Algebraic generators of the skein algebra of a surface
Ramanujan Santharoubane

Department of Mathematics, University of Virginia, Charlottesville, VA 22904-4137, USA
ramanujan.santharoubane@gmail.com
 

Maximality

in finite-valued Łukasiewicz logics defined by order filters

    Marcelo E. Coniglio^1, Francesc Esteva^2, Joan Gispert^3 and Lluis Godo^2 
 

^1 Dept. of Philosophy - IFCH and  
 Centre for Logic, Epistemology and the History of Science, 
 University of Campinas, Brazil 

coniglio@cle.unicamp.br 

^2 Artificial Intelligence Research Institute (IIIA) - CSIC, Barcelona, Spain 

{esteva,godo}@iiia.csic.es 

^3 Dept. of Mathematics and Computer Science, University of Barcelona, Spain 

jgispertb@ub.edu

    
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 Let Σ be a surface with negative Euler characteristic, genus at least one and at most one boundary component. We prove that the Kauffman bracket skein algebra of Σ over the field of rational functions can be algebraically generated by a finite number of simple closed curves that are naturally associated to certain generators of the mapping class group of Σ. The action of the mapping class group on the skein algebra gives canonical relations between these generators. From this, we conjecture a presentation for a skein algebra of Σ.




§ INTRODUCTION



 §.§ Main results

This paper is focused on finding algebraic generators of the Kauffman bracket skein algebra of a surface. Throughout this paper, we will refer to the Kauffman bracket skein algebra simply as the skein algebra. Let Σ be a compact oriented connected surface of genus at least one and with at most one boundary component. Moreover we will suppose that Σ has negative Euler characteristic. We denote by 𝒮(Σ,(A)) the skein module of Σ× [0,1] with coefficients in the field of rational function (A) and by S(Σ) the skein module over [A^± 1]. The module 𝒮(Σ,(A)) is equipped with a natural product given by stacking banded links. For γ is a simple closed curve on Σ, we write γ for the element γ×[1/2,2/3] in 𝒮(Σ,(A)) and we denote by t_γ the Dehn twist along γ. 

 
Let {γ_j }_j ∈ I be a finite set of non separating simple closed curves such that 


  * For any i,j ∈ I, the curves γ_i and γ_j intersect at most once.

  * The set { t_γ_j}_j ∈ I generates the mapping class group of Σ.

Then the algebra generated by {γ_j}_j ∈ I is 𝒮(Σ,(A)). Moreover if ξ is a non zero complex number such that ξ^4 ≠ 1 then the set {γ_j}_j ∈ I generates 𝒮(Σ)A=ξ⊗ as an algebra.

We recall that the mapping class group of Σ is π_0(Homeo^+(Σ,∂Σ)).
We will now give an interpretation of some relations that should hold for the generators in the previous theorem.  Let us fix {γ_j }_j ∈ I a set of simple closed curves on Σ satisfying the hypothesis of Theorem <ref>. Let ℚ(A) ⟨ I ⟩ be the free non commutative ℚ(A)-algebra generated by { X_j }_j ∈ I. The theorem says that there exists a surjective algebra homomorphism 
    Ψ : ℚ(A) ⟨ I ⟩→𝒮(Σ,(A))
 defined by: 
    Ψ(X_j)=γ_j/A^2-A^-2  ∀ j ∈ I

Now for j ∈ I and ϵ∈{-1,1}, let T_j ^ϵ : ℚ(A) ⟨ I ⟩→ℚ(A) ⟨ I ⟩ be the homomorphism of (A)-algebra defined by:

    T_j^ϵ (X_k)     = X_k     when  ι (γ_j , γ_k)= 0 
    
     T_j^ϵ (X_k)     =  ϵ (A^ϵ X_j X_k-A^-ϵ X_k X_j)     when  ι (γ_j , γ_k)= 1

Here ι is the geometric intersection of simple closed curves. We can easily check that Ψ exchanges the actions of the T_j and the t_j in the sense that 

