Refined and Generalized $\hat{Z}$ Invariants for Plumbed 3-Manifolds

We introduce a two-variable refinement $\hat{Z}_a(q,t)$ of plumbed 3-manifold invariants $\hat{Z}_a(q)$, which were previously defined for weakly negative definite plumbed 3-manifolds. We also provide a number of explicit examples in which we argue the recovering process to obtain $\hat{Z}_a(q)$ from $\hat{Z}_a(q,t)$ by taking a limit $ t\rightarrow 1 $. For plumbed 3-manifolds with two high-valency vertices, we analytically compute the limit by using the explicit integer solutions of quadratic Diophantine equations in two variables. Based on numerical computations of the recovered $\hat{Z}_a(q)$ for plumbings with two high-valency vertices, we propose a conjecture that the recovered $\hat{Z}_a(q)$, if exists, is an invariant for all tree plumbed 3-manifolds. Finally, we provide a formula of the $\hat{Z}_a(q,t)$ for the connected sum of plumbed 3-manifolds in terms of those for the components.


Introduction
It is well-known that Dehn surgery establishes a pivotal relation between links in R 3 and closed oriented 3-manifolds [10,16]. This gives rise to an important relation between invariants of links and 3-manifolds. A typical example is the colored Jones polynomial for framed oriented links and the Witten-Reshetikhin-Turaev (WRT) invariant for closed oriented 3-manifolds [14,17].
Khovanov homology [9] is another well-known invariant for knots and links. Furthermore, it is a categorification of the Jones polynomial, that is, the Euler characteristic of Khovanov homology is the Jones polynomial. Therefore, it is natural to ask if there is an invariant of 3-manifolds not only as the counterpart of Khovanov homology but also as a categorification of the WRT invariants. This is one of the major open questions in quantum topology.
Some progress in this direction has been made in [7,8], where a physical definition of certain new invariants of 3-manifolds, often referred to as homological blocks, Gukov-Pei-Putrov-Vafa (GPPV) invariant, or asẐ invariant, denoted byẐ a (q), was formulated by the study of 3d N = 2 supersymmetric theory obtained by compactification of 6d N = (2, 0) theory on a 3manifold. The 3d N = 2 theory depends on the choice of a Lie group G and the case G = SU (2) is mostly studied in [6,7,8] since it corresponds to the Jones polynomial. In this paper, we will also assume G = SU (2). The cases for other gauge groups, especially for G = SU(N ) are studied in [13]. The invariantẐ a (q) is a power series in q with integer coefficients and a denotes a Spin c structure of the 3-manifold. Moreover, a general relation betweenẐ invariants and WRT invariants was conjectured. The conjecture says that the limit of a certain linear combination ofẐ a (q) over all possible a as q goes to a root of unity would be equal to the WRT invariant. A rigorous mathematical definition of the invariants for general 3-manifolds is yet to be found.
A concrete mathematical formula ofẐ a (q) and its invariance are given in [6] and [7] only for weakly negative definite plumbed 3-manifolds.
The purpose of this paper is to introduce a refined and generalized version of these q-series, which is defined for a much larger class of plumbed 3-manifolds. To do this, we first construct a refinement, the (q, t)-series, by introducing a new regulator variable t, which is also an invariant of reduced plumbed 3-manifolds. This (q, t)-series has a nice property that for weakly negative definite plumbings the evaluation at t = 1 is equal to the q-seriesẐ a (q). However, for weakly positive definite plumbings we can switch the (q, t)-series into the q-series in [6] by taking t = 1 and transforming the expansion in q −1 into an expansion in q as in [3]. Also, the (q, t)-series for negative definite plumbings is reminiscent of the 2-variable series introduced in [1], which is obtained by combining the lattice cohomology and the q-series.
Another property of the (q, t)-series is that the exponents of t are intimately related to nonnegative integer solutions of quadratic Diophantine equations. Therefore, we can recover the q-series from the (q, t)-series by computing the limit t → 1 even for a certain class of strongly indefinite plumbings by solving the corresponding quadratic Diophantine equations. In cases when the limit is finite, the recovered q-series is conjecturally also an invariant of plumed 3manifolds.
The organization of the paper is as follows. In Section 2, we review some known facts about the plumbed 3-manifolds and the q-seriesẐ a (q). In Section 3, we provide the formula for theẐ a (q, t) invariants of reduced plumbed 3-manifolds, and prove that they are independent of the plumbing presentation. In Section 4, we give an analytical study on the recovering process for plumbings with two high-valency vertices by using the solutions of quadratic Diophantine equations in two variables. Also, we propose a conjecture that the recovered q-series is an invariant of arbitrary plumbed 3-manifolds. Finally, in Section 5, we provide a relation between the (q, t)-series of the disjoint union of two plumbing graphs and those of component plumbings. The Appendix contains a brief review on the solutions of quadratic Diophantine equations in two variables, the results of which are used in the main text.

Plumbed 3-manifolds and q-series
In this section we review some known facts about plumbed 3-manifolds and their invariants, and we set up notational conventions.

Plumbing graphs
A plumbing graph is a finite weighted graph Γ, that is, a graph consisting of a finite number of vertices and edges together with the data of integer weights associated to vertices. In Section 2-4, we will assume that Γ is a tree, and we will consider disconnected plumbings in Section 5.
