Dimension and Trace of the Kauffman Bracket Skein Algebra

Let $F$ be a finite type surface and $\zeta$ a complex root of unity. The Kauffman bracket skein algebra $K_{\zeta}(F)$ is an important object in both classical and quantum topology as it has relations to the character variety, the Teichm\"uller space, the Jones polynomial, and the Witten-Reshetikhin-Turaev Topological Quantum Field Theories. We compute the rank and trace of $K_{\zeta}(F)$ over its center, and we extend a theorem of Frohman and Kania-Bartoszynska which says the skein algebra has a splitting coming from two pants decompositions of $F$.


Introduction
Let F be a finite type surface and ζ a complex root of unity. The Kauffman bracket skein algebra K ζ (F ) is an important object in both classical and quantum topology as it has relations to the character variety, the Teichmüller space, the Jones polynomial, and the Witten-Reshetikhin-Turaev Topological Quantum Field Theories. We recall the definition of K ζ (F ) in Section 3.
The linear representations of K ζ (F ) play an important role in hyperbolic Topological Quantum Field Theories. In [9] we prove the Unicity Conjecture of Bonahon and Wong [4] which among other things states that generically all irreducible representations of K ζ (F ) have the same dimension equal to the square root of the dimension of K ζ (F ) over its center Z ζ (F ). Here if A is an algebra whose center is a domain Z then the dimension of A over Z, denoted by dim Z A, is defined to be the dimension of the vector space A ⊗ ZZ over the field of fractionsZ of Z. The calculation of the dimension dim Z ζ (F ) K ζ (F ) is one of the main result of this paper.
Theorem 1 ( See Theorem 6.1). Suppose F is a finite type surface of genus g with p punctures and negative Euler characteristic, and ζ is a root of unity of order ord(ζ). Let m be the order ζ 4 , then if ord(ζ) ≡ 0 (mod 4).
We show thatK ζ (F ) := K ζ (F )⊗ Z ζ (F )Zζ (F ), whereZ ζ (F ) is the field of fractions of Z ζ (F ), is a division algebra having finite dimension over its centerZ ζ (F ). Thus every element α ∈K ζ (F ) lies in a finite field extension ofZ ζ (F ) and hence has the reduced trace trZ ζ (F ) (α) ∈Z ζ (F ). We recall the definition of the reduced trace in Section 2. The second goal of the paper is to compute the reduced trace of elements of K ζ (F ).
To state the theorem, denote by S the set of all isotopy classes of simple diagrams on F , where a simple diagram is the union of disjoint, non-trivial simple closed curves on F . For each α ∈ S one can define an element T (α) ∈ K ζ (F ), such that the set {T (α) | α ∈ S } is a Cbasis of K ζ (F ) and T (α) is central if and only if α is in a certain subset S ζ of S . See Section 3.3 for details. The definition of T (α) involves Bonahon and Wong's threading map [3]. As the C-vector space K ζ (F ) has basis {T (α) | α ∈ S }, hence it is enough to compute the trace of each T (α).
Theorem 2 (See Theorem 8.1). Let F be a finite type surface and ζ be a root of 1. For α ∈ S one has Along the way we develop tools for determining when a collection of skeins forms a basis forK ζ (F ).
The last goal of the paper is to prove that there exists a splitting ofK ζ (F ) over its center coming from pairs of pants decompositions of the surface.
Theorem 3 (See Theorem 9.1). Let F be a finite type surface of negative Euler characteristic. There exist two pants decompositions P and Q of F such that for any root of unity ζ theZ ζ (F )-linear map ψ : C ζ (P) ⊗Z ζ (F ) C ζ (Q) →K ζ (F ), ψ(x ⊗ y) → xy, is aZ ζ (F )-linear isomorphism of vector spaces. Here C ζ (P) (respectively C ζ (Q)) is theZ ζ (F )-subalgebra ofK ζ (F ) generated by the curves in P (respectively in Q). Both C ζ (P) and C ζ (Q) are maximal commutative subalgebras of the division algebraK ζ (F ).
This theorem has an application in defining invariants of links in 3-manifold which will be investigated in a future work.
The paper is organized as follows. In Section 2 we survey results about division algebras that have finite rank over their center, and facts about trace, and filtrations of algebras, with a goal of applying these to the Kauffman bracket skein algebra. We follow by introducing the Kauffman bracket skein algebra in Section 3. Its basis is given in terms of simple diagrams, so we describe ways of parametrizing simple diagrams on a surface. We also introduce a residue group. In section 4 we show that after enough twisting the Dehn Thurston coordinates of a simple diagram on a closed surface stabilize to become an affine function of the number of twists. This allows us to define stable Dehn-Thurston coordinates. In Section 5 we introduce a degree map and use it to formulate a criterion for independence of a collection of skeins over its center. Section 6 computes the dimension of the Kauffman bracket skein algebra over its center, proving Theorem 1. In section 7 we find bases for commutative subalgebras ofK ζ (F ) generated by the curves in a primitive non-peripheral diagram on F with coefficients inZ ζ (F ). In Section 8 we find a formula for computing the trace, proving Theorem 2. The paper concludes in Section 9 which proves the splitting theorem (Theorem 3).

Acknowledgment. The authors thank F. Luo and D. Thurston for their comments.
This material is based upon work supported by and while serving at the National Science Foundation. Any opinion, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation. T.L. thanks the CIMI Excellence Laboratory, Toulouse, France, for inviting him on a Excellence Chair during the period of January -July 2017 when part of this work was done.

Division algebras, trace, filtrations
In this section we survey some well-known facts about division algebras, trace, and filtrations of algebras that will be used in the paper.
2.1. Notations and conventions. Throughout the paper N, Z, Q, R, C denote respectively the set of natural numbers, integers, rational numbers, real numbers, and complex numbers. Note that 0 ∈ N. Let Z 2 = Z/(2Z) be the field with 2 elements.
A complex number ζ is a root of 1 if there is a positive integer n such that ζ n = 1, and the smallest such positive integer is called the order of ζ, denoted by ord(ζ).
All rings are assumed to be associative with unit, and ring homomorphisms preserve 1. A domain is a ring A, not necessarily commutative, such that if xy = 0 with x, y ∈ A, then x = 0 or y = 0. For a ring A denote by A * the set of all non-zero elements in A. For example, C * is the set of all non-zero complex number.

