get that βγ, where γ is as in Figure <ref>(b). Since γ∈ A', we see that β∈ A'.
    
    
    
    Thus, =A'. Hence as -modules, =/J ≅ A, which has  as an -basis. 
    B̅ Positive order is used substantially in the proof. For other orientation of , the set similar to  might not be the basis of .
     The ordinary skein algebra () and the Muller skein algebra  embed naturally into the reduced skein algebra .
     Clearly the standard basis of the ordinary skein algebra and the standard basis of the Muller skein algebra (where all the states are +) are subsets of the basis  of .
    
    
     §.§ Corner elements
     Let u be a stated corner arc with both states positive  and v be the same arc  with both  states negative. Then uv=vu=1 in .
     In  we have 
    
        vu=   =  q^2    + q^-1/2  =  q^-1/2   = 1,
    
    where the second identity follows from (<ref>) and  the last follows from (<ref>). Similarly,
    
        uv =   =  q^-2   - q^-5/2  =  q^-5/2   = 1,
    
    where the second identity follows from (<ref>) and  the last follows  from (<ref>).
    
    
     §.§ Filtration
    
    
    For a finite collection  of ideal arcs or simple closed loops let F^_n() be the -submodule of  spanned by stated tangle diagrams  such that ∑_a∈ I(a, ) ≤ n. Then (F^_n())_n=0^∞ is a filtration of  compatible with the algebra 
    structure. Denote by ^() the associated graded algebra:
    
        ^() = ⊕_n=0^∞^_n(),  where ^_n() = F^_n()/F^_n-1().
    
    
    From Theorem <ref> we have the following analog of Proposition <ref>.
     Suppose  is a collection of boundary edges of .
    
    (a) The set {∈|∑_a∈ I(, a) ≤ n} is an -basis of F^_n().
    
    (b) The set {∈|∑_a∈ I(, a) = n} is an -basis of ^_n().
    
    
     §.§ Splitting theorem
    
    Suppose  ' is the result of splitting  along an interior ideal arc a. The splitting algebra embedding
    θ_a: (')
    descends to an algebra embedding
    θ̅_a: →(').
    
    
    
    Besides, if a and b are two disjoint ideal arcs in the interior of , then 
    _a ∘_b = _b∘_a 
    a'a”TLBLTRBRS'_LS'_RD_LP_RP_L
        Suppose ⊂ is a bad arc. The geometric intersection I(,a) is 0 or 1. In the first case θ_a()= is also a bad arc in '.  In the second case the splitting of , given in Figure <ref>(b), has a bad arc for both values of  ν∈{±}. It follows that θ_a() ⊂. Hence θ_a descends to an algebra homomorphism _a:  →(') and we also have (<ref>).
    
    
    It remains to show that _a is injective. Let 0≠ x∈. We have to show that _a(x) ≠0. 
    Since () is an -basis, there is a non-empty finite set S ⊂() such that
     
    x = ∑_∈ S c_,    0 ≠ c_∈.
     
    Let k=max_∈ S I(, a). Then S':={∈ S | I(,a)=k} is non-empty.
    
    Let :'→ be the projection and  ,⊂' be the boundary edges which are  ^-1(a). To simplify the notations we write F^_n and ^_n for respectively F^_n((')) and ^_n((')).
    From the formula of the splitting homomorphism, for every ∈ S,
    
        _a() ∈ F^_k ∩ F^_k ⊂  F_2k^{,}.
    
    Let P:  F_2k^{,}_2k^{,} be the canonical projection.
    Clearly if ∈ S∖ S' then P()=0. We consider  Case 1 and Case 2 below.
    
     Case 1:   There exists β∈ S such that P(_a(β))≠0.split7The split surface ', with orientations ' on '. The top left, top right, bottom left, and bottom right corners are marked respectively TL, TR, BL, BR.2.5cm
    
    Choose an orientation of a such that the induced orientation on  is positive. Then the induced orientation on  is negative, see Figure <ref>. Let ' be the orientation of ' which is positive everywhere except for the edge a' where it is negative. For ∈ S' its lift =^-1() is a  partially stated tangle diagram: it is stated everywhere except for endpoints on ∩, and the endpoints on each of  and  are ordered by '. Let ^+ be the same  except that the order on  (and hence on all edges) is given by the positive orientation.
    
    For 0≤ j≤ k let
    s_j() (respectively s_j(^+)) be the stated tangle diagram  which is   (respectively ^+) where  the states on each of  and  are increasing and having exactly j minus signs. Then s_j(^+) is either equal to 0 in (') or belongs to the basis set (').
    β̃
    
    By Proposition <ref> and then Proposition <ref> we have, for some f(,j) ∈, 
     
    P(_a()) = ∑_j=0^k   kj_q^4  s_j () =  ∑_j=0^k   kj_q^4 q^f(,j)s_j(^+). 
    Since  P(_a(β))≠0,  there is l such that s_l(^+)≠ 0 in (') and hence s_l(^+)∈(').
    
    Using (<ref>) we have
     
    P (_a((x))= ∑_∈ S'∑_j=0^k  kj_q^4 q^f(,j)  c_  s_j(^+).
    
    
    As  ∈ S' can be recovered from , if ≠β  then the two partially stated diagrams  and   are not isotopic.
     It follows that  s_l(^+)≠ s_j(^+) for all j and all ≠β. It is also clear that s_j(^+) ≠ s_l(^+) for j ≠ l.
     Hence the right hand side of (<ref>) is not 0, since the basis element s_l(^+) has non-zero coefficient, and all other elements s_j(^+) is either 0 or a basis element different from s_l(^+). Thus P (_a((x))≠ 0 and consequently _a(x)≠ 0. This completes the proof in Case 1.
     
     
    
     Case 2:  For all ∈ S we have P(_a())=0. Identity (<ref>) shows that s_j(^+)=0 for all 0≤ j≤ k and all ∈ S'.
    
    Incident with  there are two corners, the top left corner and the bottom left corner. 
     Similarly, incident with  there are the top right corner and the bottom right corner, see Figure <ref>. A corner arc of  at one of these four corners has  one end stated and one end not stated, and it is called a  negative (respectively positive) corner arc if this only state is negative (respectively positive).
    
    
    For ∈ S' and ν∈{±} let  _ν()  be the number of top left corner arcs  whose only state  is ν. Define _±(), _±(), ±() similarly.
    
    
     Suppose ∈ S'. One of the following two mutually exclusive cases happens:
    
     (i)  _-() >0 and _+() >0, or 
    
     (ii) _-() >0 and _+() >0.  Since s_0(^+) is 0 in ('), it has a bad arc. This implies either _-() >0 or _-() >0.
    
