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Relations between Witten-reshetikhin-turaev and non semi-simple sl(2) 3-manifold invariants

The Witten-Reshetikhin-Turaev invariants extend the Jones polynomials of links in S^3 to invariants of links in 3-manifolds. Similarly, in a preceding paper, the authors constructed two 3-manifold invariants N_r and N^0_r which extend the Akutsu-Deguchi-Ohtsuki invariant of links in S^3 colored by complex numbers to links in arbitrary manifolds. All these invariants are based on representation theory of the quantum group Uqsl2, where the definition of the invariants N_r and N^0_r uses a non-standard category of Uqsl2-modules which is not semi-simple. In this paper we study the second invariant N^0_r and consider its relationship with the WRT invariants. In particular, we show that the ADO invariant of a knot in S^3 is a meromorphic function of its color and we provide a strong relation between its residues and the colored Jones polynomials of the knot. Then we conjecture a similar relation between N^0_r and a WRT invariant. We prove this conjecture when the 3-manifold M is not a rational homology sphere and when M is a rational homology sphere obtained by surgery on a knot in S^3 or when M is a connected sum of such manifolds.


Introduction
In [14], Witten proposes a program to construct a topological invariant of 3-manifolds from the viewpoint of quantum mathematical physics. Reshetikhin and Turaev [12] give rigorous construction of these invariants which have become known as quantum invariants of 3-manifolds. These invariants are defined via surgery presentations of a 3-manifold. The best known example of these invariants is a weighted sum of colored Jones polynomials. The invariants of Reshetikhin and Turaev generalize to the setting of modular categories. Some of the common obstructions to applying this construction to any ribbon tensor category D include the following facts: (1) the simple objects may have zero "quantum dimension", (2) there might be infinitely many isomorphism classes of simple objects in D and (3) D might be non-semi-simple. In [5] the authors derive a general categorical setting where these obstructions can be overcome. In particular, they show that the category C of nilpotent representations of a generalized version of quantized sl(2) at a primitive r th ordered root of unity gives rise to two invariants: N r and N 0 r . In this paper we investigate the invariant N 0 r . Let C be the category mentioned above and defined in Subsection 1.2. This category has complex family of weight modules divided into typical and atypical modules. Here all the atypical modules have integral weights.
Let F be the usual Reshetikhin-Turaev invariant of links in S 3 arising from C . The invariant F has the following properties: • If L is a link whose components are all colored by simple modules of C with integral weights then F is determined by the Kauffman bracket and so is a version of the colored Jones polynomial. • If L is a link with a component colored by a typical module then F (L) = 0.
The first author's research was supported by French ANR project ANR-08-JCJC-0114-01. Research of the second author was partially supported by NSF grants DMS-1007197 and DMS-1308196.
In [6], the second two authors and Turaev give an extension of F to links colored with modules in C with non-integral weights (see also [7]). In particular, we construct an invariant F defined on links with at least one component colored by a typical module. F is a generalization of the links invariants defined by Akutsu, Deguchi and Ohtsuki in [1]. We have the following relation where L 1 is in the domain of F and L 2 is any C -colored link. From this relation it follows that F recovers F : if L is any C -colored link then where o is an unknot colored by any typical module. Thus, F is a kind of extension of the colored Jones polynomial to complex colors. Furthermore, as we will show in Corollary 15, the invariant F (K α ) of a knot K ⊂ S 3 colored by a typical module of weight α ∈ C is a meromorphic function of α whose residues at the integers are proportional to the colored Jones polynomials of K evaluated at q = exp( iπ r ). This relation allows us to re-prove the well known Symmetry Principle (see [8]) for the colored Jones polynomials of K using a mainly graphical argument detailed in Corollary 13 (see Remark 14).