2.2.
Algebras finitely generated over their centers. The following is well-known, and we present a simple proof for completeness. Proposition 2.1. (a) If k is a field and A is k-algebra which is a domain and has finite dimension over k, then A is a division algebra. (b) Let Z be the center of a domain A andZ be the field of fractions of Z. Assume A is finitely generated as a Z-module. ThenÃ = A⊗ ZZ is a division algebra.
Proof. (a) Suppose 0 = a ∈ A. The k-subalgebra of A generated by a, being a commutative finite domain extension of the field k, is a field. Hence a has an inverse. (b) Every element ofÃ can be presented in the form az −1 where a ∈ A and 0 = z ∈ Z. From here it is easy to show thatÃ is a domain and the natural map A →Ã is an embedding. Since A is Z-finitely generated,Ã is alsoZ-finitely generated, and (b) follows from (a).
The above proposition reduces many problems concerning domains which are finitely generated as modules over their centers to the case of division algebras finitely generated over their centers.

2.3.
Trace. Suppose k is a field and A is a k-algebra which is finitedimensional as a k-vector space. For a ∈ A, the left multiplication by a is a k-linear operator acting on A, and its trace is denoted by TR A/k (a). The reduced trace is defined by Again, the following is well-known.
Proposition 2.2. Suppose that k is a field, and A is division k-algebra having finite dimension over k.
(c) The function tr A/k : A → k is non-degenerate in the sense that for 0 = a ∈ A there exists b ∈ A such that tr A/k (ab) = 0. In particular, A is a Frobenius algebra.
Proof. (a) Let C be the k-subalgebra of A generated by a, then C is a field. By the definition of the minimal polynomial, As a k-vector space C has basis {1, a, . . . , a l−1 }. It follows that TR C/k = −c l−1 . Let {a 1 , . . . , a t } be a basis of A over C so that A = t i=1 Ca i . Each Ca i is invariant under the left multiplication by a, and the action of a on each Ca i has trace equal to TR C/k (a). Hence, Then tr(ab) = tr(1) = 1 = 0.

Maximal commutative subalgebras. Suppose
A is a division algebra with center k. If C ⊂ A is a maximal commutative subalgebra, then C is a field and 2.5. Dimension. Suppose a C-algebra A has center Z which is a commutative domain. LetZ be the field of fractions of Z. The dimension dim Z A is defined to be the dimension of theZ-vector space A ⊗ ZZ . A filtration compatible with the product of A is a sequence There exists a positive integer u such that for all k ≥ u, Proof. Assume a 1 , . . . , a d ∈ A form a basis of A ⊗ ZZ overZ. Let u be a number such that all a j are in F u (A). Since a 1 , . . . , a d are linearly independent over Z, the sum d j=1 Za j is a direct sum. We have The dimension of the first space in (4) is 2.6. Lattice points in polytope. Suppose R n is the standard ndimensional Euclidean space. A lattice Λ ⊂ R n is any abelian subgroup of maximal rank n. A convex polyhedron Q is the convex hull of a finite number of points in R n and its n-dimensional volume is denoted by vol(Q). Let kQ := {kx | x ∈ Q}.
Lemma 2.4. Suppose Γ ⊂ Λ are lattices in R n and Q ⊂ R n is the union of a finite number of convex polyhedra with vol(Q) > 0. Let u be a positive integer. One has Proof. Define vol(Λ) to be the n-dimensional volume of the parallelepiped spanned by a Z-basis of Λ. One has from which one easily obtains (5).

Kauffman bracket skein algebra
The Kauffman bracket skein module of a 3-manifold was introduced independently by Przytycki and Turaev. For a surface the skein module has an algebra structure first considered in [29]. In this section we recall the definition of the Kauffman bracket skein algebra of a finite type surface F , and present some results concerning its center. We also explain how to coordinatize the set of curves on F and use coordinates to define a residue group associated to F and a root of 1.
3.1. Finite type surface. An oriented surface F of the form F = F \ V, whereF is an oriented closed connected surface and V is finite (possibly empty), is called a finite type surface. A point in V is called a puncture. The genus g = g(F ) and the puncture number p = |V| totally determine the diffeomorphism class of F , and for this reason we denote F = F g,p . The Euler characteristic of F is 2 − 2g − p, which is non-negative only in 4 cases: (g, p) = (0, 0), (0, 1), (0, 2), or (1, 0).
Since the analysis of these four surfaces is simple and requires other techniques, very often we consider these cases separately.
Throughout this section we fix a finite type surface F = F g,p .
In this paper a loop on F is a unoriented submanifold diffeomorphic to the standard circle. A loop is trivial if it bounds a disk in F ; it is peripheral if it bounds a disk inF which contains exactly one puncture.
A simple diagram is the union of several disjoint non-trivial loops. A simple diagram is peripheral if all its components are peripheral.
If a : [0, 1] →F is a smooth map such that a(0), a(1) ∈ V and a embeds (0, 1) into F then the image of (0, 1) is called an ideal arc. Isotopies of ideal arcs are always considered in the class of ideal arcs.
Suppose α ⊂ F is either an ideal arc or a simple diagram and β ⊂ F is a simple diagram. The geometric intersection number I(α, β) is the minimum of |α ∩ β |, with all possible α isotopic to α and β isotopic to β. We say that α is β-taut, or α and β are taut, if they are transverse and |α ∩ β|= I(α, β).
A simple diagram α ⊂ F is even if I(α, a) is even for every loop a ⊂ F . It is easy to see that α is even if and only if α represents the zero element in the homology group H 1 (F , Z 2 ).
Very often we identify a simple diagram with its isotopy class. Denote by S = S (F ) the set of all isotopy classes of simple diagrams on F . Let S ev ⊂ S be the subset of all classes of even simple diagrams, and S ∂ ⊂ S be the subset of all peripheral ones. For convenience, we make the convention that the empty set ∅ is a peripheral simple diagram. Thus ∅ ∈ S ∂ ⊂ S ev ⊂ S .