    Assume _-() >0. Since  does not have a bad arc, we conclude that _+()=0. Then from s_k(^+)=0 we see that _+() >0. Again since  does not have a bad arc, we conclude that _-() =0. Thus we have case (i) but not case (ii).
    
    Assume _-() >0. Since  does not have a bad arc, we conclude that _+()=0. Then from s_0(^+)=0 we see that _+() >0. Again since  does not have a bad arc, we conclude that _-() =0. Thus we have case (ii) but not case (i).
    
    The cases (i) and (ii) of Lemma <ref> partition  S'= ⊔, where
    
        ={∈ S' |_-() _+()  >0 },   ={∈ S' |_-() _+() >0 }.
     If ∈ then _a()=0 in ^_k. Similarly, if ∈ then _a()=0 in  ^_k.
     Suppose ∈. For  =(ν_1,…, ν_k) the stated tangle diagram (,) is defined to be  with states on both  and  are sequence  listed from top to bottom.
    By definition,
    _a () = ∑_∈{±}^k (, ).
    
    Let (^+, )_L be 
    ^+ whose states on  is given by  but whose state on  is given by a permutation of  such that the states are increasing on . 
    By Proposition <ref> (, )  (^+, )_L    in ^_k .
    
    If  has at least one negative sign then (^+, )_L has a bad arc in the bottom left corner (because BL_+() >0) and hence is equal to 0 in ('). If  has at least one positive sign  then (^+, )_L has a bad arc in the top left corner (because TL_-() >0) and hence is equal to 0 in ('). Thus we always have (^+, )_L=0 in ('). From (<ref>) and (<ref>) we conclude that _a ()=0 in ^_k.
    
    The other case follows from the above case by noticing that if one rotates the Figure <ref> by 180^∘, then the top left corner becomes the bottom right corner.
    
    
    
    
    
    As ∅≠ S'= ⊔, one of  and  is non-empty. Without loss of generality we can assume that  is not empty.
    
    Let d = min{_-() |∈} and S” ={∈|_-() =d }. Then S”≠∅.
     
     Let P_+: (') →(') be the -linear map defined on basis elements γ∈(') by 
     
        P_+(γ) =   γ     if I(γ,)=d, I(γ, )=k, all states on  are +
        
        0    otherwise.
    
     
    For ∈ S” let  be the stated tangle diagram obtained from ^+ by first removing the d negative top left corner arcs  then providing states on  and  so that all states on  are + and the states on  are increasing and having exactly d negative signs. Since _-()=0, we see that  is an element of the basis ('). As  can be recovered from , the map → from S” to (') is injective.
    
     
    Let u be a top left corner arc whose both states are +.
     For ∈ S one has
     P_+(_a()  u^d) =  q^-(k-d)(k-d-1)/2       if ∈ S”
    
    0    if ∉S”. One has S = (⊔) ⊔ (S ∖ S'). If ∈ (S∖ S') then I(, a) <k and hence P^+(_a()) =0.
    
    If ∈ then by Lemma <ref> one has _a()=0 in ^_k which means _a() is a linear combination of elements γ∈ with I(γ,) <k. It follows that P_+(_a())=0.
    
    It remains to consider the case ∈= S”⊔ (∖ S”).
    From (<ref>),
    
    P_+(θ_a () u^d ) = ∑_∈{±}^k P_+((, )  u^d).  
    Recall that for β∈(') one defines δ_(β) as the sum of all the states of β∩. From the definition, if δ_(β) ≠ k-d then P^+(β)=0.
    If  has m negative signs where m>d then 
    
        δ_((, )   u^d)=  k -2m +d < k-d,
     
    and hence P_+((, )   u^d)=0. Thus we can assume that in the sum in (<ref>), the number of negative signs in  is ≤ d.
    
     Assume that _-()=m. Note that the m negative top left corner arcs of  are below any other components of .  Hence the number of the first m components of  must be negative since otherwise one of the m top left corner arcs is bad and (,)= 0 in (').
    We conclude that if  TL_-() >d (that is, if ∈∖ S”), then
     P_+(θ_a () u^d ) =0 . 
    Moreover, if TL_-()=d (that is, ∈ S”), then 
    
    P_+(θ_a () u^d ) = P_+((, _d)  u^d), 
    where _d∈{±}^k is the sequence whose first d components are - and all other components are +. The d negative top left  corner arcs of (,_d) are all v, the corner edge with negative states on both ends.  Hence we have
    (, )= ' v^d, where ' is the same as  except that the order on  is negative. Using the height exchange move between positive states, see Equation (<ref>),  and relation vu=1 we get that
    
        P_+((, _d)  u^d) =  q^-(k-d)(k-d-1)/2 ,
    
    which proves (<ref>) and completes the proof of the lemma.
    
    Let us continue the proof of the theorem for Case 2. From Lemma <ref> we have
    
        P_+(_a(x) u^d ) = ∑_∈ S” c_ q^-(k-d)(k-d-1)/2 ,
    
    which is non-zero since {} are distinct elements of the basis ('). The theorem is proved.
        
    
     §.§ The bigon
     The elements _μν∈() are defined in Section <ref>.
      Let  be the bigon.  There is an algebra isomorphism 
     ()≅[x^± 1] given by
     _++→  x ,  _–→ x ^-1,  _+-→ 0,  _-+→ 0.
     
                                                                                                      
     A presentation of the algebra ()≅ is given by Theorem <ref>,  with generators a=α_++, b=_–, c= _-+, d= _– and relations (<ref>) and (<ref>).  
    The only bad arcs in  are _-+=c and _+-=b. Thus ()= () / has a presentation like that of , with additional relations b=c=0. From the quantum determinant relation in (<ref>) we get ad=1 in (). 
    
    On the other hand it is easy to check that the relations b=c=0 and ad=1 imply all other relations in (<ref>) and (<ref>). Hence
    
        () ≅ a,b,c,d /(ad=1, b=c=0) ≅[a^±1].
    
    
     §.§ The triangle
     
    Let _3 be the ideal triangle, with boundary edges a,b,c as in Figure <ref>. 
    Let , β,γ be the corner arcs which are opposite respectively to  a, b and c. For μ,ν∈{±} and ξ∈{, β,γ} let ξ(μν) be the arc ξ with states μ and ν on the end points such that ν follows μ along ξ counter-clockwise (with respect to the vertex surrounded by ξ). 
    
    triangleEdges a,b,c opposite to corner arcs , β,γ2.5cm
    
    
    For an anti-symmetric n× n matrix A = (a_ij)_i,j=1^n the  quantum torus associated to A is the algebra with presentation
    
        x_i^± 1, i= 1, … n  /( x_i x_j = q^a_ij x_j x_i).
    
    For basic properties of quantum tori see for example <cit.>.
    