In [5], the authors layout a relationship between N r and N 0 r analogous to that outlined above between F and F ; we will now briefly recall this relation. The invariants N r and N 0 r are WRT-type 3-manifold invariant which are certain weighted sums of F (L) where L is a surgery presentation of M . These invariants are topological invariants of triples (a closed oriented 3-manifold M , a link T in M , an element ω in H 1 (M \ T ; C/2Z)). Here for N r the triples must satisfy some requirements of "typicality" as in the case of F . The invariant N 0 r is zero unless ω is in the image of the natural map H 1 (M ; Z/2Z) → H 1 (M ; C/2Z) induced by the universal coefficient theorem. (Compare this with the above statement that F (L) is zero if at least one component of L is colored by an atypical module.) Finally, N r recovers N 0 r (compare with Equation (1)): where (M , T , ω ) is a triple where N r does not vanish (for further details on the notion of connected sum see [5]). Since F is essentially the colored Jones polynomial, the above analogy leads us to the question: Is N 0 r related to the WRT-invariant? The purpose of this paper is to answer this question positively for certain types of triples (M, T, ω). To formulate this properly we must define the WRT-invariant of a triple (M, T, ω). Kirby and Melvin [8] and Blanchet [2] consider a WRT-type invariants of (M, ω) where ω ∈ H 1 (M, Z/2Z). In Theorem 8 we give a slight generalization of their invariants to triples of the form (M, T, ω) where T is a link in M and ω ∈ H 1 (M \ T, Z/2Z). We denote this invariant by WRT r (M, T, ω). The question above can be formulated in the following conjecture.
If G is a finite abelian group let ord(G) be the order of G, i.e. the number of elements in the set underlying G. If G is an infinite abelian group set ord(G) = 0.  Remark that if an abelian group G has a square presentation matrix A ∈ M n (Z) then ord(G) = | det(A)|. In particular, if a 3-manifold is obtained by surgery on a link in S 3 whose linking matrix is A then A is a presentation matrix for H 1 (M ; Z) thus ord(H 1 (M ; Z)) = | det(A)|.
It should be noticed that the invariant N r does not reduce to N 0 r . For example, the invariant W RT r is trivial for r = 2 and N 0 r should only depends of H 1 (M, Z). But for r = 2, the invariant N r is related to the Reidemeister torsion. This is shown in [3] where the two invariants N r and N 0 r are extended to manifolds with boundary using the setting of topological quantum field theory.
In Sections 3 and 4 we prove this conjecture in the following two cases: (1) when M is an empty rational homology sphere obtained by surgery on a knot in S 3 (or more in general a connected sum of manifolds of this type) and (2) when the first Betti number of M is greater than zero.
1. Preliminaries 1.1. Notation. All manifolds in the present paper are oriented, connected and compact unless explicitly stated. All tangles in this paper will be framed and oriented. Given a set Y , a graph is said to be Y -colored if it is equipped with a map from the set of its edges to Y .
Let r be an integer greater or equal to 2 and let q = e iπ/r . For x ∈ C, we use the notation q x for e xiπ/r and set {x} = q x − q −x . Let X r = Z \ rZ ⊂ C and define the modified dimension d : C \ X r → C by Finally, let 1.2. A quantization of sl(2) and some of its modules. Here we give a slightly generalized version of quantum sl (2). Let U H q sl(2) be the C-algebra given by generators E, F, K, K −1 , H and relations: The algebra U H q sl(2) is a Hopf algebra where the coproduct and counit are defined by Define U H q sl(2) to be the Hopf algebra U H q sl(2) modulo the relations E r = F r = 0. Let V be a finite dimensional U H q sl(2)-module. An eigenvalue λ ∈ C of the operator H : V → V is called a weight of V and the associated eigenspace is called a weight space. We call V a weight module if V splits as a direct sum of weight spaces and q H = K as operators on V . Let C be the category of finite dimensional weight U H q sl(2)-modules. The category C is a ribbon Ab-category, see [6,9,10].