Let V be the set of vertices of Γ. For each vertex v ∈ V , m v denotes the weight of the vertex v, and the degree deg(v) describes the number of edges connected to the vertex. Let s = |V | be the cardinality of the set V . Then we define the symmetric s × s matrix M = M (Γ), called linking matrix of Γ, by if v 1 , v 2 are connected by an edge, 0, otherwise.
v i ∈ V.
From Γ, we can construct plumbed 3-manifolds in the following way: we first obtain the framed link L(Γ) in S 3 = ∂B 4 by taking an unknot with framing m v for each vertex v and making these unknots forming Hopf links whenever corresponding vertices are connected by an edge, Neumann moves on plumbing graphs that realize homeomorphic 3-manifolds.
e.g., see Figure 1. By attaching two-handles to B 4 along L(Γ), we get the 4-manifold, denoted by W (Γ). It can be also obtained by plumbing disk bundles over S 2 with Euler numbers m v . Then its boundary Y = Y (Γ) = ∂W (Γ) is the closed and oriented 3-manifold obtained by Dehn surgery from L(Γ) whose first homology is Two different plumbing graphs can represent the same homeomorphism class of Y (Γ). This happens if and only if they are related by a finite sequence of Neumann moves [11] depicted in Figure 2.
For the purposes of this paper, we establish some terminology of plumbings that can be helpful for understanding the structures of plumbing graphs. Given a graph Γ, we divide the set V of vertices into three disjoint subsets V 1 , V 2 and V h with respect to the degree of vertices as following: Then we name the elements of V 1 , V 2 and V h by valency one, valency two and high-valency vertices, respectively. Assume that Γ has at least one high-valency vertex. Choose a high-valency vertex w ∈ V h among them. Then w has arms as many as its degree deg(w), where an arm means a part of graph, starting from a given high-valency vertex, possibly going through only a finite number of valency two vertices and ending at a valency one or another high-valency vertex. If an arm starts from w and ends at a valency one vertex v ∈ V 1 , then we call it a branch from w to v and denote it by Γ wv . An arm starting from a high-valency vertex w 1 ∈ V h and ending at another high-valency w 2 ∈ V h is called a bridge between w 1 and w 2 , denoted by Γ w 1 ,w 2 . For example, two graphs in Figure 2 (a) have bridges between m 1 and m 2 for the bottom one or from m 1 ± 1 to m 2 ± 1 for the top, respectively, and the top graph in Figure 2 (b) has a branch from m 1 ± 1 to ±1.
Since a branch Γ wv from w to v can be thought as a linear plumbing, the following continued fraction can give us the information of the branch. It is actually a complete invariant under the Neumann moves creating or annihilating valency two vertices along the branch. In (2.1), m w is the weight of the starting high-valency vertex of the branch Γ wv , u i s represent the weights of valency two vertices on the linear plumbing between w and v, and m v is the weight of the ending valency one vertex. If the continued fraction m w of Γ wv is an integer, the branch is called pseudo branch because it can be annihilated or removed by a sequence of Neumann moves. For example, the top graph in Figure 2 (b) has a pseudo-branch Γ m 1 ±1,±1 . 1 A high-valency vertex is w ∈ V h called a pseudo high-valency vertex if it satisfies the following condition where n w denotes a number of pseudo branches connected to the vertex w. This means that a pseudo high-valency vertex can be turned into a valency one or two vertex once we annihilate or collapse all of its pseudo branches by using Neumann moves. For a given bridge Γ w 1 ,w 2 between two high-valency vertices w 1 , w 2 ∈ V h , we also examine the continued fraction of valency two vertices laid on the bridge if they exist. The bridge is called a pseudo bridge if it contains at least one valency two vertex and the continued fraction is zero. A pseudo bridge can also be annihilated by a sequence of Neumann moves of type (a) and (c).
We define a plumbing graph to be reduced if there is at least one vertex with degree greater than two and there does not exist a pseudo high-valency vertex. One can change any plumbing tree into a reduced plumbing by removing pseudo high-valency vertices by using Neumann moves. Note that removing pseudo high-valency vertex could create another pseudo high-valency vertex, hence it is needed to keep removing all pseudo high-valency vertices as they appear until there is no pseudo high-valency vertex anymore. We also notice that a plumbing without any highvalency vertex represents a lens space.

Identification of Spin c structures
Let us review briefly the identification of Spin c structures on plumbed 3-manifolds in terms of plumbing data. The affine space Spin c (Y ) in the case of Y = Y (Γ) for a plumbing tree Γ has already been studied in the Heegard-Floer homology literature [12], and a slightly different description has been used in order to construct an invariantẐ a (q) of the plumbed 3-manifold equipped with the Spin c structure a in [6]. Here we recall Spin c structures on plumbed 3-manifolds Y (Γ) by following [6].
The Spin c structures on Y can be inherited by those on the 4-manifold W = W (Γ) with boundary Y . More explicitly, a natural identification for Spin c structures on W Spin c (W ) ∼ = 2Z s + m translates into a natural identification for the plumbed 3-manifold Y Spin c (Y ) ∼ = 2Z s + m / 2M Z s , 1 By abuse of notation, we often use the weights to denote the vertices where the meaning is clear in the context.
where m is the vector made of the weights m v for v ∈ V . Note that both identifications take the conjugation of Spin c structures to the involution a ↔ −a on the right-hand side.