    To the triangle _3 we associate the quantum torus  with presentation
    
        := ^± 1, β^± 1, γ^± 1 /( qβ  =  β, qβγ =  γβ, qγ =  γ).
    (_3)(_3)
    The cyclic group /3= τ|τ^3=1 acts by algebra automorphisms on each of the algebras (_3), (_3), and  as follows. In short  τ is rotation τ  by 2π/3 counterclockwise about the center of the triangle. This rotation induces the algebra automorphism τ of ; it also induces the algebra automorphism τ of . On  and  τ is given by
    
        τ() =β, τ(β) =γ,  τ(γ) = .
    
    The reduced skein algebra (_3) of the ideal  triangle is isomorphic to the quantum torus . The isomorphism is /3-equivariant and given by 
    (++) →,   (+-) → q^-1/2γβ,   (-+) → 0,   (–) →^-1.
    
    By <cit.> the algebra  (_3) is generated by 
        X={(ν,ν'),β(ν,ν'), γ(ν,ν')  |ν, ν'∈{±}}
     subject to the following relations and their images under τ and τ^2:
    
        β(μ,ν) (μ',ν')    = q (ν, ν')  β(μ,μ') -q^2 C^ν_μ' γ(ν', μ)
        
        (-,ν) (+,ν')    = q^2 (+,ν)  (-,ν') - q^5/2 C^ν_ν'
        (ν,-) (ν',+)    = q^2 (ν,+)  (ν',-) - q^5/2 C^ν_ν'
        (-,ν) β(ν',+)    = q^2 (+,ν) β(ν',-) - q^5/2γ(ν,ν')
        
        (ν,-) γ(+, ν')    = q^2 (ν,+) γ(-,ν') + q^-1/2β(ν',ν).
    
    As the only bad arcs are α(-,+), β(-,+),γ(-,+), the quotient  is obtained by adding the relations α(-,+)= β(-,+)=γ(-,+)=0, and from this presentation one can check that the map given by (<ref>) and its images under the action of /3 is an isomorphism.
    
    Here is an alternative, more geometric proof. 
    First in   we have 
    (++)  (–) =1 ,   β(++) (++) = q (++) β(++), 
    and all its images under /3. In fact the first identity follows from Proposition <ref> and the second follows from the height exchange identity (<ref>).
    It follows that the /3-equivariant map f: → given by
    
        f()= (++), f(^-1)= (–),  and images under /3,
    
    gives a well-defined algebra homomorphism, as all the defining relations of  are preserved under f. In  we have
    γ(+-) = q^-1/2 β(++) γ(–),
    
    which follows from the identity in Figure <ref> (where the left hand arc is stated to become (+-)). Thus all elements in the generator set  X are in the image of f. This shows that f is surjective. 
    
    Let us show f is injective. The set {^k β^m γ^n | k,m,n,∈} is an -basis of . Assume that there is a finite set S⊂^3 such that
      f (∑_(k,m,n) ∈ S c_k,m,n^k β^m γ^n) =0,  c_k,m,n∈.
    
     Multiplying the identity (<ref>) on the left by f(^k'β^m'γ^n') with large k', m', n' and using the q-commutations between , β,γ we can assume that k,m,n > 0 in (<ref>). For each (k,m,n)∈^3 let z(k,m,n) be the stated  simple tangle diagram consisting of k arcs parallel to , m arcs parallel to β, and n arcs parallel to γ, with all state positive. Note that z(k,m,n) ∈(_3).
    Clearly the map z:^3 →(_3) is injective. 
    As the diagram of f(^k β^m γ^n) can be obtained from z(k,m,n) by a sequence of height change moves of positively stated endpoints, the first identity of (<ref>) shows that 
    
        f(^k β^m γ^n)= q^g(k,m,n)   z(k,m,n)
    
    for some g(k,m,n) ∈. From (<ref>) we get
    
        ∑_(k,m,n) ∈ S c_k,m,n q ^g(k,m,n)   z(k,m,n) =0.
    
    As z(k,m,n) are distinct elements of the basis (_3), this forces all c_k,m,n=0. Hence f is injective.
    
    ℱ()
    
     §.§ The quantum trace map
     Assume that  is  triangulable, i.e.    is  not one of the following: a monogon, a bigon, a sphere with one or two punctures. 
     A triangulation of  is a collection consisting of all boundary edges and several ideal arcs in the interior of  such that
    
    (i) no two arcs in  intersect and no two are isotopic, and
    
    (ii) if a is an ideal arc not intersecting any ideal arc in  then a is isotopic to one in .
    
    It is known that if  is triangulable, then by splitting  along all interior ideal arcs in  we get a collection  of ideal triangles.
    By the splitting theorem, we get an algebra embedding of  into a quantum torus
    
        Θ: →⊗_.
    
    
    In addition to the quantum torus  we associate the quantum torus ' to the standard ideal triangle _3:
    
        ' : =  a^± 1, b^± 1, c^± 1 /( qab  =  ba, qbc =  cb, qca =   ac).
    
    One should think of a,b,c as the edges opposite to ,β,γ, see Figure <ref>. 
    
    (_3)(_3)
    The cyclic group /3= τ|τ^3=1 acts by algebra automorphisms on  '  by
    
        τ(a) =b,  τ(b) =c,  τ(c) = a.
     
    
    
    There is a /3-equivariant algebra embedding 
    ', defined by
    
        → q^1/2 bc ,   β q^1/2 ca ,   γ→ q^1/2 ab.
    
    
    
    
    Consider  the composition _q given by
    
        _q: Θ⊗_⊗_'.
    
    
    
    On the collection of all edges of all the triangles in  define the equivalence relation such thata' ≅ a” if a' and a” are glued together in the triangulation. Then the set of equivalence classes is canonically isomorphic to .
    Let () be the subalgebra of ⊗_' generated by all a'⊗ a” with a≅ a' and all boundary edges (each boundary edge is equivalent only to itself). It is easy to see that the image of _q is in (). Thus, _q restricts to
    
        _q: →().
    
     The algebra  () is a quantum torus, known as the Chekhov-Fock algebra associated to a triangulation  of , see <cit.>.
      
     The quantum trace map of Bonahon and Wong is an algebra homomorphism _q: ()→(), where () is the coarser version of () defined using only (<ref>) and (<ref>), see Subsection <ref>. 
    
    
     
    If  is a triangulation of  then the algebra embedding 
    _q: () is a refinement of the 
    the quantum trace map of Bonahon and Wong  in the sense that _q is the composition () _q.  In <cit.> an algebra homomorphism ϰ_: () →() is defined as the composition
    
        ⊗_( _3 )  ⊗_',
    
    where the map from (_3) to '  is exactly the composition (_3) →(_3)→'. It follows that ϰ_ is the composition () _q. In <cit.> it is proved that _q is the composition () ()  ϰ_→. Hence  _q is also the composition () _q.
    