Kauffman bracket skein algebra.
is an embedding of a disjoint union of oriented annuli in F × [0, 1]. By convention the empty set is considered as a framed link with 0 components and is isotopic only to itself.
For a non-zero complex number ζ, the Kauffman bracket skein module of F at ζ, denoted by K ζ (F ), is the C-vector space freely spanned by all isotopy classes of framed links in F ×[0, 1] subject to the following skein relations Here the framed links in each expression are identical outside the balls pictured in the diagrams, and the arcs in the pictures are supposed to have blackboard framing. If the two arcs in the crossing belong to the same component then it is assumed that the same side of the annulus is up.
Theorem 3.1. [24,26] The set S of isotopy classes of simple diagrams is a basis of K ζ (F ) over C.
For two framed links L 1 and L 2 in S × [0, 1], their product, L 1 L 2 , is defined by first isotoping L 1 into F × (1/2, 1) and L 2 into F × (0, 1/2) and then taking the union of the two. This product gives K ζ (F ) the structure of a C-algebra, which is in most cases non-commutative. Let Z ζ (F ) be the center of K ζ (F ).
Theorem 3.2. Let F be a finite type surface and ζ a root of 1.
The module K ζ (F ) is finitely generated as a Z ζ (F )-module.
is the field of fractions of Z ζ (F ). From Proposition 2.1, we have the following corollary.
They satisfy the product to sum formula, Suppose α ∈ S is a simple diagram. Some components of α may be isotopic to each other. Let C 1 , . . . , C k be a maximal collection of components of α such that no two of them are isotopic. Then there are positive integers (l 1 , . . . , l k ) such that α is the union of l j parallel copies of C j with j = 1, . . . , k. In other words, α = k j=1 C l j j in K ζ (F ). Let If α, β ∈ S then in general αβ ∈ S . However, if I(α, β) = 0 where α, β ∈ S , then α and β can be represented by disjoint simple diagrams, and hence αβ ∈ S . In particular, if α ∈ S and β ∈ S ∂ , then αβ = βα ∈ S . Further, if α ∈ S , then α k ∈ S .
[9] Let F be a finite type surface and ζ be a root of 1.
Recall that Z ζ (F ) is the center of the skein algebra K ζ (F ).
The point is while K ζ (F ) has a C-basis parameterized by S , its center Z ζ (F ) has a basis parameterized by S ζ .
Remark 3.6. The Chebyshev basis is important in the theory of quantum cluster algebras and quantum Teichmüller spaces of surfaces. It was first used, for the case when F is a torus, by Frohman-Gelca [7]. The famous positivity conjecture states that the Chebyshev basis is positive [15,27] For each n ∈ N define F n = F A n (K ζ (F )) to be the C-subspace of K ζ (F ) spanned by α ∈ S such that k i=1 I(α, a i ) ≤ n. The following is well-known and its variants were used extensively in the study of skein algebras, see e.g., [9,14,16].
Proof. It is clear that F n ⊂ F n+1 and F n = K ζ (F ). It remains to show that F n F l ⊂ F n+l . Suppose α, β ∈ S and α ∈ F n , β ∈ F l . The product αβ is obtained by placing α above β. Using the skein relation (6) we see that αβ = c γ γ, where each γ is a diagram obtained by a smoothing of all the crossings in αβ and hence I(γ, a i ) ≤ I(α, a i ) + I(β, a i ). It follows that αβ ∈ F n+l .