    Ψ(T_j^ϵ X) = t_γ_j^ϵ (Ψ(X))     for all   X ∈ℚ(A) ⟨ I ⟩,  j ∈ I, and  ϵ∈{-1,1}

Let Γ(Σ) be the mapping class group of Σ and let Γ̅(Σ) be the group Γ(Σ)  modulo its center. Suppose that I = {1, … , N} and let us denote each t_γ_j simply by t_j. 
Note that the map t_j^ϵ↦ T_j^ϵ does not extend to an action of Γ̅(Σ) on ℚ(A) ⟨ I ⟩. Indeed ℚ(A) ⟨ I ⟩ is a free non commutative algebra and the relations in Γ̅(Σ) satisfied by the Dehn twists {t_j^±1}_j ∈ J are clearly not satisfied by the operators {T_j^± 1}_j ∈ J. Hence the relations between the {t_j^±1}_j ∈ J give relations between the generators {γ_j}_j ∈ J in 𝒮(Σ,(A)). Suppose that Γ̅(Σ) has the following presentation with respect to the generators {t_j }_j ∈ I: 

    Γ̅(Σ) = ⟨  t_1, … , t_N   |   R_1(t_1, …, t_N) = ⋯ = R_K(t_1, …, t_N) = 1 ⟩
 where K is an integer and the R_k(t_1,…,t_N) are some words in {t_j^± 1}_j∈ I. Let ℛ be the bi-ideal of ℚ(A) ⟨ I ⟩ generated by the elements



    R_k (T_1, …, T_N) X_i -X_i,    for   1 ≤ i ≤ N  and  1 ≤ k ≤ K  
        T_j T_j^-1 X_i-X_i,    for   1 ≤ i,j  ≤ N   
       X_i X_j - X_j X_i,     for  ι (γ_i , γ_j)= 0

We define 
    𝒜(Γ(Σ)) =ℚ(A) ⟨ I ⟩/ℛ
 which is a quotient of ℚ(A) ⟨ I ⟩ on which the actions of the T_j^± 1 extend to a canonical action of Γ̅(Σ).
A direct consequence of Theorem <ref> is the following:

The canonical map 
    𝒜(Γ(Σ)) →𝒮(Σ,(A))
 is surjective.



 
There exists a presentation of Γ̅(Σ) for which 𝒜(Γ(Σ)) is isomorphic to 𝒮(Σ,(A)) as a non commutative (A)-algebra.





 §.§ Notes and references


Bullock in <cit.> was the first to find algebraic generators of the skein algebra of a surface. His generators are over [A^± 1] and not over (A). The number of his generators is exponential in the genus of the surface whereas here we have a linear number (by chosing the right generators of Γ(Σ)).

Finite generation was also prove by Abdiel and Frohman (see <cit.>). In <cit.>, Frohman and Kania-Bartoszynska studied the skein algebra when A is evaluated at a root of unity. They prove that it is generated over its center by a pair of subalgebras from pants decomposition. Their generators have some similarities with the one in the current paper.


Presentations of skein algebras of surfaces are only known in genus zero and one. In <cit.>, Bullock and Przytycki found such a presentation for the one holed torus, the four holed sphere and twice holed torus. They related some of these algebras to non standard deformations of lie algebras.

When A is specialized to -1, it was shown by Bullock (see <cit.>) and Przytycki-Sikora (see <cit.>) that the skein algebra of a surface is isomorphic to the ring of algebraic functions of the SL(2,) character variety of the surface. Moreover, for A = √(-1), Marché (see <cit.>) gave an homological interpretation of the skein algebra of the surface. Note that the map Ψ defined in Equation (<ref>) is not defined if A is specialized to a 4-th primitive root of unity. It is possible to see that if we specialize A at a 4-th root of unity in the algebra 𝒜(Γ(Σ)) we find something different from the algebras studied by  Bullock, Marché and Przytycki-Sikora.

Humphries generators (see <cit.>) and Lickorish generators (see <cit.>) are examples of generators of the mapping class groups satisfying the hypothesis of Theorem <ref>. Moreover presentations for both of these generating sets are know, we refer to book <cit.> of Farb and Margalit for more details.


We consider Γ(Σ) quotiented by its center because the center of the mapping class group acts trivially on the skein algebra of the surface.




§ ACKNOWLEDGEMENTS

I want to thank C. Frohman, J. Marché and G. Masbaum for helpful conversations. 