    
    Besides giving another proof of the existence of the quantum trace map, Theorem <ref> shows that the kernel of _q is the ideal generated by relations (<ref>), (<ref>), and the ideal ^bad.
    
    
     §.§ Co/module structure for 
         The Hopf algebra structure of () descends to a Hopf algebra of (). We identify ()≡[x^± 1] using the isomorphism of  Proposition <ref>.  Then  Δ(x) = x ⊗ x and ϵ (x)=1.
                        
    Arguing exactly as in Subsection <ref>, one sees that for each surface  and each edge e of , the algebra  has both a left and a right [x^±1]-comodule algebra structure (which is equivalent to a ℤ-valued grading counting the number of + and - states of each skein along e): 
     (a) The map _e: →⊗() gives  a right comodule-algebra structure over the Hopf algebra [x^±1]. Similarly  gives gives  a left comodule-algebra structure over the Hopf algebra [x^±1]. 
     
    (b)  If e_1,e_2 are two distinct boundary edges, the coactions on the two edges commute, i.e.  for instance 
        (Δ_e_2⊗𝕀)∘Δ_e_1=(Δ_e_1⊗𝕀)∘Δ_e_2.
    
    
    In the reduced setting, though, Theorem <ref> does no longer hold:
    indeed, with the notation in there, if β∈() is a basis element intersecting a cutting edge exactly once, then its image θ(β) under the cutting morphism is θ(β)=β'_+++β'_– where β'_++,β'_–∈(') are identical except for their states on β'∩ (c_1∪ c_2). But it is not difficult to check that β'_++ is balanced: 
        Δ_c_1(β'_++)=β'_++⊗α_+++β'_+-⊗α_-+=β'_++⊗α_++=_c_2Δ(β'_++)
     because the class of α_-+=α_+-=0∈(), still β'_++ is not in the image of θ. 
    
                                                          
      𝕆√(_e)
    
    § THE CLASSICAL CASE: TWISTED BUNDLES
    
    In this section we will suppose that  is a connected, oriented surface with a non-empty set of boundary edges and let  be the positive orientation of ∂ i.e. that induced by the orientation of . We will prove that if q^1/2=1 then () is isomorphic to the algebra of regular functions on the affine variety of “twisted bundles” on . A similar result for the case when ∂=∅ is well known (see for instance <cit.>). 
    
    Fix an arbitrary  Riemannian metric and let U be the unit tangent bundle over , with the canonical projection
     π:U→.
    A point in U is  a pair (p,v),  where p∈, v∈ T_p, v=1. For each immersion α:[0,1]→ its  canonical lift is the path (α(t),α̇(t)/α̇(t))) in U. In particular, since each edge e of ∂ is oriented by , it has a canonical lift e⊂∂ U; we will denote ∂:=∪_e⊂∂ẽ.  If we let -e be the edge oriented in the opposite way, then we get a different lift which we will denote (-e)^∼. Let  -∂=∪_e⊂∂ (-e)^∼ and ±∂=∂∪ -∂.
    
    
     For a point x∈ the fiber =π^-1 (x) is a circle, and we will orient it  according to the orientation of . It is clear that the free homotopy class of  does not depend on x. 
     
     For each boundary edge e choose a point x∈ e.  
     Let v∈ T_x(e) be the unit tangent vector with orientation . Then both (x,v) and (x,-v) are in π^-1(x), and the half circle of π^-1(x) going from (x,v) to (x,-v) in the positive direction is denoted by . The exact position of x on e will not be important in what follows. 
    
    
    Let X be a path connected topological space and {E_i}_i∈ I disjoint contractible subspaces of X.  The fundamental groupoid π_1(X,{E_i}_i∈ I) is the groupoid (i.e. a category with invertible morphisms) whose objects are {E_i, i∈ I} and whose morphisms are the homotopy classes of oriented paths in X with endpoints in ∪_i∈ I E_i.
    A morphism of groupoids if a functor of the corresponding categories. 
    
    Recall that a group is a groupoid with only one object.
    
    With the above notation, let E⊂ X be a contractible subspace disjoint from ∪_i∈ I E_i. Then given a morphism ρ:π_1(X,{E_i}_i∈ I)→ G for some group G, an oriented path γ connecting some E_i to E, and an arbitrary g∈ G there is a unique extension ρ':π_1(X,{E_i}_i∈ I∪{E})→ G of ρ such that  ρ'(γ)=g and  ρ'(α)=ρ(α) for all α∈ Mor(E_i,E_j) for some i,j. 
    
    We sketch a proof. Let β be a homotopy class of an oriented path connecting E to some E_j; write β=(β∘γ)∘γ^-1. Observe that β∘γ∈ Mor(E_i,E_j) thus ρ is defined on it; hence define ρ'(β)=ρ(β∘γ)· g^-1. Similarly if β is a path connecting E_j to E define ρ'(β)=g·ρ(γ^-1∘β). Finally if β is an endomorphism of E define ρ'(β)=g·ρ(γ∘β∘γ^-1)· g^-1. We leave to the reader to verify that this is indeed a functor with the required properties. 
    
    
    We shall be interested in two particular groupoids:  π_1(,∂) and π_1(U, ∂). Note that π:(U,∂)→ (,∂) induces a surjective morphism π_* of groupoids. 
    
    
    A flat twisted SL_2(ℂ)-bundle(“twisted bundle” in what follows to keep notation short) on  is a morphism ρ:π_1(U;∂)→SL_2(ℂ) such that ρ()=-Id.
    
    By Lemma <ref> we extend ρ to a morphism (with the same notation) ρ:π_1(U;±∂)→SL_2(ℂ) such that for every boundary edge e,
    
        ρ()=([  0 -1;  1  0 ]).
    
    Since  is not a closed surface, its fundamental group π_1() is a free group.
    
    Suppose that ∂≠∅. Then the set tw() of twisted bundles on  is the affine algebraic variety SL_2(ℂ)^n+k where 
        n=-1+#{e⊂∂}  and k=rank(π_1()).
     
    In particular the algebra χ() of its regular functions is generated by the matrix entries of each of the copies of SL_2(ℂ).  
    Since U is trivial, the fundamental groupoids π_1(;∂) and π_1(U,∂) are isomorphic. More explicitly, we claim that there are non canonical injective morphisms of fundamental groupoids s_*:π_1(;∂)→π_1(U,∂).
    To build one, pick any non-zero vector field on  which is positively tangent to the edges of ∂: it exists because we are not prescribing its behavior near the (non compact) cusps. This trivializes U as × S^1; let s:→×{1} be a section of π:U→. 
    