3.5.
Coordinates and residues, open surface case. When F = F g,p has negative Euler characteristic, one can parameterize the set S of simple diagrams on F by embedding it into the free abelian group Z r , where r = 6g − 6 + 3p. This embedding depends on an object that we call the coordinate datum. In this subsection we describe this embedding for open surfaces.
Suppose F = F g,p with p ≥ 1 and F has negative Euler characteristic, By definition, a coordinate datum of F is an ordered ideal triangulation, which is any sequence {e 1 , . . . , e r } of disjoint ideal arcs on F such that no two are isotopic. Here r = 6g − 6 + 3p = −3χ(F ). Such ideal triangulations always exists and we fix one of them. The ideal arcs (e i ) r i=1 cut F into triangles. Recall that I(α, e i ) is the geometric intersection number. Define It is known that ν is injective, and its image A := ν(S ) consists of all (n 1 , . . . , n r ) ∈ N r such that (11) whenever e i , e j , e k are edges of a triangle, n i + n j + n k is even and n i ≤ n j + n k .
We call ν(α) the edge-coordinates of α with respect to the coordinate datum. Let S : A → S be the inverse of ν. That is, S(n) is the simple diagram whose edge coordinates are n. Note that 0 = ν(∅) ∈ A, and A is closed under addition. Hence A is a submonoid of Z r . For a submonoid X of Z r let X be the subgroup of Z r generated by X, then A is the subset of Z r consisting of (n 1 , . . . , n r ) such that whenever e i , e j , e k are edges of a triangle, n i + n j + n k is even.
Proof. (a) follows from the description (11) of A. (b) Consider the triangulation as a cellular decomposition ofF which has p zero-cells and r one-cells. Identify (Z 2 ) r with the set of all maps from one-cells to Z 2 , and let C 1 ⊂ be the set of all one-cocycles. Let f : Z r → (Z 2 ) r be the reduction modulo 2, then A = f −1 (C 1 ). Therefore Z r /A ∼ = (Z 2 ) r /C 1 and hence (12) [ The following sequence is exact In an exact sequence of finite groups, the alternating product of orders of groups is 1. Hence Plugging this value of |C 1 | in (12), we get the result.
The following follows easily from the definition.
Proposition 3.10. If α, β ∈ S and I(α, β) = 0, then αβ ∈ S and Let S • ⊂ S be the set of all simple diagrams containing no peripheral loops, with the convention that the empty diagram is in S • , and let A • = ν(S • ). Every simple diagram can be presented in a unique way as the product of an element in A ∂ and an element in A • . By Proposition 3.10 every x ∈ A can be presented uniquely as (14) x Using the decomposition (14) for x 1 and x 2 , we get The uniqueness of (14) shows that is the homology class of α, is a surjective monoid homomorphism and extends to a surjective group homomorphismh : 2 . Suppose ζ is a root of 1. From Theorem 3.5 and the bijection ν : S → A, we see that the center Z ζ (F ) has a C-basis parameterized by A ζ while K ζ (F ) has a basis parameterized by A. Hence we want to study the quotient A/A ζ . Define the ζ-residue group Proof. Case 1: ord(ζ) = 0 mod 4. Then A ζ = A ∂ + mA, and In all cases we have This was proved in [2,9] for a slightly different filtration but the easy proof there works also for this case. In [9], an explicit formula for j(n, n ) is given.
3.6. Coordinates and residues, closed surface case. Let F be a closed oriented surface of genus g > 1. A coordinate datum of F consists of an ordered pants decomposition P and a dual graph D defined as follows.
An ordered pants decomposition of F is a sequence P = (P 1 , . . . , P 3g−3 ) of disjoint non-trivial loops such that no two of them are isotopic. The collection P cuts F into 2g − 2 pairs of pants (i.e., thrice punctured spheres). A dual graph D to P is a trivalent graph embedded into F having exactly 2g − 2 vertices, one in the interior of each pair of pants, and 3g − 3 edges (e i ) 3g−3 i=1 such that e i intersects P i transversally in a single point and is disjoint with P j for j = i.
For technical simplicity we assume that D does not have an edge with endpoints in the same vertex, in other words each pair of pants has 3 different boundary components. Such a coordinate datum always exists, and we fix one.
Let N (D) be a regular neighborhood of D and Ω = ∂N (D) be its boundary. We assume that for each pair of pants C the intersection C ∩N (D) is a regular neighborhood (in C) of D ∩C, and Ω∩C consists of 3 arcs as in Figure 1. We call C ∩ N (D) the red hexagon of C. Figure 1. The pair of pants C bounded by loops P i , P j , P l . The trivalent graph is D ∩C. The bold arcs a i , a j , a l are Ω ∩ C. The red hexagon contains the trivalent graph and is bounded by a i , a j , a l and parts of P i , P j , P l .
The curves {P i } 3g−3 i=1 and the system of red hexagons allow to define the Dehn-Thurston coordinates as in [20], which is an injective map The remaining 3g − 3 coordinates of ν(α) are the twist coordinates, where ν i+3g−3 (α) is the twist coordinate of α at the loop P i . We use the same conventions for the DT coordinates as in [9]. Our description relates to those of other authors as follows: Our dual graph is embedded as a spine of the complement of the red hexagon of [20]; in relation to [10] it is embedded so that it misses the windows and is disjoint from the triangular model curves. The approach in [23] is the same as in [10]. All those conventions result in the same DT coordinates of simple diagrams.
If P i , P j , P l bound a pair of pants then n i + n j + n l is even. If Since these conditions are linear, A is a submonoid of Z 6g−6 . Let A be the group generated by A.
Proof. Identify Z 6g−6 = Z 3g−3 ⊕ Z 3g−3 and let p 1 be the projection from Z 6g−6 onto the first summand. Note that ν(P j ) = 0 ⊕ δ j , where 0 ∈ Z 3g−3 is the zero element and δ j ∈ Z 3g−3 is the element all of whose coordinates are zero except the j-th entry which is 1. It follows that is an ideal triangulation, giving rise to edge-coordinates of simple diagrams onN (D), and the set of all such edge-coordinates is denoted by A . Note that P i , P j , P k bound a pair of pants if and only if e i , e j , e k are edges of an ideal triangle. Condition (*) and Lemma 3.9(a) show that where the second identity follows from Lemma 3.9 (b).
Let m = ord(ζ 4 ). Define the ζ-residue group R ζ (F ) and the number D ζ (F ) just like in the case of open surfaces (noting p = 0). That is, (20) R ζ (F ) = A/A ζ which depends on a coordinate datum of F . The formulas for D are the same: Let ζ be a root of 1. For any Dehn-Thurston datum of F , Proof. The proof is almost identical to that in the case of open surfaces. (a) Case 1: ord(ζ) = 0 mod 4. In this case A ζ = mA, and R ζ (F ) = A/mA. Since rkA = 6g − 6, we have |A/mA|= m 6g−6 .
Case 2: ord(ζ) = 0 mod 4. In this case A ζ = mA ev . First note that |A/A ev |= 2 2g . The proof of this fact is identical to that of Proposition 3.11. Now we have When F is closed we don't have a nice product formula like the identity in Proposition (3.13). However, this identity still holds for the class of triangular simple diagrams defined as follows.
A simple diagram α ⊂ F is triangular with respect to P if it is Ptaut and for every pair of pants C each connected component of α ∩ C is an arc whose two endpoints are in two different components of ∂C. In particular, α cannot have a component isotopic to any P i .
Let S ∆ ⊂ S be the subset consisting of triangular simple diagrams, and A ∆ = ν(S ∆ ). Then n = (n 1 , . . . , n 6g−6 ) ∈ A is in A ∆ if and only if • whenever P i , P j , P k bound a pair of pants, n i ≤ n j + n k , and In [9] we proved a stronger result, giving the exact value of j in (23).

Stable Dehn-Thurston coordinates
Throughout this section F is a closed surface with g = g(F ) ≥ 2, and with a fixed coordinate datum (P, D). Proposition 3.16 only works for triangular simple diagrams. We show that after enough twisting by Ω a simple diagram becomes triangular and its DT coordinates become an affine function of the number of twists.  Let α ⊂ F be a simple diagram. There exists η(α) ∈ Z 6g−6 such that if k is large enough then h k Ω (α) is triangular with respect to P, and In particular, the last 3g − 3 coordinates of µ(α) are equal to 0.
Note that the action of any element of the mapping class of the surface on diagrams extends linearly to yield an automorphism of the skein algebra, In specific, it makes sense to talk about h k Ω (x) for x ∈ K ζ (F ).
We present the proof of Theorem 4.1 in Subsection 4.3.