Another natural identification can be obtained by using the map where δ is the vector of the degrees of the vertices in Γ, and u = (1, 1, . . . , 1).

The q-series
The q-series has been firstly proposed in [7] aimed to categorify the WRT invariant. It was originally defined for plumbed 3-manifolds coming from negative definite plumbing graphs. For a later convenience, we recall the formula of the q-seriesẐ a (q) for weakly negative definite plumbed 3-manifolds following [6] and [7]. Let Γ be a plumbing graph satisfying the weakly negative definite condition, that is, the linking matrix M = M (Γ) is invertible and M −1 is negative definite on the subspace of Z s spanned by the high-valency vertices. Using identifications (2.2), fix a representative a ∈ 2Z s + δ of a class a ∈ 2Z s + δ / 2M Z s .
Then the q-seriesẐ a (q) is defined bŷ where Note that all rational functions in (2.4) should be understood as the symmetric expansion, that is, the average of the expansion as z v → 0 and z v → ∞. For example, given a high-valency vertex w ∈ V h with deg(w) = 2 + p w , we have where A(r, p) is a binomial coefficient A(r, p) = r p = (r + 1)(r + 2) · · · (r + p − 1) (p − 1)! .
Remark 2.1. The weakly negative definite condition should be imposed to ensure thatẐ a (q) is well-defined. In fact, together with this condition, (2.5) implies that the exponents of q inẐ a (q) has a lower bound, and that only finitely many terms can contribute to the same exponent of q.
Remark 2.2. The condition that M is invertible means the corresponding plumbed 3-manifold is a rational homology sphere. We will mostly be interested in the case where M is invertible. The q-series for plumbings satisfying det M = 0 was dealt in [4].

The (q, t)-series invariants
In this section we construct the formula of the (q, t)-seriesẐ a (q, t) for a certain class of plumbed 3-manifolds by introducing a regulator t into the q-seriesẐ a (q), previously defined only for weakly negative definite plumbings.

Definition
Let Γ be a reduced plumbing graph, that is, a tree plumbing that has at least one high-valency vertex, but does not have any pseudo high-valency vertex. We assume that its linking matrix M = M (Γ) is invertible. We keep our notations as in Section 2.
Definition 3.1. The (q, t)-seriesẐ a (q, t) for a reduced plumbing, equipped with some Spin c structure a ∈ 2Z s + δ, is defined bŷ Let us elaborate on the various elements of this formula. For the factors in front of the CT z operation, the π = π(M ) and σ denote the number of positive eigenvalues and the signature of the linking matrix M , respectively. The theta function is the same as the one in (2.5) for the q-series, and CT z denotes the operation of taking the constant term of the Laurent series in z v ∈ V .
The S is a set of vectors ξ ∈ {±1} V h that satisfy the following property where ξ w denotes the element in a vector ξ corresponding to the high-valency vertex w ∈ V h , and π(Γ w 1 ,w 2 ) is the number of positive eigenvalues of the linking matrix associated to Γ w 1 ,w 2 . One can easily see that the set S for a given Γ is well-defined as long as Γ is a tree. Moreover, the cardinality |S| of the set S is equal to 2 to the power of the number of high-valency vertices minus the number of pseudo bridges in Γ. A vector ξ ∈ S is called a chamber for Γ. For a given chamber ξ, the discriminant function at this chamber is defined by Here I w , associated to each w ∈ V h , is the set of valency one vertices that are end-points of all branches starting from w. And the discriminant D The function ϕ(v) returns a sign, determined by the branch Γ wv , as follows: if the branch Γ wv from w to v is not a pseudo-branch, then ϕ(v) = 0. If the branch Γ wv is a pseudo-branch, by definition, the weights of the vertices laid on the branch should satisfy for some finite integer m w . Then we define where Γ m w is the graph that consists of the single vertex weighted by m w . Recall that the notation π(Γ wv ) denotes the number of positive eigenvalues of the linking matrix corresponding to Γ wv , so it is immediate to see (−1) For completeness, we defineẐ a (q, t) =Ẑ a (q) if there is no high-valency vertex in Γ. The (q, t)-series is well-defined even when Γ does not satisfy the weakly negative (or positive) definite because there are only finitely many contributions to any monomial t nt q nq .
Remark 3.2. Comparing (3.2) with (2.6), one can see that the new variable t is introduced as the regulator that appears in the standard ζ-regularization process. Moreover, if Γ is weakly negative definite, then we can recoverẐ a (q) fromẐ a (q, t) by taking t = 1. Remark 3.3. In [6], two-variable series F K (x, q) for knot complements is introduced as the analogue of the invariantsẐ a (q). It is also possible to construct the t-deformation of F K as the analogue ofẐ a (q, t) for knot complements in the same way. We note that another version of t-deformation and a-deformation has been discussed in [5].

Invariance
For the (q, t)-seriesẐ a (q, t) to be an invariant of the plumbed 3-manifold Y equipped with the Spin c structure a, it should be independent on the presentation of Y as a plumbing, that is, it has to be unchanged under the Neumann moves in Figure 2.
Proposition 3.4. If two reduced plumbings represent a same 3-manifold, then there exists a sequence of Neumann moves such that none of those Neumann moves creates any pseudo highvalency vertices.
