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Splice Diagram Singularities and The Universal Abelian Cover of Graph Orbifolds

Given a rational homology sphere M, whose splice diagram satisfy the semigroup condition, Neumann and Wahl were able to define a complete intersection surface singularity called splice diagram singularity from the splice diagram of M. They were also able to show that under an additional hypothesis on M called the congruence condition, the link of the splice diagram singularity is the universal abelian cover of M. In this article we generalize the congruence condition to the class of orbifolds called graph orbifold. We show that under a small additional hypothesis, this orbifold congruence condition implies that the link or the splice diagram equations is the universal abelian cover. We also show that any two node splice diagram satisfying the semigroup condition, is the splice diagram of a graph orbifold satisfying the orbifold congruence condition.


Introduction
The topology of an isolated complex surface singularity is determined by its link, which all turn to be among the class of 3-manifolds called graph manifolds, that is the manifolds which only have Seifert fibered pieces in their JSJ-decomposition. There are several graph invariants of graph manifolds which is used to study them. The first is the plumbing diagram of a plumbed 4-manifold X such that our graph manifold M is the boundary of X, this does give a complete invariant if one assumes the plumbing diagram is in a normal form (see [Neu81]), of which there are several different. Now the plumbing diagrams can be quite large and does not always display the properties of the manifold clearly, so we are interested in an other invariant called splice diagram. Splice diagrams where original introduced in [EN85] and [Sie80] for integer homology sphere graph manifolds. It was then later generalized by Neumann and Wahl to rational homology spheres in [NW02]. They used it extensively in [NW05b] and [NW05a], and especially their use in [NW05a] is of interest to us.
In [NW05a] they use the splice diagram of a singularity link M satisfying what they call the semigroup condition, to construct a set of equations called splice diagram equations defining an isolated complete intersection surface singularity X. They then showed that if M satisfy an additional hypothesis called the congruence condition, the link of X is the universal abelian cover of M . In [Ped10a] I showed that the splice diagram of any graph manifold M always determines the universal abelian cover of M . So combining these to result one gets a nice description of the universal abelian cover of a graph manifold M , as the link of complete intersection, provided that there is some graph manifold M ′ with the same splice diagram satisfying the congruence condition. This already implies that the congruence condition might not be needed, moreover the following splice diagram • have no manifolds with it as its splice diagram satisfying the congruence condition, even though it satisfy the semigroup condition and it is the splice diagram of 4 different manifolds. Nonetheless one can construct a plumbing diagram of the abelian cover use the algorithm derived from my proof of Theorem 6.3 in [Ped10a] which is explained in more detail in [Ped10b] where this example is explicitly constructed, and one can find a dual resolution graph for the resolution of the splice diagram equation of Γ by hand, and it shows that also in this case is the link of the splice diagram singularity the universal abelian cover. This again indicates that the congruence condition is not needed. Even more interesting is the next example. The following splice diagram • . satisfy the semigroup condition, and the universal abelian cover is the link of the splice diagram singularity. But there are no manifolds with Γ ′ as its splice diagram. So what is the link the universal abelian cover of? To prove Theorem 6.6 of [Ped10a] I had to generalize the notion of splice diagram to a class of 3-dimensional orbifolds which I called graph orbifolds, and Γ ′ is the splice diagram of several graph orbifold.
This leads to the purpose of this article, to generalize the congruence condition to graph orbifolds, which is done in Section 6. Show that, under a small extra hypothesis, the link of at splice diagram equation of Γ(M ) is the universal abelian cover of M if M satisfy the orbifold congruence condition in Section 7. In section 8 we show that this is indeed an extension of the results of [NW05a], by given any two nodes splice diagram Γ satisfying the semigroup condition constructing a graph orbifold M satisfying the orbifold congruence condition, with Γ as its splice diagram. Section 2 introduces graph orbifolds, Section 3 introduces splice diagram, and Section 4 the splice diagram equations. Section 5 introduces the discriminant group which is needed in the definition of the congruence condition.

Graph Orbifolds
To generalize the conditions for the splice diagram equation to define the universal abelian cover, we have to extend the notion of splice diagrams to graph orbifolds, so in this section we define graph orbifolds.