4.2.
Piecewise affine functions. A function f : R k → R l is affine if there is an l × k matrix A and a vector B such that   Proof. This is a special case of Proposition A1 of [10,Section 4].
In order to prove the next proposition we need to explore the topology of Dehn-Thurston datum (P, D) of a closed surface F . To define a geometric intersection of a simple diagram S with the graph D we add the assumption that S misses the vertices of D. That is, I(S, D) is the minimum cardinality of S ∩ D where S is isotopic to S, misses the vertices of D and is transverse to its edges. Proof. Given Dehn-Thurston datum for F let A j denote the annuli which are collars of the pants curves P j , and let Q i be the pairs of pants that are the complement of ∪A j in F . These are the shrunken pairs of pants, versus the pairs of pants C j as defined in section 3.6. By assumption D is transverse and minimizes its intersection with the boundaries of the annuli A j . We say that a simple diagram is in standard position if • its intersection with the Q i is isotopic to standard model curves in the complement of D, • its intersection with ∂A j is disjoint from ∂A j ∩ D, • it minimizes its intersection with D ∩ A j , for each j. If a simple diagram is in standard position then its twists coordinates are given by its signed intersection numbers with D ∩ A i .
We are most interested in the case that S is triangular. In Figure 2 we show the triangular model curve d 12 and another curve d 12 that will play a role in the following. For each pair of boundary components of each pair of pants Q i there are two curves like this, the model curve d ij and its mate d ij . Recall that model curves describe possible ways in which a simple diagram in standard position intersects a pair of pants (see, e.g., [9]). We say a triangular diagram S is in special position if: • It realizes I(S, ∂A j ) for all j; • It realizes I(S, D); • It does not intersect D ∩ ∂A j for any j; • It minimizes the intersection S ∩ D ∩ Q i for all i, among all S satisfying the first condition. It is easy to see that if S is a triangular diagram in special position then it intersects each Q j in curves parallel to model curves d kl and the new curves d kl .
To move a curve from special position to standard position, each curve of type d kl needs to be isotoped to a curve of type d kl . In the Figure 3 we show a curve of the form d kl in the process of being pushed into standard position. Its intersection with D needs to be pushed inside the annuli. As the result the intersection of S with D ∩ A j in each of the annuli on either end of d kl is incremented by 1.  (a) There exists an additional collection of 6g − 6 loops P j with j = 3g − 2, . . . , 9g − 9, and a piecewise affine function G : Z 9g−9 → Z 6g−6 such that for all α ∈ S , (29) ν(α) = G I(α, P 1 ), . . . , I(α, P 9g−9 ) .  [20,Proposition 4.4]. In [20] it was proved that the Dehn-Thurston coordinates can be expressed as homogeneous continuous functions of the intersection numbers. But the explicit functions appeared there are actually piecewise linear.
Note that Ω∩C consists of 3 arcs, a (j, l)-arc a i , a (k, i)-arc a j , and a (j, i)-arc a k , see Figure ??. For each s = i, j, l let m s be the intersection number of α with the component of Ω containing a s . By (30) and (31), for k ≥ k 0 we have Hence if m i > 0 then, for sufficiently large k we have It remains to show that for k large, h k Ω (α) does not have a component isotopic to P i . Suppose h k 1 Ω (α) has a component isotopic to P i for some k 1 . As P i has non-trivial intersection with the components of Ω which contains a j and a l , we have m j , m l > 0. From (32) it follows for large k we have ν i (h k Ω (α)) > 0, implying h k Ω (α) does not have components isotopic to P i . This completes the proof of Theorem 4.1.

4.4.
More on Theorem 4.1. We callν(α) := (µ(α), η(α)) ∈ Z 12g−12 the stable DT-coordinate of α ∈ S with respect to the coordinate datum (P, D). Recall that A is the subgroup of Z 6g−6 generated by the monoid A. Let 0 ∈ Z 3g−3 be the element having all 0's as entries, and δ i ∈ Z 3g−3 be the element having all 0's as entries except a 1 in the i-th entry.
(e) Note that ν(P j ) = ( 0, δ j ). After twisting once along Ω its twist coordinate becomes −δ j , Since h Ω (P j ) makes no bigons with P j and its intersection with D is contained inside the annulus around P j , therefore its DT coordinates change linearly, and the result follows.

Independence over the center
We formulate a criterion for independence of a collection of elements of K ζ (F ) over the center. Throughout this section we fix a finite type surface F = F g,p with negative Euler characteristic equipped with coordinate datum. One can define the coordinates ν : S → Z r , where r = 6g − 6 + 3p. The set of possible coordinates A = ν(S ) is a submonoid of Z r . Let A denote the subgroup of Z r generated by A. For a root of unity ζ we also define the submonoid A ζ and its group A ζ as in Section 3.
For a ring R we denote by R * the set of non-zero elements of R. Since K ζ (F ) is a domain, K ζ (F ) * is a monoid under multiplication. Theorem 5.1. Given a finite type surface F = F g,p with negative Euler characteristic and a fixed coordinate datum, let ζ be a root of 1. There exists a degree map is a surjective monoid homomorphism.
We will construct the map deg in later subsections. We want to mention an important corollary that we will use in the future. (a) If x 1 , . . . , x d ∈ K ζ (F ) * such that deg ζ (x 1 ), . . . , deg ζ (x d ) are pairwise distinct, then x 1 , . . . , x d are linearly independent over Z ζ (F ).
Proof. (a) Suppose z 1 , . . . z d ∈ Z ζ (F ) * . From the assumption, the elements deg(z 1 x 1 ), . . . , deg(z d x d ) are pairwise distinct in A. By Property (ii) of degree maps, the sum which is equal to D ζ (F ) by Propositions 3.12 and 3.15.

Lead term.
Since S is a C-basis of K ζ (F ), for every x ∈ K ζ (F ) * there is a unique set supp(x) ⊂ S such that x has the presentation If ≤ is a total order on S , then c α α, where α = max supp(x), is called the (≤)-lead term of x, and we can write where G < (α) is the C-span of {β ∈ S | β < α}. where |n|= n i . Let be the total order on S and A induced from the lexicographic order on Z r+1 via the embeddings The order makes A an ordered group. Define deg : K ζ (F ) * → A by Let the order ≤ on S and Z r be the one induced from the lexicographic order of Z r+1 via the embeddings Since the first component is used to define the filtrations appeared in Proposition 3.16, for n, n ∈ A ∆ , there is j such that where G < (n + n ) is a C-span of {α ∈ S | α < S(n + n )}. This holds only for triangular n, n . There is a better order on S .
The order on Z r ×Z r induced from the lexicographic order of Z 2r+2 viaκ satisfies the lemma.
From the definition, c α α is the ( )-lead term of x ∈ K ζ (F ) * if and only if c α h k Ω (α) is the (≤)-lead term of h k Ω (x) for sufficiently large k. This yields an ordering that gives us control of lead terms of all diagrams, not just triangular ones.
From here we have the following.
Lemma 5.4. Suppose x ∈ K ζ (F ) * , α k ∈ S and 0 = c k ∈ C such that for sufficiently large k we have A crucial property of the order is that its lead term is a monoid map.
Define the map deg : We show that this is in fact a degree mapping in the sense of subsection 5.1.
Proof. (a) follows from Lemma 5.5 and the fact that A × A, equipped with , is an ordered monoid.
Lemma 5.8. The monoid homomorphism deg : Proof. Let η(S ) be the Z-span of η(S ). One has to show that η(S ) ⊃ A. From the description of A in Section 3.6 we see that A = (A) 1 ⊕ Z 3g−3 , and (A) 1 is the set of all n = (n 1 , . . . , n 3g−3 ) ∈ Z 3g−3 such that whenever P i , P j , P l bound a pair of pants, n i + n j + n l is even.