Proof . Since two reduced plumbings Γ and Γ realize the same 3-manifold, there exists a sequence of plumbings G = {Γ 0 = Γ, Γ 1 , Γ 2 , . . . , Γ n−1 , Γ n = Γ } such that any adjacent two plumbings Γ i−1 and Γ i are related by a Neumann move depicted in Figure 2 for i = 1, . . . , n. It is important to note that in this proof we consider a Neumann move as an action applied to a certain vertex on a branch or on a bridge, instead of regarding it as a transformation from a plumbing graph to another.
As a first step, we pick up a subsequence of Neumann moves such that the first move in the subsequence creates a pseudo branch or a pseudo bridge, the last move deletes the pseudo branch or bridge, and all the moves in between are actions on the branch or bridge. It is clear that the subsequence is redundant, therefore, we are going to remove all such subsequences. Notice that another subsequence with this property can be appeared after removing a subsequence, so we keep removing subsequences until they will not appear anymore. We also note that the remaining sequences of Neumann moves are still well-defined since all the moves in this subsequence are limited on a redundant branch or bridge.
If the relevant plumbings are all reduced after removing subsequences, then the proof is done. Therefore, without loss of generality, let us assume that there exists at least one non-reduced plumbing in the sequence G. For simplicity, suppose that all non-reduced plumbings in the sequence G contain only one pseudo high-valency vertex, because the cases with more than one pseudo high-valency vertex can be extended in a similar way. Among non-reduced plumbings, choose the non-reduced plumbings Γ j and Γ k with the smallest index j and the largest one k. This means that Γ j−1 and Γ k+1 are reduced plumbings. Furthermore, since we have removed all redundant subsequences of Neumann moves, there are only two possible ways to create the pseudo-high valency vertex in Γ j . The first is that the pseudo high-valency vertex in Γ j is created by adding a pseudo branch to a valency two vertex on a pseudo bridge in Γ j−1 . The other is that a high-valency vertex with degree more than 3 in Γ j−1 is split into two high-valency vertex connected by a pseudo bridge, one of which is the pseudo high-valency vertex.
Since two cases are related to a pseudo bridge, the key is to collapse the pseudo bridge by using Neumann moves. More explicitly, we insert a plumbing Γ j−1 between Γ j−1 and Γ j where Γ j−1 is obtained by collapsing the pseudo bridge on which the pseudo high-valency vertex is laid in Γ j , only if Γ j−1 has the corresponding pseudo bridge. Then we get Γ i from Γ i for i = j, j + 1, . . . , k by collapsing the relevant pseudo bridge. We also insert a plumbing Γ k+1 between Γ k and Γ k+1 by rebuilding a pseudo bridge as same as the one in Γ k+1 if it exists in Γ k+1 .
Theorem 3.5. The seriesẐ a (q, t) defined in (3.1) is an invariant for the reduced plumbed 3-manifolds.
Proof . SinceẐ a (q, t) is equal toẐ a (q) for the plumbings without high-valency vertices by definition, we can assume that V h (Γ) is not an empty set. We use a prime to distinguish the quantities associated to the top graphs in Figure 2. For example, M is the linking matrix for the bottom graph Γ, and M denotes the one for the top graph Γ . According to the result of Proposition 3.4, it is enough to consider Neumann moves that does not create a pseudo high-valency vertex.
Consider the move (a) in Figure 2, with the signs on top being −1. By linear algebra, it is immediate to see that we have σ = σ − 1, π = π, hence the quantity 3σ − v m v does not change. Furthermore, the factors in front of the CT z operation in (3.1) does not change either.
For the bottom graph Γ, let us write a vector ∈ Z s as a concatenation such that 1 describes the left part of the graph including the vertex m 1 and 2 describes the right part. Then we can construct a vector for the top graph Γ by Moreover, it is shown in [6] that has the property ∈ 2M Z s+1 + a whenever ∈ 2M Z s + a, where a is a representative of the Spin c structure a for Y (Γ ) that is the counterpart of a through the isomorphism Y (Γ) ∼ = Y (Γ ). Taking these into account, when the move happens inside a branch Γ wv from a high-valency vertex w ∈ V h to a valency one vertex v ∈ I w , observe that there is no change for ϕ(v) whether the branch is pseudo or not. This implies that the discriminant function does not change, hence we obtain the same result.
For the case when the move happens inside a bridge between two high-valency vertices, two discriminants for Γ and Γ have the same structure, therefore they yield the sameẐ a (q, t).
Let us consider now the case of the move (a) with the sign +1 for the top graph in a similar way. As before, the quantity 3σ − v m v is still unchanged, but we have π = π + 1, so (−1) π switches the sign. Given = 1 , 2 for the bottom graph, a vector for the top graph For the move on a bridge that is not pseudo, the discriminant does not change, but due to the change of sign in 2 , the theta function changes as if we do the substitutions z v → z −1 v for all the vertices v corresponding to 2 . From this, we obtain extra −1 sign, that cancels out the disagreement of signs by (−1) π . Note that the powers of t remains the same by the substitutions If the move happens on a pseudo-bridge, the proof is similar, but the only difference is that due to the minus sign in − 2 for the choice of the chamber for the top graph should be opposite than that for the bottom, because the move increases by 1 the number of the positive eigenvalues of the linking matrix associated to the bridge.