Definition 2.1. A graph orbifold is a 3-dimensional orbifold M , in which there exist a finite collection {T i } i∈1,...,n of smoothly embedded tori, such that M − n i=1 T i is a collection of S 1 orbifold fibrations over orbifold surfaces.
We will only consider graph orbifolds which has compact closure which is also a graph orbifold, and the boundary components will always be smoothly embedded tori. Notice that if M is smooth then M is a graph manifold, since any S 1 orbifold fibration over an orbifold surface with smooth total space is Seifert fibered.
Next we want to describe how a graph orbifold look locally, so we will look at a S 1 orbifold fibration over orbifold surface π : M → Σ. If x ∈ Σ is not an orbifold point then there is a disk neighborhood D of x such that M | π −1 (x) is a trivial fibered solid torus. So the interesting situation is when x ∈ Σ is an orbifold point. This means that a neighborhood U of x is homeomorphic to Ê 2 /( /α ) for a α > 1, where the group acts as rotation. Then M | π −1 (U) is homeomorphic to ∈ /α acts by [k](z, s) = (e 2πikv/α z, e −2πikq/α s) where we consider U = , q | α and gcd(v, α) = 1. Let β ′ satisfy β ′ v ≡ −1 mod α, and let β = β ′ q. We call the pair (α, β) the Seifert invariants of the singular fiber over x if we choose 0 ≤ β < α. Notice if gcd(α, β) = 1 then the action on (U × S 1 ) is free and hence π −1 (U ) is smooth. We call q = gcd(α, β) for the orbifold degree of the singular fiber. We will say that any non singular fiber has orbifold degree 1, which off course follows from the Seifert invariants of a non singular fiber being (1, 0). We denote by N (α,β) a solid torus neighborhood of the singular fiber with Seifert invariants (α, β), and by abuse of notation we call N q a solid torus neighborhood of a singular fiber of orbifold degree q.
One can now construct an unique decomposition of a compact graph orbifold M , the following way. Let q 1 , . . . , q n be the orbifold degrees of all the singular fibers which has orbifold degree greater than 1, and let N qi be solid torus neighborhoods of the corresponding singular fiber as above. Then M ′ = M − n i=1 N qi is a graph manifold with boundary, and hence the JSJ-decomposition of M ′ gives us a collection of Seifert fibered manifolds with boundary M ′ j 's. We then makes the pieces of the decomposition of M by gluing back the N qi in their original places. This is unique since the JSJ decomposition of M ′ is unique. We will call this decomposition for the JSJ-decomposition of M .
Let K ⊂ M be an orbifold curve with Seifert invariants (α, β), then the gluing of N (α,β) into M ′ = M − N (α,β) is completely determined by a simple closed curve in T = ∂M ′ of slope β/α. Let q = gcd(α, β) be the orbifold degree of K and let α ′ = α/q and β ′ = β/q. Then (α ′ , β ′ ) determines a Seifert fibration of the solid torus N (α ′ ,β ′ ) , notice that the Seifert fibration of N (α ′ ,β ′ ) and the orbifold structure on N (α,β) determines the same closed curve in ∂M ′ since β ′ /α ′ = β/α, therefore if M K = M ′ N (α ′ ,β ′ ) using the same gluing, M K and M has the same topology, even though M and M K is not equal as graph orbifolds since the singular fiber K has different Seifert invariants. If one replaces M with M Ki for all curves K i which has orbifold degree greater than one, we get a graph manifold M which have the same topology as M , we call M the underlying manifold of M .
In the definition of splice diagram of a graph manifold M , one uses the order for the first singular homology group of some graph manifolds constructed from M . Now the above implies that if we just use the singular homology groups when we define the splice diagram of a graph orbifold M , then one just get the splice diagram of the underlying manifold. So if using orbifold to extend the theory has to have any meaning, we need another homology theory, which reduces to singular homology in M is smooth, but sees the orbifold structure otherwise. This is going to be what we will call orbifold homology. Since we only need the first orbifold homology group, it suffices to define H orb 1 (M ) to be the abelianazation of the orbifold fundamental group π orb 1 (M ). The orbifold fundamental group is a more classical object, and have been studied a lot, especially in the geometrization of 3 dimensional orbifolds (see e.g. [Sco83]). Since π orb (M ) = π(M ) if M is smooth it follows that H orb where q is the orbifold degree of the central fiber, i.e. gcd(α, β).