    The above isomorphism allows to provide an isomorphism from the set of twisted bundles to morphisms from π_1(;∂) →SL_2(). Indeed to each twisted bundle ρ:π_1(U;∂)→SL_2() we associate ρ': π_1(;∂) →SL_2() defined as ρ'=ρ∘ s_* . Reciprocally given ρ':π_1(;∂) →SL_2() we extend it to ρ:π_1(U;∂)→SL_2() by setting ρ()=-Id and ρ|_π_1(×{1})=ρ'.To conclude we now argue that the set of morphisms ρ': π_1(;∂) →SL_2() is in bijection with SL_2(ℂ)^n+k. Indeed fix a set of immersed smooth paths α_1,… ,α_n⊂ connecting a fixed edge e_0⊂∂ to each other edge of ∂ as well as a set of paths whose endpoints are in e_0 representing generators g_1,… , g_k of π_1(;e_0) (which is free because ∂≠∅). Since the fundamental group of  is free, the list of values (ρ'(α_1),…, ρ'(α_n),ρ'(g_1)…, ρ'(g_k))∈SL_2(ℂ)^n+k provides the sought non canonical bijection.
    Let _n be the n-polygon with vertices numbered in the orientation sense from 0 to n-1; then tw(_n)=SL_2(ℂ)^n-1 where the n-1 matrices are given by the holonomies of the diagonals connecting the edge v_0v_1 to each other edge. Then χ(_n)= O(SL_2)^⊗ n-1 and in particular χ()=O(SL_2) and χ(_3)=O(SL_2)⊗ O(SL_2) (where by “equal” we mean “non-canonically isomorphic to”).
     The notion of flat twisted SL_2(ℂ)-bundle is closely related to the one considered in <cit.>.
    
    
     §.§ Trace functions for non oriented curves
     We will identify the states of a stated tangles with vectors in ^2 as follows:
     
        +:=([ 1; 0 ]), -:=([ 0; 1 ]).
    
     If x⃗,y⃗∈ℂ^2 let (x⃗|y⃗) denote the determinant of the matrix whose first column is x⃗ and second is y⃗. 
    
    We will say an immersion  a:[0,1]→ is in  good position if a(0), a(1) ∈∂ 
    and the tangent vectors ȧ(0), ȧ(1)are positively tangent to ∂.
    
    An immersion  :[0,1]→ is  transversal if (0), (1) ∈ and  is transversal to  at 0 and 1. One can bring such a transversal 
       to an arc  a in good position by an isotopy (relative 0 and 1) in a small neighborhood of (0) and (1).  The canonical lift of a will be denoted by  and is called the  good lift of . Note that the homotopy class of  is uniquely determined by , and we will consider  as an element of π_1(U;∂).
    Note that the good lift of the inverse path  α^-1, defined by ^-1(t)=α(1-t),  is not the inverse of , since before    lifting one has to isotope α^-1 to a good position. 
    sign
    
    A  stated transversal immersion is a transversal immersion whose end points are stated {±}.
     Let ρ be a twisted bundle on . 
    
    Assume α:[0,1]→ is a stated transversal immersion with state   at (0) and η at (1).  Define the trace of α by
     
        (α):=(η| ρ(α̂)·ϵ).
    
     
    Assume β:[0,1]→ is an immersed closed curve (i.e. β(0)=β(1) and β has the same tangent at 0 and 1). Define the trace  of β by
    
        (β)=(ρ(β')),
    
    where β' is any smooth closed curve isotopic to β such that β'(0)∈.
    
    In the first case if α' is homotopic to α through stated transversal immersions and has the same states as α then (α')=(α) (indeed the homotopy lifts to a homotopy of α and α'). 
    In the second case it is easy to see that (β) does not depend on the choice of β', as the images under ρ of any two such β' are conjugate in SL_2() and hence have the same trace.
    
    Let α:[0,1]→ be a stated transversal immersion with state   at (0) and η at (1).
    
        If ρ(α̂)=([ a b; c d ]),  then()=(η,ρ(α)·ϵ) =
        η\ϵ    +     -
        
        +     c     d
        
        -     -a     -b
        .
    We remark that the matrix on the right, expressing the values of the traces for an immersed transverse stated arc α is ρ(√()^-1)ρ(α̂) (see Equation (<ref>)). 
      The notion of trace here is similar to the one introduced in <cit.>, where trace is defined only for oriented arcs. The novelty here is the good lift, which is used to define traces for unoriented arcs and the use of twisted bundles as representations of fundamental groupoids.
    
    
    
    When  is stated, we provide ^-1 with states so that the state of  ^-1(t) is equal to the state of (1-t) for t=0,1.
     Suppose ρ is a twisted bundle on .
    (a) Let  be a stated transversal immersion.  One has
    
        ρ (α^-1)=- ρ (α̂)^-1.
     
     As a consequence, (α)=(α^-1).
     
     (b) Let β:[0,1]→ be an immersed closed curve such that β(0)∈. Then
      
        ρ (β)^-1=ρ (β^-1).
     
      
     As a consequence, if γ is any immersed closed curve then (γ) = (γ^-1).
     (a). A direct inspection shows that the homotopy class of the closed simple loop in U given by the concatenation α^-1∘α̂ is  (see the left hand side of Figure <ref>). 
    The first equality follows as by definition ρ is a functor such that ρ()=-Id. 
    
    To prove that (α)=(α^-1), 
    
    we compute the traces using the notation and content of Example <ref>:
    
        If ρ(α^-1)=-ρ()^-1=([ -d  b;  c -a ]),  then()=(η',ρ(α)·ϵ') =
        η'\ϵ'    +     -
        
        +     c     -a
        
        -     d     -b
        .
    
    But since the state at α^-1(0) is η and that at α^-1(1) is ϵ, we get the claim in this case by directly comparing with the transpose of the matrix of values provided in Example <ref>. 
    
    (b). 
    Observe that if β is the black curve depicted in the r.h.s. of Figure <ref>, then β^-1 is regularly homotopic to the dotted curve β' in the same picture. By construction β̇'̇(0)=β̇(0) and β'∘β is regularly homotopic to an eight-shaped immersed curve in a disc. 
    Then β^-1 is homotopic to β̃'̃ in U and it holds ρ(β̃'̃∘β̃)=Id, thus ρ(β^-1)=ρ(β̃)^-1.
    The last statement now follows because ρ(β̃)∈SL_2() so that (ρ(β̃))=(ρ(β̃)^-1). 
    
    
    Suppose α=∪α_i, where each _i is either a stated transversal non-oriented arc or a non-oriented immersed closed curved. Define
    
        (α):=∏_i(α_i).
    