If the move appears on a non-pseudo branch Γ wv , then the mechanism is same as one for the move on a bridge, so we get the same result. However, if the move appears on a pseudobranch Γ wv , then we need to care about the change of ϕ(v) for the discriminant function because the move (a) with the sign +1 increases the number π(Γ wv ) of positive eigenvalues of the linking matrix associated to the branch by 1. In this case, v is the only vertex corresponding to 2 , hence the change of the sign for v produces the extra sign, but it gives the same powers of t because of the change of ϕ(v). Also, the extra sign remedies the change of (−1) π . Now, we consider the move (b) when the sign for the blow-up vertex is −1. We have π = π, σ = σ − 1, and the quantity 3σ − v m v for the top graph is 1 lower than that for the bottom one. This yields an extra factor q −1/4 for the top graph.
For the bottom graph Γ, we write vectors as where 1 denotes the vertex with weight m 1 . Then we define corresponding vectors for the top graph Γ ± = 2 , 1 ± 1, ∓1 .
By simple linear algebra, one can observe that The extra factor +1 in (3.4) gives rise to q 1/4 for the top graph that cancels with q −1/4 coming from the factor in front of the operation CT z .

S.J. Ri
Let us consider the case where the vertex w decorated with m 1 is a high-valency vertex in the bottom graph. We assume that the high-valency vertex does not have a pseudo bridge. One can follow the similar argument for the case when it has a pseudo bridge.
The corresponding part of the discriminant function for Γ is given by where D ± are the contributions from vertices in I w corresponding to 2 , and p is determined by deg(w) = 2 + p. From the role of the operation CT z in (3.1), it follows that 1 has values in the form of 1 = ±(2r + p) for non-negative integers r. Without loss of generality, suppose that 1 = 2r + p. Then the vectors = 2 , 2r + p pick up monomials in t given by For the top graph, the discriminant function has the following portion where z 0 is the variable for the newly introduced blow-up vertex in Γ . Corresponding to = 2 , 2r + p , we have vectors of the form Therefore, the monomials chosen by those vectors are This means that we get the same answer forẐ a (q, t).
For the case when the move applies to a valency one vertex v 1 ∈ I w for some w ∈ V h , it is essential to check that the discriminant for Γ can be obtained from the one for Γ by the substitution z 1 → z 0 since we have ϕ(v 1 ) = ϕ(v 0 ). Then, vectors for Γ are described by and the corresponding vectors for Γ are given by Putting all together, we get the same result.
Move (b) with the sign +1 is similar, but we use the vectors of the form Let us move on to the move (c). Here we have π = π + 1 that yields an extra sign, and the factor 3σ − v m v is unchanged. For the top graph Γ , we write vectors as where 1 is corresponding to the left side of the graph (not including v 1 ) and 2 is corresponding to the right side of the graph (not including v 2 ). From we define a vector for the bottom graph Γ as Let v 1 , v 0 , v 2 be the vertices shown in the top graph and v b the vertex shown in the bottom one. There are four different cases depending on where those vertices are placed in the graph.
First, we consider the case when v 1 , v 0 , v 2 are laid on a non-pseudo bridge, but at least one of v 1 and v 2 is not a high-valency vertex. Then the discriminant functions for Γ and Γ have the same structure, and the minus sign in front of 2 in = 1 , 1 , 0, 2 , − 2 produces the opposite sign for monomials in t to the sign from = 1 , 1 − 2 , 2 . This sign difference is recovered by the extra sign from (−1) π . Thus we get the same result forẐ a (q, t).
If the vertices v 1 , v 0 , v 2 are laid on a pseudo bridge, we should put into our consideration the change of choice of chambers for the top graph since the move creates a positive eigenvalue of the linking matrix.
The second case is when v 1 , v 0 , v 2 form a bridge between v 1 and v 2 , that is, v 1 and v 2 are both high-valency vertices, and v 0 is a valency two vertex. This case is just the one depicted in Figure 2 (c). For the bottom graph Γ, we can assume that the vertex v b does not have a pseudo bridge, then the corresponding portion of the discriminant function is given by where D 1 and D 2 denote the discriminant functions of valency one vertices in I v b corresponding to 1 and 2 , respectively. Then the vectors of the form = 1 , 2r + p, 2 choose the following monomials For the top graph, since the elements ξ v 1 and ξ v 2 of a chamber ξ should be opposite, we have where p 1 = deg(v 1 ) − 2 and p 2 = deg(v 2 ) − 2 such that p = p 1 + p 2 . Associated to = 1 , 2r + p, 2 , we have the following vectors Here the sum of r 1 and r 2 should be equal to r. From such vectors, we obtain − 1 2 r 1 +r 2 =r A(r 1 , p 1 )A(r 2 , p 2 )t 2r 1 +2r 2 +p 1 +p 2 = − 1 2 A(r, p)t 2r+p , where we have used A(r, p) = r 1 +r 2 =r A(r 1 , p 1 )A(r 2 , p 2 ). Thirdly, let us consider when v 1 , v 0 , v 2 are vertices on a branch from v 1 to v 2 through v 0 . Then, the discriminant function for the bottom graph Γ has the following part The vectors = 1 , b = 1 , 2r + p pick up the factors of the form On the other hand, for the top graph Γ we have where we have used ϕ(v 2 ) = −1 since the branch Γ v 1 v 2 is a pseudo-branch and the move increases the number of positive eigenvalues by 1. Corresponding to = 1 , 2r + p , we have two vectors = 1 , 2r + p + 1, 1 and = 1 , 2r + p − 1, −1 , which produce the following factors: The minus sign in the right side of the above equation cancel with the extra sign from (−1) π . Therefore, we obtain the same result. At last, if v 1 , v 0 , v 2 are a part of a branch, then we use the similar approach to the third case. Now we present a simple example of the (q, t)-series for a 3-manifold realized by the plumbing shown in Figure 3. It is known as Poincaré homology sphere. It is also an example of a Brieskorn 3-sphere, denoted by Σ(2, 3, 5). The overline denotes the reverse of the orientation compared to the standard one.