§ QUICK REVIEW OF THE SKEIN ALGEBRA

 For any compact oriented manifold M (maybe with boundary), we denote by 𝒮(M) the Kauffman bracket skein module with coefficients in [A^± 1]. We recall that it is the free [A^±1]-module generated by the set of isotopy classes of banded links in the interior of M quotiented by the following so-called skein relations.  First
 
    L_× = A  L_∞ + A^-1  L_0
 where L_×, L_0, L_∞ are any three banded links in M which are the same outside a small 3-ball but differ inside as in Figure <ref>. In this case, the triple L_×, L_0, L_∞ is called a Kauffman triple. The second relation satisfied in 𝒮(M) is 
    L ∪ D=-(A^2+A^-2)   L
 where L is any link in M and D is a trivial banded knot. We define 𝒮(M, (A)) to be the (A)-vector space 𝒮(M)⊗(A) where the tensor product is made over [A^± 1].



Let Σ be a compact oriented connected surface, we denote by 𝒮(Σ,(A)) the space 𝒮(Σ× [0,1],(A)). Stacking banded links on top of each other gives 𝒮(Σ,(A)) a structure of (A)-algebra.


A multicurve is a disjoint union of non null-homotopic simple close curve inside Σ. For γ⊂Σ a multicurve we write γ for the banded link γ×[1/2,2/3] in 𝒮(Σ,(A)) and we will still call this banded link a multicurve. A well known theorem is the following:


The set of isotopy classes of multicurves on Σ is a basis of the (A)-vector space 𝒮(Σ,(A)).

In particular this theorem clearly implies that simple closed curves generate 𝒮(Σ,(A)) as an algebra. Recall that Γ(Σ) = π_0(Homeo^+(Σ,∂Σ)) acts canonically on  𝒮(Σ,(A)). Recall also that if γ⊂Σ is a simple closed curve, we denote by t_γ the Dehn twist along γ. The following lemma is standard and can be obtained by applying the skein relation (<ref>).
 
Let α and β be two simple close curves intersecting once. We have in  𝒮(Σ,(A)):

    t_α^ϵ (β) =  A^ϵαβ-A^-ϵβαϵ(A^2-A^-2)









§ PROOF OF THEOREM <REF>

Let Γ(Σ) be the mapping class group of Σ. Let {γ_j }_j ∈ I be a set of simple closed curves satisfying the hypothesis of the Theorem <ref> and let ℬ be the sub-algebra of 𝒮(Σ,(A)) generated by {γ_j}_j ∈ I.
 
 ℬ is stable by the action of Γ(Σ).
 
 
 Since { t_γ_j}_j ∈ I generate Γ(Σ), it enough to prove that for any j,k ∈ I we have t_γ_j^± 1(γ_k) ∈ℬ. If γ_j does not intersect γ_k then t_γ_j^± 1(γ_k) = γ_k ∈ℬ. Now if γ_j intersects γ_k once then, by Lemma <ref>, we have 
    t_γ_j^± 1(γ_k)= A^± 1γ_j γ_k-A^∓ 1γ_k γ_j/±(A^2-A^-2)∈ℬ





If  𝒞_0 denotes the set of non separating simple closed curves then 𝒞_0 ⊂ℬ. 



Let γ∈𝒞_0, there exists ϕ∈Γ(Σ) such that ϕ(γ_1) = γ. Since γ_1 belongs to ℬ which is stable by the action of Γ(Σ) (see previous lemma), we have γ∈ℬ.





If  𝒞_1 denotes the set of  separating simple closed curves then 𝒞_1 ⊂ℬ. 



Suppose that the genus of Σ is g ≥ 1. Let δ_1,…,δ_g be the following curves 

    < g r a p h i c s >

where δ_g is trivial when Σ does not have boundary. Let j ∈{1,…,g} and let z_j,z_j' be the two following non separating curves in the torus with two boundary components defined δ_j,δ_j-1 : 

    < g r a p h i c s >

By applying the skein relations we have :

    z_j' z_j = A^2 x_j x_j'+A^-2 y_j y_j'+ δ_j+ δ_j-1
 where δ_0= -A^2-A^-2 and x_j,x_j',y_j,y_j' are non separating curves. By Lemma <ref>, z_j', z_j, x_j,x_j',y_j,y_j' ∈ℬ so by an induction on j we can prove that for all 1 ≤ j ≤ g we have δ_j ∈ℬ. 


Now if γ is a separating curve, there exists ϕ∈Γ(Σ) and j_0 such that γ = ϕ(δ_j_0). Since ℬ is stable by the action of Γ(Σ), we have γ∈ℬ.



 
𝒮(Σ,(A)) is algebraically generated by simple closed curves. Hence combining Lemma <ref> and Lemma <ref>, we can conclude. Moreover if ξ is a non zero complex number such that ξ^2-ξ^-2≠ 0, we can still conclude by Lemmas <ref>, <ref> and <ref> that ℬ = S(Σ) A=ξ⊗.