    If q^1/2=1 the map :()→χ() sending a stated skein to its trace is an isomorphism of algebras. The same  holds for :(_n)→χ(_n) for every n≥ 2. 
    
    Applying Theorem <ref> at q^1/2=1 we get an explicit algebra isomorphism ϕ:()→ O(SL_2); by Example <ref>, we know that χ() is isomorphic to O(SL_2). Furthermore, by  Example <ref>, the map  is an algebra isomorphism. 
    One argues similarly for _n: applying Corollary <ref> at q^1/2=1 we get an explicit algebra isomorphism ϕ:(_n)→ O(SL_2)^⊗ n-1; by the proof of Corollary <ref> a system of algebra generators of (_n) is easily seen to be the arcs connecting the edge e_0 to each other edge and stated arbitrarily. By Example <ref>, the map  on these generators provides a system of generators of O(SL_2)^⊗ n-1 which by Example <ref> is isomorphic to χ(_n). 
    
    
     §.§ Splitting theorem for trace functions
    
    
    In all this subsection, let c⊂ be an ideal arc oriented arbitrarily, let ' be the result of cutting  along c and let :'→ be the projection and :U'→ U the projection induced on the unit tangent bundles.  Let ^-1(c)=c_1∪ c_2⊂∂' so that c_1 has the positive orientation and c_2 the negative one with respect to the orientation induced by that of ' on the boundary. 
    For each c_i let c_i⊂ U' be its canonical lift and (-c_i)^∼ the canonical lift of -c_i. Similarly let c be the canonical lift of c in U and (-c)^∼ be the canonical lift of -c.
    
    
    
    Each [α]∈π_1(U;∂) can be written as a composition [α_k]∘ [α_k-1]∘⋯∘ [α_1] of homotopy classes of immersed paths α_i:[0,1]→ U such that α_i({0,1})⊂∂∪c̃ and α_i∩π^-1(c)=∂α_i∩c. 
    Such a decomposition is unique up to insertion/deletions of compositions  [α']∘ [α'^-1] for [α']∈π_1(U;c) and replacement of [α_j] by [α”'_j]∘ [α”_j]∘[α'_j] for some [α'_j],[α”'_j]∈π_1(U;∂∪c) and [α_j”]∈π_1(U;c) such that [α_j]=[α”'_j]∘ [α”_j]∘[α'_j] (or reciprocally).
    
    Observe that π^-1(c)⊂ U is homeomorphic to an annulus A=ℝ× S^1 so that c=ℝ×{1}. Represent the class [α] by a smooth curve α:[0,1]→ U so that it is transverse to A; then homotope it so that it intersects A exactly along c: this provides an instance of the claimed splitting. If α':([0,1],{0,1})→ (U,∂) is another smooth representative of the same class intersecting A exactly along c, let h(t,s):[0,1]× [0,1]→ U be a smooth homotopy between α and α' which is transverse to A. Then h^-1(A) is a disjoint union of arcs and circles embedded in [0,1]× [0,1] with boundary in [0,1]×{0,1} containing a finite number of maxima and minima with respect to the height function given by the second coordinate s. Pick a finite number of heights s_0=0<s_1<… <s_n=1 so that each strip [0,1]× [s_i,s_i+1] contains at most one maximum or minimum of the diagram of h^-1(A).  
    Each immersed path α_s_i(t):=h(t,s_i):[0,1]→ U intersects A transversally a finite number of times and we can then modify h locally around h^-1(A)∩ ([0,1]×{s_i}) without inserting new maxima and minima so that α_s_i(t) intersects A only along c. 
    Then the homotopies h|_[s_i,s_i+1] transform the immersed path α_s_i into α_s_i+1 by the moves described in the thesis: passing through a minimum replaces a smooth curve α with a composition α'∘α”∘α”' were all of α,α',α”,α”' intersect A only along c and in their boundary; passing through a maximum has the converse effect. Finally a strip containing no maxima and minima corresponds to a finite number of moves consisting in rewriting α∘α' with α∘β∘β^-1∘α' where β∈π_1(U;c) is the homotopy class represented by the restriction of h to a “vertical arc” of h^-1(A)∩ [0,1]× [s_i,s_i+1] (i.e. an arc joining [0,1]×{s_i} and [0,1]×{s_i+1}). 
    
    
    If ρ':π_1(U';∂')→SL_2() is a twisted bundle, then by Lemma <ref> we can extend it to a twisted bundle ρ”:π_1(U';∂'∪ (-c_2)^∼)→SL_2() by setting ρ”(√(_c_2))=([  0 -1;  1  0 ]) where √(_c_2) is the path connecting c_2 and (-c_2)^∼ by following in the positive direction the fiber π^-1(x) for some x∈ c_2. 
    
    There is a surjective map i^*:tw(')→ tw() defined as follows. 
    Given α∈π_1(U;∂), decompose it as α=α_k∘α_k-1∘⋯∘α_1 where each α_i∈π_1(U;∂∪c) intersects π^-1(c) at most in its endpoints and exactly along c (such a decomposition exists by Lemma <ref>). 
    Then for each ρ'∈ tw(') let 
    
        i^*(ρ')(α)=ρ”(α'_k)ρ”(α'_k-1)⋯ρ”(α'_1)
     where α'_i=^-1(α_i) is the lift of α_i to π_1(U';∂'∪ (-c_2)^∼).
    Passing to the algebras χ() and χ(') of regular functions on the algebraic varieties tw() and tw('), i^* induces an injective algebra morphism i:χ()↪χ(') which we will call the “cutting morphism” associated to c.
    
    By Lemma <ref> to check that i^* is well defined it is sufficient to check that for each α the choice of the decomposition does not affect the result of i^*(ρ')(α). But this is evident if we make an exchange α_2∘α_1↔α_2∘α'∘α'^-1∘α_1 or α↔α'∘α”∘α”' as in the statement of Lemma <ref> because ρ' is a functor.   
    
    To prove surjectivity observe that by Lemma <ref> we can extend any morphism ρ:π_1(U;∂)→SL_2(ℂ) to ρ:π_1(U;∂∪c∪ (-c)^∼)→SL_2(ℂ) setting in particular 
        ρ(√(_c))=([  0 -1;  1  0 ]).
    
    Then if α'∈π_1(U';∂'∪ (-c_2)^∼) define ρ': π_1(U';∂'∪ (-c_2)^∼)→SL_2() by ρ'(α')=ρ(_*(α')) where  _*:π_1(U';∂'∪ (-c_2)^∼)→π_1(U;∂∪ (-c)^∼) is the morphism induced by the continuous map :U'→ U. Then letting ρ' be the restriction of ρ' to π_1(U';∂') we have that by construction ρ=i^*(ρ'). 
    Surjectivity of i^* implies the injectivity of i.
    