The linking matrix for the plumbing is given by As H 1 Σ(2, 3, 5), Z ∼ = 0, there is a unique value a = 0 to consider. The formula (3.1) for the manifold reads aŝ Due to the operation CT z , the values of v should be 1 = ±(2r + 1), and v = ±1 for v = 2, 3, 4.

Recovering the q-series
Let Γ be a reduced plumbing graph that realizes a 3-manifold Y = Y (Γ). By applying the formula (3.1), we have the invariantsẐ a (q, t) for Y , which can be expressed bŷ where ∆ a ∈ Q, S is the set of all possible chambers for Γ, and C ξ,n (t) ∈ Z[[t]] is in general the formal power series in t as the coefficient of q n inside a chamber ξ ∈ S. As we have seen in the example of Section 3, C ξ,n (t) turns out to be a finite polynomial in t if Γ has the weakly negative (or positive) definite property. Furthermore, by simply setting t = 1, we can recover the q-series from the (q, t)-series. Being inspired by this, it would be nice if we can recover the q-series by taking a limit t → 1 − . However, without weakly definite property, C ξ,n (t) is in general infinite series and lim t→1 − C ξ,n (t) might be ill-defined. This means that it would be not easy to recover the q-series for such plumbings. By abuse of notation, we will denote the limit from below t → 1 − by t → 1. We notice that C ξ,n (t) is convergent for |t| < 1 due to the boundedness of its coefficient polynomial.
Fortunately, there are some examples of plumbings, for which it is possible to recover the q-series by computing the limit t → 1. Since the (q, t)-seriesẐ a (q, t) for reduced plumbings are invariant under Neumann moves, its recovered version,Ẑ R a (q) = lim t→1Ẑa (q, t) , are also invariant, therefore indeed define a topological invariant of plumbed 3-manifolds.
Remark 4.1. The recovered q-seriesẐ R a (q) is exactly same as previously defined q-seriesẐ a (q) when plumbings are weakly negative definite. However, there are some examples where the recovered q-series are not equal to previously defined q-seriesẐ a (q), for example, weakly positive definite plumbings [3,6]. Furthermore, when plumbings have strongly indefinite property, q-seriesẐ a (q) for them have not been defined, whileẐ R a (q) still exists in certain cases. Example 4.2. The first example we consider is to compute the (q, t)-series for two reduced plumbings, shown in Figure 4, that are equivalent by a Neumann move of type (c) in Figure 2.
The interesting point here is that the plumbing depicted Figure 4 (a) has the weakly negative definite property but another plumbing in Figure 4 (b) which is obtained by applying one Neumann move to the plumbing (a) does not have the weakly negative definite property. This means the plumbing (a) has the q-series by applying the formula (2.3) while the plumbing (b) does not have it. We are going to overcome such drawback by the recovering process. By the formula (2.3), the q-series for the plumbing (a) is given bŷ where we notice that the determinant of the linking matrix for the plumbing is 1 and it has the unique q-series. The formula (3.1) for the plumbing (a) reads aŝ which gives us the following result Z 0 (q, t) = q 417 2 t 2 1 − q 29 − q 211 + q 252 − q 393 + q 442 − q 667 + q 726 + · · · . (4.2) It is clear that the limit t → 1 of (4.2) is the same as q-series in (4.1). Moreover, one can also check that (q, t)-series for the plumbing (b) is equal to (4.2), which is obvious by Theorem 3.5. We note that q-series for the plumbing (b) does not exist because of non-weakly negative definite property, but we can associate the recovered q-series to it. Example 4.3. Let us consider a reduced plumbing depicted in Figure 5. As we will see in Section 5, this plumbing represents a 3-manifold which is diffeomorphic to the disjoint union of two 3-manifolds. In order to compute the (q, t)-series, the first step is as usual to get the linking matrix and its inverse: Then the application of the formula (3.1) to the plumbing leads to the following result: Now let us compute lim t→1Ẑ0 (q, t) to recover the q-series. To do this, we need to find out the analytical property of the series C ξ,n (t), the coefficients of the term q n in (4.3) from the contribution of the chamber ξ ∈ S. By the role of the action CT z in (3.1) it follows that the vectors = ( 1 , 2 , 3 , 4 , 5 , 6 ) should be of the form where ξ 1 and ξ 2 are the coordinates of the chamber vector ξ = (ξ 1 , ξ 2 ) ∈ {±1} 2 . Therefore, C ξ,n (t) is given by where is the vector of the form = ξ 1 (2r + 1), ξ 2 (2s + 1), ε 3 , ε 4 , ε 5 , ε 6 and the convention denotes that the only r, s and ε i satisfying the quadratic equation Q( ) = 0 contribute to the summations. By swapping the order of sums in (4.4), we get which means that given a chamber ξ ∈ S and chosen ε i 's, we have the sum of monomials in t with powers of 2r + 2s + 2 from all non-negative integer solutions of the quadratic equation in two variables r and s moreover, the product ξ 1 ξ 2 ε 3 ε 4 ε 5 ε 6 determines the sign of the sum over r and s. Therefore, we naturally move on to the solutions of the quadratic Diophantine equations (QDEs). In Appendix A, we briefly review the algorithm to solve the QDEs in two variables for convenience. In general, the quadratic form from a plumbing with two high-valency vertices is a quadratic equation with rational coefficients, but multiplying it by 4 det M we can obtain the one with integer coefficients. For a fixed chamber ξ ∈ S and chosen ε i 's, we denote the quadratic equation with integer coefficients by By putting the solutions of quadratic equations into (4.5), we obtain the explicit formula for C ξ,n (t), for example, n = 2 as follows: where ξ ++ = {+1, +1} and ξ +− = {+1, −1}. We notice that C ξ ++ ,n (t) = C ξ −− ,n (t) and C ξ +− ,n (t) = C ξ −+ ,n (t) due to the quadratic form. Therefore, it is enough to evaluate the limit of the form lim t→1 ∞ k=0 t a(k+b) 2 +c , for a > 0.