§ INTERPRETATION OF THE RELATIONS IN THE SKEIN ALGEBRA

Let {γ_j }_j ∈ I be a set of simple closed curves on Σ satisfying the hypothesis of Theorem <ref>. Recall that ℚ(A) ⟨ I ⟩ is the free non commutative ℚ(A)-algebra generated by { X_j }_j ∈ I. 

For X,Y ∈ℚ(A) ⟨ I ⟩ we define [X,Y]_A := AXY - A^-1 YX.

Recall that the maps {T_j^ϵ}_j ∈ I are defined by (<ref>) and (<ref>) in the introduction. Recall also that given a presentation of Γ̅(Σ) with respected to the generating set {t_γ_j}_j ∈ I, we defined 𝒜(Γ(Σ)) via Equation (<ref>) (see the introduction). By definition, any relation satisfied by the {t_γ_j}_j ∈ I (which may not appear in the given presentation) gives some relation in 𝒜(Γ(Σ)).
Let us focus on the following relations: 

    T_j^-1 T_i^-1 T_j^-1 T_i T_j T_i X_a -X_a=0,    for  ι (γ_i , γ_j)= 1  and  a ∈ I 
        T_j T_j^-1 X_i-X_i=0,                   for   i,j ∈ I   
       X_i X_j - X_j X_i=0,                     for  ι (γ_i , γ_j)= 0

Note that these relations hold but are not a complete set of relations in 𝒜(Γ(Σ)). The first relation comes from the braid relations in the mapping class group. 
 
In 𝒜(Γ(Σ)), the relation (<ref>), (<ref>) and (<ref>) are equivalent to:

    [[X_j,X_i]_A,X_j]_A = X_i ,       for  ι (γ_i , γ_j)= 1  
    
    X_i X_j - X_j X_i=0,       for  ι (γ_i , γ_j)= 0



Let i,j ∈ I, note that if ι(γ_i,γ_j) =0, the relation (<ref>) in empty and if ι(γ_i,γ_j) =1, this relation gives  [[X_j,X_i]_A,X_j]_A = X_i.

Let i,j ∈ I such that ι(γ_i,γ_j) =1. Because of (<ref>), the relation (<ref>) can be rewritten

    T_i T_j T_i X_k = T_j T_i T_j X_k
  for all k ∈ I.
It is easy to check that this relation is implied by (<ref>) and (<ref>).


We did not include T_j^-1 T_j X_ i = X_i in the relations defining 𝒜(Γ(Σ)) because they give [X_j,[X_i,X_j]_A]_A = X_i for ι(γ_i,γ_j) =1 which is the same as [[X_j,X_i]_A,X_j]_A = X_i.




 §.§ The case of the one-holed torus

Let Σ be a surface of genus one surface with one boundary component. Its mapping class group is the braid group B_3 whose presentation is ⟨ t_1,t_2   |   t_1 t_2 t_1 = t_2 t_1 t_2 ⟩. Here is t_1 is Dehn twist along the canonical meridian of Σ and t_2 is the Dehn twist around the longitude of Σ. Note that these two curves satisfy the hypothesis of Theorem <ref>. The center of this group is the group generated by (t_1 t_2 t_1)^2 and Γ̅(Σ) is PSL_2() with following presentation 
    Γ̅(Σ) = ⟨ t_1,t_2  |  t_1 t_2 t_1 = t_2 t_1 t_2,   (t_1 t_2 t_1)^2=1⟩
  In this case 𝒜(Γ(Σ)) is a non commutative algebra generated by X_1 and X_2. Because of Proposition <ref>, the only relations between X_1 and X_2 are:

    [X_1,[X_2,X_1]_A]_A = X_2,   [X_2,[X_1,X_2]_A]_A = X_1,  (T_1 T_2 T_1)^2X_1 = X_1,   (T_1 T_2 T_1)^2 X_2 = X_2

It is easy to check that the two last relations are implied by the two first one. Therefore 
    𝒜(Γ(Σ)) = ⟨ X_1, X_2 | [X_1,[X_2,X_1]_A]_A = X_2,   [X_2,[X_1,X_2]_A]_A = X_1 ⟩
 From <cit.> the skein module of the one-holed torus is isomorphic to 𝒜(Γ(Σ)). Therefore Conjecture <ref> holds for the one holed torus.
10

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