5.5.
More on deg ζ . The degree map yields a characterization of central skeins and allows the exploration of the independence of diagrams.
Proposition 5.9. Suppose F = F g,p has negative Euler characteristic and a coordinate datum. Let ζ be a root of 1, with m = ord(ζ 4 ).
(b) By part (a), we have C n ∈ S ζ . From the definition of S ζ (see Section 3.3) one has C n = βγ m where β ∈ A ∂ and γ ∈ A. Since there are no peripheral elements among the C i we must have β = ∅ and C n = γ m . This proves n ∈ mN k . Moreover γ = C n/m .
If ord(ζ) = 0 mod 4, then the definition of S ζ requires γ ∈ S ev . Hence in this case C n/m ∈ S ev . 6. Dimension of K ζ (F ) over Z ζ (F ) 6.1. Formulation of result. Recall that for a finite type surface F = F g,p and a root of unity ζ with m = ord(ζ 4 ), if ord(ζ) ≡ 0 (mod 4).
Theorem 6.1. Suppose F is a finite type surface with negative Euler characteristic and ζ is a root of 1, then dim Z ζ (F ) K ζ (F ) = D ζ (F ).
Remark 6.2. Let us discuss the cases excluded by Theorem 6.1, namely the cases when the Euler characteristic of F g,p is non-nagative. There are four such cases: the sphere with zero, one or two punctures and the torus. The skein algebras of the first three are commutative so they have dimension 1 over their respective centers. For the torus in the case where n is odd this was done in [1] and the dimension is m 2 . The case when m has residue 2 on division by 4 is similar and the dimension is m 2 . Finally, when n is divisible by 4 the dimension is 4m 2 .
Corollary 6.3. Let F = F g,p be a finite type surface with negative Euler characteristic equipped with a coordinate datum. Suppose X is ã Proof. Let B ⊂ X be such that deg ζ is a bijection from B to R ζ . By Corollary 5.2, B isZ ζ (F )-linearly independent. Thus dimZ ζ (F ) X ≥ |R ζ |= D ζ (F ) = dimZ ζ (F )K ζ (F ), and hence X =K ζ (F ).

By Corollary 5.2 we have dim
To prove Theorem 6.1 we need to prove the converse inequality 6.2. Proof of Theorem 6.1, open surface case. Assume p > 0. Fix a coordinate datum (a triangulation) {e 1 , . . . , e r }.
From Theorem 3.5 it follows that It follows that Hence by Lemma 2.3, there is a positive integer u such that where the first identity follows from (5) and the second one follows from Proposition 3.12. This proves Theorem 6.1 for open surfaces.
6.3. Piecewise-rational-linear functions. A function f : For X ⊂ R k a function h : X → R l is piecewise-rational-linear if there is a piecewise-rational-linear function from R k to R l restricting to h.
It is clear that sums of piecewise-rational-linear functions are piecewiserational-linear, and that a piecewise-rational-linear function h is positively homogeneous, i.e. h(tx) = th(x) for all real t ≥ 0.
A rational convex polyhedron is the convex hull of a finite number of points in Q n . The following properties are easy consequences of the definition.
Proposition 6.4. Suppose h 1 , . . . , h l : R n → R are piecewise-rationallinear, c 1 , . . . , c l ∈ Q and (a) If Q is not bounded, then Q contains a set of the form {tx | t ∈ R ≥0 } for some non-zero x ∈ Q n . We call such set a rational ray.
(b) If Q is bounded, then Q is the union of a finite number of rational convex polyhedra.
Fix a coordinate datum (P, D) which gives the DT coordinate map ν : S → Z 6g−6 .
Here P = (P 1 , . . . , P 3g−3 ) is a pants decomposition. The set A = ν(S ) consists of all points (x 1 , . . . , if P i , P j , P l bound a pair of pants, then x i + x j + x l is even. Let Q ∞ be the set of all (x 1 , . . . , x 6g−6 ) ∈ R 6g−6 satisfying conditions (i) and (ii) above. Note that we allow points in Q ∞ to have real coordinates.
For any submonoid X of Z 6g−6 , let X be the subgroup generated by X. The set Q ∞ was introduced so that A = A ∩ Q ∞ . Lemma 6.5. (a) For any subset Q ⊂ Q ∞ one has follows from (iii) above.
Proof. (a) follows, for example, from the computations in Section 4 of [20]. (b) The more general fact: "For any simple closed curve α, the function A → R, defined by n → I(S(n), α), is piecewise-rationallinear" is well-known. It was formulated as Theorem 3 in [28] without proof. Here is a short proof based on [23]. First if α is one of P i with i ≤ 3g − 3 then the statement is obvious as I(S(n), P i ) = n i . Suppose now α is an arbitrary simple closed curve. Choose a coordinate datum (P , D ) such that α is a curve in P . By [23], the change from DT coordinates associated with (P , D ) to the one associated with (P, D) is piecewise-rational-linear. The result follows.
It follows that there is a piecewise-rational-linear h : R 6g−6 → R such that I(S(n), P i ) for all n ∈ A.
totally determine α ∈ S , the set A ∩ kQ is finite for any k ≥ 0.
Lemma 6.7. The set Q is the union of a finite number of convex polyhedra. Moreover, Q has positive volume in R 6g−6 .
Proof. Let us prove that Q is bounded. Suppose to the contrary that Q is not bounded. By Proposition 6.4(a), Q contains a rational ray, which in turns contains infinitely many points whose coordinates are even integers. Since each such point is in A, the set A ∩ Q is infinite, a contradiction. Thus Q is bounded, and by Proposition 6.4 (b), Q is the union of a finite number of convex polyhedra.