First, we change the variable by using t = e − , then we have We recall the Euler-Maclaurin formula where N i and N f are real numbers such that N f − N i is a positive integer number, B j and b j denote Bernoulli polynomials and numbers, respectively, j is any positive integer and {x} denotes the fractional part of a real number x. Applying the Euler-Maclaurin formula, we have where we have used the error function's Maclaurin series By substituting (4.7) into (4.6), we obtain Even though the term π is singular at → 0, this singularity will disappear when we take a sum over all possible chambers. Thus, we see that the coefficient of q 21 8 in the recovered q-series is finite and equal to −5.
By using a similar approach, we obtain the following recovered q-serieŝ where the superscript denotes the recovered q-series.
In the previous example, we have seen that for some plumbings without weakly definiteness the (q, t)-series can be recovered into the q-series by computing the limit t → 1. However, this does not hold for all strongly indefinite plumbings. The next example shows a such case.
Example 4.4. We consider a reduced plumbing shown in Figure 6. We note that just for simplicity we choose a plumbing whose linking matrix has determinant 1 such that there is a unique Spin c structure a = 0.  Figure 6. A plumbing graph that has the strongly indefinite property and whose linking matrix has determinant equal to 1.
For this plumbing, the linking matrix and its inverse are given by From the linking matrix, we obtain the following quantities Since there is no pseudo bridge in the plumbing, the set S of all possible chambers is given by Therefore, the formula (3.1) readŝ The application of the formula (4.8) gives the following result: We note that all the coefficients of q n in this result are infinite series in t because of the strong indefiniteness of the linking matrix. For a chosen chamber ξ = (ξ 1 , ξ 2 ) ∈ {±1} 2 , we know that the vectors = ( 1 , 2 , 3 , 4 , 5 , 6 ) should be of the form and C ξ,n (t) is given by According to the algorithm in Appendix A, the nonnegative integer solutions of are determined by a finite number of families of solutions, and each family {(r k , s k )} k of solutions has the following form: where k ≥ k 0 ∈ Z + . Here the real numbers α r , α s , β r , β s , η r , η s are constants which do depend on the choice of the chamber ξ and the vector ε. However, the A does not depend on ξ and ε. This is because all QDEs Q ξ, ε = 0 from different ξ and ε can be transformed into the same generalized Pell's equation. From the requirement for non-negative solutions, it follows that α r and α s are positive real numbers and A > 1.
For each family of solutions, we obtain a series in t by (4.9) as following: In order to compute the limit t → 1, we use the substitution t → e − By applying doubly exponential series transformations of Ramanujan [2], we can estimate the above series as follows: where γ is the Euler's constant γ = 0.57721 . . . and B k denotes the kth Bernoulli number defined by Now let us analyse the terms in the right side of (4.10). First, the second term proportional to log has a singularity at → 0, but this term will be cancelled out by taking a sum of C ξ,n (t) over all choices of chambers ξ ∈ S, which allows us to ignore this term. Next, the infinite series of oscillating terms has a singularity as well. Furthermore, this terms do not disappear by a sum of C ξ,n (t). Thus, we conclude that due to such series of oscillating terms the limit t → 1 is ill-defined and the recovered q-series does not exist. Example 4.5. In this example, we consider a non-reduced plumbing shown in Figure 7, which can be obtained by Neumann moves from the one in Figure 3. Hence, the 3-manifold realized by the plumbing graph in Figure 7 is homeomorphic to the Brieskorn sphere Σ (2,3,5). One can easily check that the high-valency vertex with weight 3, placed in the right side of the plumbing, is a pseudo high-valency vertex because the branches are both pseudo. We have already computed the (q, t)-series (3.5) in the last part of Section 3. Now we are going to compute the (q, t)-series for the non-reduced plumbing in Figure 7 and compare it with the result in (3.5).
The linking matrix and its inverse of the plumbing are given by The formula (3.1) readŝ is changed by Neumann moves, but the recovered q-series might be still preserved. This property indeed holds for non-reduced plumbings with at most two high-valency vertices.