    
    The following proposition tells us that the trace functions behave exactly as the skeins under cutting along an ideal arc (see Theorem <ref>).
     Suppose  is a stated transverse smooth simple curve intersecting transversally c. Then ':=^-1() is a transverse smooth simple curve 
     which is stated at every boundary point except for newly created boundary points, which are points in ^-1 (c)∩'=(c_1∪ c_2)∩'.
       A lift in ' of  is a stated transverse smooth simple curve β in ' which is ' equipped with states on ^-1 (c∩) such that if x,y∈^-1 (c∩) with (x)=(y) then x and y have the same state. If |c∩|=k, then   has 2^k lifts in '.
    
    
    Let α be a stated transverse smooth simple curve intersecting transversally c. Then
    
        i((α))=∑(β)
    
     where the sum is taken on all the lifts in ' of α (i.e. as in Theorem <ref>). Furthermore if c'⊂ is another ideal arc disjoint from c and i' is the associated cutting morphism, it holds i'∘ i=i∘ i'.
    
    Since by Proposition <ref> we already know that i is a well defined injective algebra morphism, it is sufficient to check the statement for a system of stated transverse smooth curves {γ_i∈ I} which generate χ() as an algebra. By the proof of Lemma <ref>, we can choose a finite system of such γ_i such that |γ_i∩ c|≤ 2, ∀ i. 
    Let α∈{γ_i, i∈ I} be represented by a smooth immersion α:[0,1]→ intersecting transversally c with states st(α(0))=ϵ, st(α(1))=η; if |α∩ c|=0 the statement is true. 
    If |α∩ c|=1 then α=α_2∘α_1 where α_i are transverse smooth simple curves with α_1(1)=α_2(0)∈ c and are partially stated by st(α_1(0))=ϵ, st(α_2(1))=η. Furthermore, up to switching α to α^-1 we can suppose (α̇_1(1),ċ) form a positive basis of .   
    
    
    Let then A_i (resp. A) be the 2× 2 matrix expressing the values of (α_i) (resp. (α)) with states in {±} as in Example <ref>; then, as remarked in the example A_i=ρ(√(_c))^-1ρ(α_i) (resp. A=ρ(√(_c))^-1ρ(α)) so that Equation (<ref>) rewrites in this case as A=A_2· A_1.
    
    Now since the orientation induced by ^-1(c) is negative on c_2 then in U the good lift α̂ of α is homotopic to (α_2)∘√(_c)^-1∘(α_1) where α_i are depicted in the left hand side of Figure <ref>, therefore 
        A=ρ(√(_c))^-1ρ(α)=ρ(√(_c))^-1ρ(α_2)∘ρ(√(_c))^-1∘ρ(α_1)=A_2· A_1
    
    and the claim is proved. 
    
    Suppose now that |α∩ c|=2 where α is a stated smooth immersion transverse to c; by the proof of Lemma <ref> we can suppose that the sign of the intersections of α and c is opposite and we can split α as α_3∘α_2∘α_1 where α_i are transverse smooth immersions with α_1(1)=α_2(0),α_2(1)=α_3(0) and partially stated so that st(α_1(0))=ϵ and st(α_3(1))=η. Furthermore, up to switching α and α^-1 we can suppose that (α̇_1(1),ċ) form a positive basis of .
    
    As above let A_i=ρ(_c)^-1·ρ(α_i) and A=ρ(_c)^-1·ρ(α) and Equation (<ref>) is equivalent to A=A_3· A_2· A_1. 
    Then again, as shown in the right hand side of Figure <ref>, α is regularly homotopic in U to (α_3)∘√(_c)^-1∘(α_2)∘√(_c)^-1∘(α_1). Therefore we have:
    
        A=ρ(_c)^-1·ρ(α)=ρ(_c)^-1·ρ(α_3)·ρ(_c)^-1·ρ(α_2)·ρ(_c)^-1·ρ(α_1)=A_3· A_2· A_1
    
    and the thesis follows. 
    
    
     §.§ The classical limit of stated skein algebras
      Suppose q^1/2=1. The map sending a skein to its trace induces an algebra isomorphism 
        :()→χ().
     We first claim that the relations (<ref>),(<ref>),(<ref>),(<ref>) with q^1/2=1 are satisfied by the trace functions.
    By Lemma <ref> the claim is true for bigons. 
    But by Proposition <ref> cutting induces an injective algebra map, thus to verify local relations we can verify them in a bigon containing the disc where the relations are depicted: this proves the claim in general. 
    
    The algebra isomorphism is proved as follows: 
    pick an ideal triangulation of  and apply to each edge of the triangulation Proposition <ref> on the side of χ() and Theorem <ref> on (). We get the following diagram of algebra morphisms of which the horizontal lines are injective and which is commutative by Proposition <ref> and Theorem <ref>: 
    ()  @> ↪  >>  ⊗_i(T_i)
    
     @V
         VV  @V V    V    
    χ()    @> ↪  >>   ⊗_i χ(T_i) .
    
    
    Since by Lemma <ref> the right vertical arrow is an isomorphism we conclude. 
    
    9999915pt
    
    [AGS]AGS A. Alekseev, A.Y. Grosse, V. Schomerus,  Combinatorial quantization of the Hamiltonian Chern- Simons theory I-II. hep-th/9403066 and hep-th/9408097 and Commun. Math. Phys. 172, 317-358 (1995).
    
    [Ba]Barrett J. Barrett,  Skein Spaces and Spin Structures, Math. Proc. Camb, Philos. Soc. 126 (1999), 267-275.
    
    [BBJ]BBJ D. Ben-Zvi, A. Brochier and D. Jordan,  Integrating Hopf algebras over surfaces, arXiv 1501.04652 (2017).
    
    
    [BHMV]BHMV
    C. Blanchet, N. Habegger, G. Masbaum, and P. Vogel,   Three-manifold invariants derived from the Kauffman bracket,  Topology  31 (1992), no. 4, 685–699.
    
    [BR]BR E. Buffenoir, Ph. Roche,  Two Dimensional Lattice Gauge Theory Based on a Quantum Group, Commun. Math. Phys. 170 (1995), 669–698.
    
    [Bul]Bullock D. Bullock,  Rings of Sl_2()-characters and the Kauffman bracket skein module, Comment. Math. Helv. 72 (1997), no. 4, 521–542.
    
    
    
    [BFK]BFK D. Bullock,
    C. Frohman, and J.  Kania-Bartoszynska,  
    Topological Interpretations of Lattice Gauge Field Theory,  Commun. Math. Phys   198 (1998), 47–81.
    