Commutative subalgebras ofK ζ (F )
In this section we study commutative subalgebras generated by collections of disjoint loops and describe their bases.
For a finite type surface F and a root of unity ζ recall thatZ ζ (F ) is the field of fractions of the center Proposition 7.1. Suppose C 1 , . . . C k are non-peripheral, non-trivial, disjoint, pairwise non-isotopic loops on a finite type surface F = F g,p of negative Euler characteristic. Let ζ be a root of unity with m = ord(ζ 4 ). Let C be theZ ζ (F )-subalgebra ofK ζ (F ) generated by C 1 , . . . , C k . For n = (n 1 , . . . , n k ) Assume that after a re-indexing {C 1 , . . . , C t } is a basis for H. The set Proof. (a) In this case (C i ) m ∈ S ζ for all each i. By Theorem 3.5, Hence B spans C as a vector space overZ ζ (F ). By Corollary 5.2, to prove that B is linearly independent it is enough to show that deg ζ (x), x ∈ B, are distinct. Assume deg ζ (C n ) = deg ζ (C n ). Let m be the k-tuple all of whose entries are m. Since deg ζ is a monoid homomorphism and deg ζ (C m ) = 0, we have deg ζ (C m−n+n ) = 0. By Proposition 5.9 (b), for each i we have m − n i + n i = 0 (mod m).
Since 0 ≤ n i , n i ≤ m − 1, the only way this can happen is if n i = n i .
There are j 1 , . . . j l ≤ t such that the simple diagram β = C i ∪ C j 1 ∪ . . . ∪ C j l is even. This implies that β m ∈ S ζ . Hence T (β m ) ∈ Z ζ (F ) by Theorem 3.5. Using the definition of T (β m ), The element in the square bracket is in C 0 . It follows that T m (C i ) ∈ C 0 , which implies that the degree of C i over C 0 is less than equal to m for each i ≥ t + 1. Hence B 1 := {C n t+1 t+1 . . . C n k k | n i < m} spans C over C 0 . Combining the spanning sets B 0 and B 1 , we get that B spans C over Z ζ (F ).
Let us show that deg ζ (x), x ∈ B, are distinct. Suppose deg ζ (C n ) = deg ζ (C n ). Let m = (m 1 , . . . , m k ) where m i = 2m for i ≤ t and m i = m for i > t. Then deg ζ (C m ) = 0. It follows that deg ζ (C m−n+n ) = 0. By Proposition 5.9(b), we have m − n + n ∈ mZ k . This forces n i = n i for i > t as in this case m i − n i + n i is sandwiched between 1 and 2m − 1. Further C (m−n+n)/m is even by Proposition 5.9 (b). Since C 1 , . . . , C t are linearly independent over Z 2 in H 1 (F , Z 2 ), for each i ≤ t, (m i − n i + n i )/m is even. As m i = 2m and 0 ≤ n i , n i < 2m, this forces n i = n i . Thus deg ζ (x), x ∈ B, are distinct, and by Corollary 5.2 the set B is linearly independent overZ ζ (F ). if ord(ζ) = 0 (mod 4) and k = t.
Moreover u is transcendental over Q.
Proof. Since C = C (C k ), the degree of C k over C is which is equal to m using the formula for dimZ ζ (F ) C and dimZ ζ (F ) C given by Proposition 7.1. In the proof of Proposition 7.1 we see that T m (C k ) = u ∈ C . Hence T m (x) − u is the minimal polynomial of C k over C .
Suppose u = T m (C k ) is algebraic over Q. Since m > 0 this implies C k is algebraic over Q. But {C i k , i ≥ 0} is a subset of S , which is a C-basis of K ζ (F ) and hence the non-trivial Q-linear combination of these elements is never 0. This shows u is transcendental over Q.

Calculation of the reduced trace
Let F be a finite type surface and ζ a root of unity. Since K ζ (F ) is finitely generated as a module over its center Z ζ (F ), it has a reduced trace. The goal of this section is to find a formula for computing it.
By Theorem 3.5 the set {T (α) | α ∈ S } is a C-basis of K ζ (F ). Therefore it is enough to calculate tr(T (α)) for each α ∈ S . Theorem 8.1. Let F be a finite type surface, ζ be a root of 1, and First consider the case when F g,p has non-negative Euler characteristic. The skein algebras of F g,p for g = 0 and p = 0, 1, 2 are commutative, so the result is trivial. For F 1,0 and n not divisible by 4 this is proved in [1]. The remaining case of K ζ (F 1,0 ) and n divisible by 4 can be proved using similar methods. Hence we will assume that F has positive Euler characteristic.
8.1. Lemma on traces. Recall that T l (x) is defined in Section 3.3.
Lemma 8.2. Suppose k 1 ⊂ k 2 are finite field extensions of a field k 0 and x 1 ∈ k 1 , x 2 ∈ k 2 . (a) If TR k 2 /k 1 (x 2 ) = 0 then TR k 2 /k 0 (x 1 x 2 ) = 0. (b) Assume the minimal polynomial of x 2 over k 1 is T l (x) − u, where u ∈ k 1 is transcendental over Q, and l ≥ 2. For 0 < s < l we have Proof. (a) A property of the trace is that for any x ∈ k 2 we have see eg [22]. With x = x 1 x 2 , we have it is enough to show that TR k 2 /k 1 (T s (x 2 )) = 0. Let t be the smallest positive integer such that l|ts. Note that t ≥ 2. Denote m = ts/l. Define u 0 = 1 and u i = T i (u) for i ≥ 1.
Claim. The minimal polynomial of y := T s (x 2 ) over k 1 is P = T t (x) − u m . Assume the claim for now. Since t ≥ 2 and T t is either even or odd polynomial, the second-highest coefficient of T t −v is 0. By Proposition 2.2(a), we have TR k 2 /k 1 (T s (x 2 )) = 0. Thus (b) follows from the claim.
Proof of the Claim. First note the P (y) = 0. In fact, we have which shows P (y) = 0. Let us show that no polynomial Q(x) of degree < t can annihilate y. Since {T i (x)} forms a basis, we can write Since T l (x) − u is the minimal polynomial of x 2 , we have Let R i (x) be the remainder obtained upon dividing T is by T l (x) − u, then we must have To finish the proof we need another lemma: is the Chebyshev polynomial of the second kind defined recursively by Proof of Lemma 8.3. One can easily check that Using T m T n = T m+n + T |m−n| , we get Suppose ds = ql + r, with 0 ≤ r < l. By Lemma 8.3, Note that (*) there is no index j ∈ [0, d − 1] such that js has remainder r when divided by l.
Consider two cases: (i) r = l − r and (ii) r = l − r.
. From (*) we see that no index j = d contributes to the term T r (x) in (61). Hence S q (u) − S q−1 (u) = 0, contradicting the fact that u is transcendental over Q.
(ii) r = l − r. There is exactly one index j ∈ [0, d − 1] such that js has remainder l − r when divided by l, which is j = (t − d). Suppose (t − d)s = q l + (l − r), then By looking at the coefficients of T r (x) and T l−r (x) in (61), we get (with c = c t−d ) Multiply the first by S q (u), the second by S q −1 (u), and sum up the two, we get S q (u)S q (u) − S q −1 (u)S q−1 (u) = 0 contradicting the fact that u is transcendental over Q. Proof. Note that if z ∈ Z ζ (F ) then tr(z) = z, and more generally Hence we assume that T (α) ∈ Z ζ (F ) and we will show tr(T (α)) = 0.
. . , C k are non-trivial loops, no two of which are isotopic, then T (α) = k i=1 T m i (C i ). If a component C i is peripheral then C i ∈ Z ζ (F ), and (66) shows that one reduces to the case when non of C i is peripheral.
From the product to sum formula (9), we have Let C be theZ ζ (F )-subalgebra ofK ζ (F ) generated by C 1 , . . . , C k and C be the subalgebra generated by C 1 , . . . , C k−1 . Consider several cases.
(ii) Choose a subset B of A * such that deg ζ | B : B → deg ζ (A * ) is bijective. By (i), B spans A. By Proposition 5.2(a), B is linearly independent overZ ζ (F ). Hence B is a basis.