We will now recall the definition of the duality morphisms and the braiding of the category C . Let V and W be objects of C . Let {v i } be a basis of V and {v * i } be a dual basis of V * = Hom C (V, C). Then are duality morphisms of C . In [10] Ohtsuki defines an R-matrix operator defined on V ⊗ W by where q H⊗H/2 is the operator given by for weight vectors v and v of weights of λ and λ . The braiding c V,W : For each n ∈ {0, . . . , r − 1} let S n be the usual (n + 1)-dimensional irreducible highest weight U {1} . Since q is a root of unity and F r = 0 we can consider a larger class of finite dimensional highest weight modules: for each α ∈ C we let V α be the r-dimensional highest weight U H q sl(2)-module of highest weight α + r − 1. The modules V α has a basis {v 0 , . . . , v r−1 } whose action is given by All the modules V α have a vanishing quantum dimensions. They are divided into typical and atypical modules: Atypical modules: If k ∈ X r = Z\rZ ⊂ C then V k is indecomposable but not irreducible, however it is still absolutely irreducible (i.e. End C (V k ) = C Id V k since any endomorphism must map the highest weight vector v 0 to a multiple of itself). In particular, if k ∈ {0, . . . , r − 1} then the assignment sending the highest weight vector s 0 of S r−1−k to the vector v k of V k determines an injective homomorphism ı : . . , 0} then the assignment sending the highest weight vector v 0 of V j to the highest weight vector s 0 of S r−1+j induces a surjective homomorphism π : V j → S r−1+j . Typical modules: If α ∈ C \ X r then V α is irreducible and so absolutely irreducible. We call such modules typical. Let A be the set of typical modules. For g ∈ C/2Z, define C g as the full sub-category of weight modules with weights congruent to g mod 2. Then it is easy to see that {C g } g∈C/2Z is a C/2Z-grading in C (see [5]).
1.3. The link invariants F and F . The well-known Reshetikhin-Turaev construction defines a C-linear functor F from the category of C -colored ribbon graphs with coupons to C . When L is a C -colored framed link then F (L) can be identified with a complex number. When L is a framed link whose components are all colored by S n then F (L) is the Kauffman bracket with variable specialization A = q 1/2 = e iπ/2r , so it is a version of the colored Jones polynomial specialized at the root of unity q = e iπ/r (for details see Section 1.4).
Vanishing quantum dimensions make the functor F trivial on any closed C -colored ribbon graph that have at least one edge colored by a typical module. In [6], the definition of F is extended to a non-trivial map F defined on closed C -colored ribbon graphs with at least one edge colored by a typical module. We will now recall how one can compute F . Let T W be any C -colored (1-1)-ribbon graphs with both ends colored by the same element W of C . If W is absolutely irreducible then F (T W ) is an endomorphism of W that is determined by a scalar T W : F (T W ) = T W Id W . Let L be a closed C -colored ribbon graph with an edge colored by a typical module V α . By cutting this edge we obtain a C -colored (1-1)-tangle T Vα whose open edges are colored by V α . Then we define F (L) = d(V α ) T Vα . It can be shown that F (L) does not depend on the choice of the edge to be cut and yields a well defined invariant of L (see [6]).
We will use the following proposition latter.