    [BP]BP D. Bullock and J. Przytycki,  Multiplicative structure of Kauffman bracket skein module quantizations, Proc. Amer. Math. Soc. 128 (2000), no. 3, 923–931.
    
    [BW]BW F. Bonahon and H. Wong,   Quantum traces for representations of surface groups in SL2(C),  Geom. Topol.  15 (2011), no. 3, 1569–1615.
    
       
    
    [CF1]CF
     L. Chekhov and V. Fock,   Quantum Teichmüller spaces (Russian) Teoret. Mat. Fiz.  120  (1999), no. 3, 511–528; translation in Theoret. and Math. Phys.  120 (1999), no. 3, 1245–1259.
    
     [CF2]CF2
    L. O. Chekhov and V. V. Fock,  Observables in 3D gravity and geodesic algebras, in:
    Quantum groups and integrable systems (Prague, 2000), Czechoslovak J. Phys. 50 (2000),
    1201–1208.
    
    
     [CM]CM L. Charles and J.  Marché,  Multicurves and regular functions on the representation variety of a surface in SU(2), Comment. Math. Helv.  87 (2012),  409–431.
    
    [Co]Co F. Costantino,  Integrality of Kauffman brackets of trivalent graphs, Quantum Topology Volume 5, Issue 2 (2014), 143-184.
    
    
    
    [GK]GK E. Getzler, M.M. Kapranov,   Modular operads,    Compositio Math. 110 (1998), no. 1, 65-126.
    
    
    
    
    [FKL]FKL C. Frohman, J. Kania-Bartoszynska, and T. T. Q. Lê, 
     	  Dimension and Trace of the Kauffman Bracket Skein Algebra, preprint arXiv:1902.02002, 2019.
    
     
     [FR]FR Fock, V.V., Rosly, A.A.,  Poisson structure on moduli of flat connections on Riemann surfaces
    and r-matrices. Moscow Seminar in Mathematical Physics, Amer. Math. Soc. Transl. Ser. 2, 191:67–86, 1999.
    
    
    
    
    [Kau]Kauffman L. Kauffman,  States models and the Jones polynomial, Topology,
     26 (1987), 395–407.
    
    [Ka]Ka M. Kashiwara,  Global crystal bases of quantum groups, Duke Math. J. 69 no.2, 455-485, 1993.
    
    [Kas]Kashaev R. Kashaev,   Quantization of Teichmüller spaces and the quantum dilogarithm, Lett. Math. Phys.  43 (1998), no. 2, 105–115.
    
    [Kass]Kassel
    C. Kassel,   Quantum groups, Springer Verlag, New York, 1995, Graduate Texts in Mathematics, No. 155.
    
    [KS]KS A. Klymik, K. Schmudgen,  Quantum groups and their representations, Springer Verlag, Texts
    and Monographs in Physics, 1997.
    
    [KQ]KQ J. Korinman and  A. Quesney,  
    	 Classical shadows of stated skein representations at roots of unity, preprint arXiv:1905.03441, 2019.
    
    [Ko]Ko J. Korinman,  Quantum groups and braiding operators in quantum Teichmüller theory , preprint arXiv:1907.01732 , 2019.
    
     [Le1]Le:QT T. T. Q. Lê,  Quantum Teichmuller spaces and quantum trace map,  J. Inst. Math. Jussieu   18 (2019), no. 2, 249–291. See also  preprint  arXiv:1511.06054, 2015.
    
     [Le2]Le:TDEC T. T. Q. Lê,  Triangular decomposition of skein algebras,  Quantum Topology,  9 (2018), 591–632. See also preprint  arXiv:1609.04987, 2016.
    
    [LP]LP T. T. Q. Lê and J. Paprocki,
     On Kauffman bracket skein modules of marked 3-manifolds and the Chebyshev-Frobenius homomorphism,  preprint arXiv:1804.09303, 2018. 
    
    
    [Lus]Lusztig G.  Lusztig,  
     Introduction to quantum groups.  Progress in Mathematics,  110.  Birkhäuser, 1993. xii+341 pp.
    
    [Maj]MajidFoundation S. Majid,  Foundations of quantum group theory, Cambridge University Press (1995) 1-640 pp. (2nd edn.) 2000.
    
    [Marc]Marche
    J. Marché,  The skein module of torus knots,  Quantum Topol.  1 (2010), no. 4, 413–421.
    
    [Mark]Mar M. Markl,  Modular envelopes, OSFT and nonsymmetric (non-Σ) modular operads, Journal of Noncommutative Geometry, 10 (2016), 775-809, EMS.
    
    [Oh]Oh T. Ohtsuki,  Quantum invariants, Series on Knots and everything, vol. 29, World Scientific, 2002.  
    
    [Mu]Muller G. Muller,  Skein algebras and cluster algebras of marked surfaces, Preprint arXiv:1204.0020, 2012. To appear in Quantum Topology.
    
    [Mu]MS G. Muller and  P.Samuelson,   Character algebras of decorated SL_2(C)-local systems.  Algebr. Geom. Topol.  13 (2013), no. 4, 2429–2469.
    
    [Pr]Prz J. Przytycki,  Fundamentals of Kauffman bracket skein modules, Kobe J. Math. 16 (1999) 45–66.
    
    [PS1]PS1 J. Przytycki and A. Sikora,  On the skein algebras and Sl_2()-character varieties, Topology 39 (2000), 115–148.
    
    [PS2]PS2 J. Przytycki and A. Sikora,   Skein algebras of surfaces, Trans. Amer. Math. Soc.,  371 (2019), 1309–1332. See also preprint arXiv:1602.07402.
    
    [RT]RT N. Reshetikhin and V. Turaev,  Ribbon graphs
     and their invariants derived from quantum groups,  Comm. Math.
     Phys.,  127 (1990), pp.  262–288.
    
    [SW]SW A. Sikora and B.W. Westbury,   Confluence Theory for Graphs,  Alg. Geom. Topol.  7 (2007) 439–478.
    
    [Thu]Thurston D. P. Thurston,  Positive basis for surface skein algebras, Proc. Natl. Acad. Sci. USA   111 (2014), no. 27, 9725–9732.
    
    [Tu1]Turaev0 V. Turaev, 
    Conway and Kauffman modules of a solid torus, 
     J. Soviet. Math.,  52
    (1990), 2799–2805.
    
    [Tu2]Turaev V. Turaev,  Skein quantization of Poisson algebras of loops on surfaces, Ann. Sci. Sc. Norm. Sup. (4) 24 (1991), no. 6, 635–704.
    
    
    [Tu3]Tu:Book V. Turaev,  Quantum invariants of knots and 3-manifolds, de Gruyter Studies in Mathematics, 18. Walter de Gruyter & Co., Berlin, (1994).