Pants subalgebra decomposition
In this section we give a splitting of the localized skein algebra as a module over its center. Throughout, F = F g,p is a finite type surface with negative Euler characteristic, with or without punctures, ζ is a root of unity, and m = ord(ζ 4 ).
9.1. The Splitting. Recall that a pants decomposition of F is a collection of curves P = {C 1 , . . . , C 3g−3+p } such that each component of the complement of P in F is a planar surfaces of Euler characteristic −1.
Given a pants decomposition P of F , let C ζ (P) be theZ ζ (F )-subalgebra ofK ζ (F ) generated by the curves in P. By Proposition 7.1 and Theorem 6.1 dimZ ζ (F ) C ζ (P) = dimZ ζ (F )K ζ (F ). Hence C ζ (P) is a maximal commutative subalgebra of the division al-gebraK ζ (F ). In [8] the first two authors constructed a splitting of K ζ (F ), when F has at least one puncture and ord(ζ) = 0 (mod 4).
Here we prove that this decomposition works for all surfaces and all roots of unity.
Theorem 9.1. Let F be a finite type surface of negative Euler characteristic. There exist two pants decompositions P and Q of F such that for any root of unity ζ theZ ζ (F )-linear map (69) ψ : C ζ (P) ⊗Z ζ (F ) C ζ (Q) →K ζ (F ), ψ(x ⊗ y) → xy, is aZ ζ (F )-linear isomorphism of vector spaces.

Pants decompositions. Let C[∂]
be the C-subalgebra of K ζ (F ) generated by peripheral loops. For a pants decomposition P let C[P] be the C-subalgebra of K ζ (F ) generated by loops in P. For a set U ⊂ Z r let U be the Z-span of U . Lemma 9.2. Suppose F g,p is a finite type surface with negative Euler characteristic. There exist a coordinate datum for F g,p and two pants decompositions P and Q such that   and two pants decompositions P, Q such that the Z-span of ν(P), ν(Q) and A ∂ has index 2 4g−5+2p in Z 6g−6+3p . In other words, if C 1 , . . . , C 6g−6+3p it the set of curves consisting of components of P, Q and the p peripheral loops, then Pants decompositions for a surface of genus 3 and 1 puncture are shown in Figure 4. Additional curves needed for more than one puncture are shown in Figure 5 for the case of four punctures. For a detailed description of those curves see [8].
By Lemma 3.9, the index of A in Z 6g−6+3p is also 2 4g−5+2p . Hence A is equal to the Z-span of ν(P), ν(Q) and A ∂ . To prove Lemma 9.2 consider cases when p > 0 and p = 0. Thus the right hand side of (70) is the Z-span of A ∂ , ν(P), ν(Q), which by Lemma 9.3 is equal to A.
(b) Suppose p = 0. Let Σ be a compact planar surface with g + 1 boundary components, thenΣ = Σ \ ∂Σ is a finite type surface of type F 0,g . Let (a i ) 3g−3 i=1 , Q 1 , Q 2 be respectively the ideal triangulation, the pants decompositions P and Q of Lemma 9.3 for the surfaceΣ. Let D ⊂Σ be the trivalent graph dual to the system (a i ) 3g−3 i=1 . We can assume that the topological closureā i of a i in Σ is a proper embedding of [0, 1] into Σ and that the 6g − 6 endpoints of all 3g − 3 arcsā i are distinct. Take another copy Σ of Σ and assume that ϕ : Σ → Σ is a diffeomorphism. Let F be the result of gluing Σ with Σ along the boundary by the identification x ≡ ϕ(x) for every x ∈ ∂Σ. Let P = (P 1 , . . . , P 3g−3 ) where P i =ā i ∪ ϕ(ā i ) and Q = (Q 1 , . . . , Q 3g−3 ) be the collection of components of Q 1 , ∂Σ, and ϕ(Q 2 ), in some order. We claim that the coordinate datum (P, D) and the two pants decompositions P, Q satisfy (70). Note that we can take Σ = N (D), and Ω = ∂Σ.
Since the left hand side is a subgroup of the right hand side, we only need to show that they have the same index in Z 6g−6 . Let 0 ∈ Z 3g−3 be the 0 vector and δ i ∈ Z 3g−3 is the vector whose entries are all 0 except for the i-th one which is 1. By Proposition 4.7(e), we have η(P i ) = ( 0, −δ i ). Hence η(P) = { 0} ⊕ Z 3g−3 . It follows that index of η(Q) + η(P) in Z 6g−6 is equal to the index of η(Q) 1 in Z 3g−3 , where η(Q) 1 is the Z-span of (3g − 3)-tuples which are the first 3g − 3 coordinates of η(Q i ), i = 1, . . . , 3g − 3. Since each Q i has 0 intersection with Ω, by Proposition 4.7 one has η(Q i ) = ν(Q i ). The first 3g − 3 coordinates of ν(Q i ) are given by I(Q i , P j ) 3g−3 j=1 . Hence the index of η(Q) 1 is Z 3g−3 in det I(P i , Q j ) 3g−3 i,j=1 , which is equal to 2 2g−3 by (72).