Proof. In this proof we use the language of coupons, for more details see [13]. In particular, a morphism f : V → W can be represented by a coupon c(f ), which is a box with arrows: From the discussion above about atypical modules we have the injection ı : S r−1−k → V k and surjection π : V j → S r−1+j , for k ∈ {0, . . . , r − 1} and j ∈ {1 − r, . . . , 0}. Thus, as explained in the previous paragraph we can consider the ribbon graphs T V k • c(ı) and c(ı) • T S r−1−k . Since the category of C -colored ribbon graphs is a ribbon category we have that T V k • c(ı) and c(ı) • T S r−1−k are equal as ribbon graphs, so their images are equal under F . Combining this with the fact that F (T V k ) and F (T S r−1−k ) are scalars endomorphisms we have 1.4. Comparison with the Jones polynomials. In this paper, by the colored Jones polynomial, we mean the Kauffman bracket version which is an invariant of framed oriented links independent of their orientation. Let L = L 1 · · · L k ⊂ S 3 be a framed link and J(L) ∈ C[q ± 1 2 ] be its Jones polynomial determined by the following skein relations: More generally, if each L i is colored by an integer n i ≥ 0, then roughly speaking, one defines the n th -colored Jones polynomial J n (L) as a linear combination of Jones polynomials of links obtained by taking parallels of each component of L at most n i times. More precisely, one identifies the tubular neighborhood of each component L i with the product S 1 × [−1, 1] × [−1, 1] (using the framing of L i and an arbitrary orientation), and defines links and adopts the notation that L k i · L h i = L k+h i . Then one recursively defines a linear combination of links parallel to L i as follows: is defined as the linear combination of the Jones polynomials of the links obtained by replacing L i with T ni (L i ). Clearly the above defined Jones polynomial corresponds to the case when n i = 1 for all i. The following holds: Proposition 3. Let L = L 1 · · · L k ⊂ S 3 be a framed oriented link and let n = (n 1 , ..., n k ) be a tuple of integers all greater or equal to 0. Let L n be the link L such that L i is colored by n i , for all i. Similarly, let L S be the link L such that L i is colored by S ni where S ni is the simple module defined in Subsection 1.2. Then Proof. First, assume that n i = 1, for all i. In this case, we will prove that the relations of Equations (6) and (7) hold. We start by recalling that S 1 is spanned by two vectors s 0 , The second relation of Equation (7) is a consequence of the formula for the quantum dimension qdim(S 1 ) = −q − q −1 given above. The first relation follows from the fact that the inverse of the twist on S 1 is given by the action of [5]). To see that Equation (6) holds, recall the braiding c S1,S1 is defined by where R is the R-matrix and τ is the permutation x ⊗ y → y ⊗ x. Since E 2 and F 2 act by zero on S 1 we have c S1,S1 and c −1 S1,S1 are determined by τ • q •τ , respectively. Thus, Equation (6) follows from the following direct computations: Finally, to prove the statement in general it is sufficient to remark that the standard tensor decomposition of S ⊗n 1 as a sum of copies of S i with i ≤ n is still valid for n < r in C . To prove this it is sufficient to remark that if 2 ≤ n < r then S n−1 ⊗ S 1 S n ⊕ S n−2 and arguing by induction. Hence the formula (8) expressing T n (L) translates this decomposition algebraically expressing F (L) (with L-colored by n) as a linear combination of invariants of cables of L whose components are all colored by S 1 . Thus, the theorem follows.
1.5. The 3-manifold invariants N 0 r and WRT. In this subsection, we fix an integer r ≥ 2 with r / ∈ 4Z. We start by recalling some definitions given in [5]. Let M be a compact connected oriented 3-manifold, T a C -colored ribbon graph in M and ω ∈ H 1 (M \ T, C/2Z). Let L be an oriented framed link in S 3 which represents a surgery presentation of M . The map g ω defined on the set of edges of L ∪ T with values in C/2Z defined by g ω (e i ) = ω(m i ), where m i is a meridian of e i , is called the C/2Z-coloring of L ∪ T induced by ω. (1) We say that (M, T, ω) is a compatible triple if for each edge e of T its coloring is in C gω(me) where m e is a meridian of e. Recall the set H r = {1 − r, 3 − r, . . . , r − 1} defined in (3). For α ∈ C \ Z we define the Kirby color Ω α as the formal linear combination If α is the image of α in C/2Z we say that Ω α has degree α. We can "color" a knot K with a Kirby color Ω α : let K(Ω α ) be the formal linear combination of knots k∈Hr d(α + k)K α+k where K α+k is the knot K colored with V α+k . If α ∈ C/2Z \ Z/2Z, by Ω α , we mean any Kirby color of degree α. Let ∆ − and ∆ + be the scalars given by: Next we recall the main theorems of [5]. For α ∈ C \ X r , let u α be the unknot in S 3 colored by V α . Let ω α be the unique element of H 1 (S 3 \ u α ; C/2Z) such that (S 3 , u α , ω α ) is a compatible triple.
where Σ a = r−2 2 n=0 q 2(n 2 +an) is part of a quadratic Gauss sum. These terms can be computed using standard results on quadratic Gauss sum.