Proposition 4.6. Assume that the (q, t)-series for a given plumbing can be finitely recovered q-series in a sense that the limit t → 1 is finite. Then the recovered q-series is preserved by arbitrary Neumann moves even for non-reduced plumbings with at most two high-valency vertices.
Proof . Since the (q, t)-series gets changed only when a Neumann move creates a pseudo highvalency vertex, it is sufficient to show that the recovered q-series remains unchanged for such a Neumann move. Therefore, we are going to consider a case shown in Figure 8, where the left plumbing (a) is a reduced one with one high-valency vertex weighted by m 1 and the right plumbing (b) has an extra pseudo high-valency vertex created by a Neumann move of type (b) in Figure 2. The cases for other types of Neumann moves creating a new pseudo high-valency vertex can be handled in a similar way. Let Γ be the left plumbing and Γ be the right one in Figure 8. We use a prime to denote the quantities for the right plumbing. Let us write a vector by where 1 corresponds to the left part of the graph Γ including the vertex m 1 . As we have already seen in the proof of Theorem 3.5, we have corresponding vectors for Γ ± = 1 , ±1, 2 , ∓1 , which give the exactly same contribution as . However, due to the existence of pseudo highvalency vertex in Γ , we have the following extra vectors corresponding to for s ∈ N. It is immediate that those infinitely many extra vectors generate extra t-series. Therefore, we need to show that the limit t → 1 of those extra t-series is equal to zero. Let us consider the extra t-series coming from the vectors 1 . The case for the vectors 2 is similar. For some n ∈ Z and ∆ a ∈ Q, the t-series coefficient of q ∆a+n from those vectors is determined by where p r = deg(v 1 ) − 2 and Q denotes the following quadratic equation This means that those extra t-series would be zero after taking the limit t → 1 if the non-negative solutions of two quadratic equations Q 1,± are expressed by the families of solutions with the same patterns. Indeed, this is the case. As we have mentioned in Appendix A, the first step to solve general quadratic Diophantine equation (A.1) is to transform it into the form of generalized Pell's equation (A.3) by using the following transformations Therefore, the solutions of (A.1) are expressed by the factors D, E and F together with the solutions of the generalized Pell's equation. We have already seen that two quadratic equations Q 1,± are transformed into the same generalized Pell's equation. Moreover, it is an algebraic exercise that these equations have the same transforming factors D, E and F . Thus, the solutions of two quadratic equations have the same patterns, which completes the proof.
Based on Proposition 4.6, we naturally suggest the following conjecture: Conjecture 4.7. The recovered q-series, obtained from the (q, t)-series in the sense of t → 1, is an invariant of 3-manifolds realized by arbitrary (i.e., possibly non-reduced) plumbings.
Remark 4.8. As we have seen in Sections 3 and 4,Ẑ a (q, t) is defined for reduced plumbings and it has a nice property that it is not only an invariant of 3-manifolds realized by reduced plumbings but also a refined version ofẐ a (q), i.e., lim t→1Ẑa (q, t) =Ẑ a (q) for weakly negative definite plumbings. Furthermore, it provides a key to defineẐ a (q) even for strongly indefinite plumbings if its limit as t → 1 exists. For non-reduced plumbings, it is dependent on the presentation of plumbings and it undergoes changed by the Neumann moves. However, as we demonstrates in Example 4.5 and Proposition 4.6, the limit lim t→1Ẑa (q, t), if exists, is still an invariant for non-reduced plumbings with at most two high-valency vertices. Thus, if Conjecture 4.7 holds, then we could construct an invariant of all tree plumbed 3-manifolds by obtaining the recovered q-series under the assumption of the existence of the limit t → 1.

Connected sum of plumbed 3-manifolds
Let Γ 1 and Γ 2 be plumbing graphs and Y 1 , Y 2 be 3-manifolds realized by Γ 1 , Γ 2 , respectively. It is well-known that the connected sum Y 1 Y 2 can be realized by the disjoint union Γ 1 Γ 2 . The following proposition [11] says that the disjoint union of finite number of plumbing graphs is equivalent a certain single plumbing graph.
LetẐ a j (Γ j ; q, t) be the (q, t)-series for Γ j with Spin c structure a j , where j = 1, 2. Then, according to Proposition 5.1, the connected sum of Y 1 = Y (Γ 1 ) and Y 2 = Y (Γ 2 ) can be realized by a plumbing Γ in the form of Figure 9. Just for simplicity, we assume that Γ j is connected to the vertex e through a valency one vertex of a non-pseudo branch in Γ j . This assumption is just to avoid creating a new high-valency vertex or a new pseudo bridge.
We now present a formula for the (q, t)-seriesẐ a (Γ; q, t).
It is worth to notice here that the series is the symmetric expansion of a rational function 1 q 1/2 − q −1/2 , that is just the inverse of the q-series of 3-sphere.

A Quadratic Diophantine equation in two variables
We are going to review here how to solve the general quadratic Diophantine equation ax 2 + bxy + cy 2 + dx + ey + f = 0, (A.1) where the coefficients are all integers, i.e., a, b, c, d, e, f ∈ Z. We assume that not all of a, b, c are zero. The classical method of solving (A.1) was firstly given by Lagrange over 300 years ago. Let D = b 2 − 4ac, E = bd − 2ae and F = d 2 − 4af . We will examine (A.1) in several cases, mainly depending on the values of D.
At first, suppose D = 0, then without loss of generality, we can assume a = 0. Equation (A.1) can be written as