Kirby and Melvin [8] and Blanchet [2] consider invariants of (M, ∅, ω) where ω ∈ H 1 (M, Z/2Z). The following theorem is a slight generalization of these invariants (here we use the conventions of this paper and not the conventions of [8,2]). Proof. In [8], for T = ∅ and r even, this invariant is considered in a slightly different form. Also, in Remark II.4.3 of [2] for T = ∅, the existence of this invariant is discussed. Indeed, the Reshetikhin-Turaev functor applied on graphs colored by the module S 1 ∈ C 1 satisfies the Kauffman skein relation for A = q 1 2 = exp( iπ 2r ). It follows that if L ⊂ S 3 is a framed link whose components are colored by elements of {S 0 , . . . , S r−2 } then F (L) is the meta-bracket (see [2,4]) evaluated at the element corresponding to the coloring of L at A = q For a complete proof of the theorem, one can also apply Theorem 3.7 of [5] to the modular category obtained as the quotient of the subcategory of C generated by S 1 by its ideal of projective modules. Indeed, this category is obviously a Z/2Z-modular category relative to ∅ with modified dimension qdim and trivial periodicity group.
In particular, when ω = 0 one gets an invariant of manifolds also known as the SO (3)

Relations between F and the colored Jones polynomial
Recall the r-dimensional modules V α , α ∈ C, given in Subsection 1.2. Using the basis given in Equation (5) and its dual basis we identify V α and V * α with C r . With these identifications we can identify certain Hom-spaces with spaces of matrices. For example, we can make the following identifications: End C (V α ) = M at r×r (C) and Hom(C, V α ⊗ V * α ) = M at 1×r 2 (C). We say a function g : C → C is a Laurent polynomial in q α if there exists a Laurent polynomial f ∈ C[x, x −1 ] such that g(α) = f (q α ). The action of the basis given in Equation (5) implies that all the entries in the matrices ρ Vα (E), ρ Vα (F ), ρ Vα (H) and ρ Vα (K) are Laurent polynomial in q α .
Lemma 11. All the entries in the image of the maps are Laurent polynomials in q α . Also, for each entry f ij in the image of the map f : C × C → M at r 2 ×r 2 (C), (α, β) → q −αβ/2 q −(r−1)(α+β)/2 c Vα,V β there exists a two variable Laurent polynomial g ij (x, y) such that f ij (α, β) = g ij (q α , q β ).
Proof. The first statement follows from the formulas for b, d, b and d given in Subsection 1.2. For example, the entry in image of g d corresponding to . The second statement follows from the form of the R-matrix given in Equation (4). In particular, if v i and v j are any basis vectors of V α and V β , respectively then where c is an integer which does not depend on α or β. Also, Since the coefficients in the last two equalities are Laurent polynomial in q α and q β , the desired result about the function f follows.
The above lemma has the following corollaries.
Corollary 12. Let T (Vα 1 ,...,Vα n ) be a (1-1)-tangle with n components whose i th component is colored by V αi , α i ∈ C. Then the function g T : C n → C given by (α 1 , . . . , α n ) → T (Vα 1 ,...,Vα n ) is a holomorphic function in C n . In particular g T is continuous. so it is enough to consider F (T (Vα 1 ,...,Vα n ) ). The value of F (T (Vα 1 ,...,Vα n ) ) is computed by decomposing a projection of T (Vα 1 ,...,Vα n ) into building blocks made of cups, caps, vertical edges and crossings. Then the building blocks are associated with the duality morphisms, identity and the positive and negative braidings, respectively. These morphisms are tensored and composed according to the projection of T (Vα 1 ,...,Vα n ) . Lemma 11 implies that the contributions from a duality morphism corresponding to a cup or cap on the i th component is a Laurent polynomial in q αi . Lemma 11 also implies the all contributions of a crossing between the i th and j th components are Laurent polynomials in q αi and q αj times a factors of q −αiαj /2 q −(r−1)(αi+αj )/2 . Thus, the map g T (α 1 , . . . , α n ) = T (Vα 1 ,...,Vα n ) is a Laurent polynomials in the variables q α1 , . . . , q αn times a integral powers of q −αiαj /2 q −(r−1)(αi+αj )/2 and so g T is holomorphic.
Vα be a (1-1)-tangle with zero framing whose closure is K 0 Vα . Then there exists a Laurent polynomial K(X) ∈ C[X, X −1 ] such that T 0 Vα = K(q α ) and Proof. As in the proof of Corollary 12 the function g T (α) = T 0 Vα is a Laurent polynomial in q α times an integral power of q α 2 /2 . From the form of the map g c in Lemma 11 the integral power of q α 2 /2 is equal to the number of positive crossing minus the number of negative crossing in the projection of T 0 Vα . Since the framing of K α is zero this power is zero. Thus, g T (α) is a Laurent polynomials in q α and so there exists a K(X) ∈ C[X, X −1 ] such that T 0 α = K(q α ). Now we can use the duality and the braiding to compute the value of the twist: Then Equation (10) follows from the above discussion and the definition of F : Next we will show that K(q α+r ) = K(q α ). Consider the one dimensional space τ = C with the U H q sl(2)-module structure given by Ev = F v = 0, Hv = rv for any v ∈ τ . The quantum dimension of τ is (−1) r+1 . From the form of the R-matrix we have: Hence for a 0-framed knot K colored with τ , one has F (K) = F (unknot) = (−1) r+1 . Let T 0 be the zero framed tangle underlying T 0 Vα . Let T 0 τ be T 0 colored with τ . Since T 0 τ has zero framing then T 0 τ = 1. Now F (T 0 Vα+r ) is equal to the endomorphism associated to T 0 labeled with V α ⊗ τ or equivalently the 2-cabling of T 0 where the two components are labeled by V α and τ , respectively. We can use the third equality in Equation (11) to unlink the component labeled with τ from the component labeled with V α . Therefore, since T 0 Vα+r has zero framing we have Finally, Equation (10) and the above formulas for θ α and d(α) imply: Remark 14. Corollary 13 with Proposition 2 imply the well known symmetry principle relating the colored Jones polynomial associated to S k−1 with the one associated to S r−1−k for k ∈ {1, . . . , r − 2}.
Corollary 15. Let K be a knot and let K Vα be K colored by V α . The function g K : C \ X r → C defined by α → F (K Vα ) is a meromorphic function on the whole plane C. Moreover, the residue at each pole is determined by the colored Jones polynomial.
Proof. Recall that where Vα is the (1-1)-tangle obtained from cutting K Vα . From Corollary 12 it follows that α → (−1) r−1 r−1 j=1 {j} T Vα is a holomorphic function in the entire plane C. Also, it is clear that α → r−1 j=1 {α + r − j} is a holomorphic function in the entire plane C which is zero when α ∈ Z \ rZ. Therefore, the quotient of these two functions is a meromorphic function whose set of poles is Z \ rZ.
All of these poles are simple and so the residue can be computed as follows. Let n ∈ Z \ rZ. The residue at n of the 2r-periodic meromorphic function d is given by xr sin π(n+x) r sin (π(n + x)) = (−1) r−1+n r π sin nπ r .
So the residue of g K at n is equal to Res(g K , n) = Res(d, n) T Vn = (−1) r−1+n r π sin nπ r T Vn .
To finish the proof we will show that the above formula of Res(g K , n) can be rewritten in terms of the colored Jones polynomial. To do this we have two cases. First, suppose n = k + 2mr with k ∈ {1, . . . , r − 1} and m ∈ Z. By Corollary 13 and Proposition 2 we have Combining the fact that qdim( and Proposition 3 we have Thus, Similarly, if n = k + 2mr with k ∈ {1 − r, . . . , −1} and m ∈ Z then one can show that Res(g K , n) = − r π sin π r J r−1+k (K)| q=e iπ/r .

Surgery on a knot in the 3-sphere S 3
In this section we prove Conjecture 1 when M is an empty closed manifold obtained by surgery on a non-zero framed knot in S 3 : Theorem 16. Suppose that K is a knot in S 3 with non-zero framing f . Let M be the manifold obtained by surgery on the knot K and ω ∈ H 1 (M, Z/2Z). Then Remark 18. The three invariants W RT , N 0 r and M → ord(H 1 (M, Z)) are multiplicative with respect to the connected sum of 3-manifolds. Hence Theorem 16 implies that Conjecture 1 is also true for a connected sum of manifolds, each obtained by surgery on a knot in S 3 .
The rest of this section is devoted to the proof of Theorem 16.
Proof of Theorem 16. First we improve the results of [5, Section 2.4] and derive a formula for N 0 r (M, ∅, ω). We still denote by ω the integer in {0, 1} whose class modulo 2 is the value g ω (K) of the cohomology class on the meridian of K and let e ∈ {0, 1} be such that e = r − 1 + ω ∈ Z/2Z.
For α ∈ C \ X r , recall the function P(α) = k∈Hr F (K V α+k ) of [5, Section 2.4] (as above, K V means K colored by V ). The function P is continuous and so can be naturally extended to all of C. Indeed, let DK (Vα,V β ) be the 2-cable of K whose components are colored with V α and V β such that α or β is in C \ X r . From Lemma 11 we have that the map (α, β) → q − f 2 (α 2 +β 2 +2αβ) F (DK (Vα,V β ) ) is a rational function in 1 q rα −q −rα C[q ±α , q ±β ]∩ 1 q rβ −q −rβ C[q ±α , q ±β ] (also see the proof of Corollary 12). Thus, this function is a Laurent polynomial in C[q ±α , q ±β ]. In addition, if α + β ∈ C \ X r then F (DK (Vα,V β ) ) can be computed by coloring K with V α ⊗ V β k∈Hr V α+β+k . Combining the statements of this paragraph we have is a continuous function of (α, β), which we extend to all of C × C. Next we give a formula for N 0 r in terms of P. By sliding the unknot o α on K we obtain a computable presentation of (M, ∅, ω)#(S 3 , o α , ω α ) as in Theorem 6. This produces the link DK (Ωe−α, Vα) where Ω e−α = h∈Hr d(e − α + h)V e−α+h is a Kirby color of degree ω − α. By definition of N 0 r , Finally, as N 0 r (M, ∅, ω) does not depend on α we have Next we use the last formula and the continuity of P to write a multiple of N 0 r . In particular, let S be the following limit: In this sum, for fixed n the only part of the interior sum which varies is q for k, ∈ H r with k + = 2n. Here the possible values of are integers from max(1 − r, 1 − r + 2n) to min(r − 1, r − 1 + 2n) and so the sum of q , over these values, is equal to q n {r − |n|} Therefore, we have the following expression for S: where c = rf {1}∆ sign(f ) and ϕ e (k) = (q k − q e ){k} T V k . From Corollary 13, ϕ e is 2r-periodic. Furthermore, Proposition 2 implies that for k ∈ {1, . . . , r − 1}, one has ϕ e (k) + ϕ e (−k) = (q k − q e − q −k + q e ){k} T S r−1−k = {k} 2 T S r−1−k .