This can then be used to show the following relating orbifold homology of M and M K defined above which is Proposition 5.2 of [Ped10a] Proposition 2.3. Let K ∈ M be an orbifold curve of degree q, then |H orb 1 (M )| = q|H orb 1 (M k )|.
It then follows that |H orb 1 (M )| = Q|H orb 1 (M )| = Q|H 1 (M )|, where Q = K q K with the product is taking over all orbifold curves K of orbifold degree q K . It should be mentioned that our H orb 1 (M ) is part of a homology theory for orbifolds in general see [ALR07].

Splice Diagrams
A splice diagram is a tree with no vertices of valence 2, which is decorated by signs on vertices who have valence greater than 2, we call such vertices nodes, and non negative integer weights on edges adjacent to nodes. We will call vertices of valence 1 for leaves, and will in general not distinguish between a leaf and the edge connecting the leaf to a node.
We will now explain how to assign a splice diagram as an invariant to any graph orbifold. Let M be a rational homology sphere (ÉHS) graph orbifold, we the construct the splice diagram Γ(M ), by first taking a node for each piece of the JSJdecomposition of M , the connect two nodes if the corresponding S 1 -fibered pieces of the decomposition are glued to create M . We will in general not distinguish between a node in Γ(M ) and the corresponding S 1 -fibered piece. This will result in a tree since the pieces of the JSJ-decomposition of M corresponds to the pieces of the JSJ-decomposition of the underlying manifold M , and since M is a ÉHS it follows from the comments after Proposition 2.3, that M is a ÉHS, and hence its JSJ-decomposition gives a tree like structure. We add a leaf to a node for each singular fiber of the node.
Next we want to add the decorations. First the signs at a node v is going to be the sign of the linking number of two non singular fibers at v, for precise definition of this see section 2 of [Ped10a], there we do only define it for manifolds, but the definition carries over to orbifolds, even though one has to be careful since linking number is not going to be symmetric any more in the case of orbifolds. One can calculate the linking number in the underlying manifold, and it follows from Lemma 6.4 that signs will be the same.
Last thing to define is the edge weights. Let v be a node and e an edge adjacent to v, then we will do the following construction to define the edge weight d ve adjacent to v on e. Let T ⊂ M be the torus corresponding to the edge e, cut M along T and let M ′ be the piece not containing v. Let M ve = M ′ T (S 1 × D 2 ) by gluing a meridian of the solid torus to the image of a fiber of v in ∂M ′ . Then d ve = |H orb 1 (M ve )|. The above definition defines a splice diagram where all weight at leaves are greater than 1, we call such splice diagram reduced, we will not always assume that our splice diagram are reduced, i.e. we will sometimes allow weights at leaf to be 1. A leaf of weight 1 will correspond to a non singular fiber, so if one has a non reduced splice diagram of a graph orbifold M , one gets the reduced splice diagram of M by removing all leaves of weight 1.
A general edge between two nodes in a splice diagram looks like to such an edge we associate a number called the edge determinant, which we define as r 0 r 1 − ε 0 ε 1 k0 i=1 n oi k1 j=1 n 1j , where ε i is the sign on the i'th node. This number is important since it helps determining which graph manifolds arises as singularity links, by the following theorem from [Ped10a] Theorem 3.1. Let M be a ÉHS graph manifold with splice diagram Γ(M ). Then Not all combinatorial splice diagram arises as the splice diagram of a graph orbifold. We will next introduce an important condition which the splice diagram of any graph orbifold satisfy. Let Γ be a splice diagram, and let v and w be two vertices in Γ. Then one defines the linking number l vw of v and w to be the product of all edge weights adjacent to but not on the shortest path from v to w. Similarly l ′ vw is defined the the same way except we exclude the weights adjacent to v and w. Let e be an edge at v then Γ ve is the connected subgraph of Γ one get by removing v, which includes the edge e. We then define ideal generator at v in direction of e d ve , to be the positive generator of the following ideal in In section 12.1 in [NW05a] Neumann and Wahl proves that every splice diagram coming from a singularity link satisfy the ideal condition, and the proof also works in the general setting of any graph orbifold. So satisfying the ideal condition is a necessary condition for a splice diagram to come from graph orbifold, in the one node case the ideal condition is void and we show in section 8 the in the two node case the ideal condition is also sufficient, unfortunately this is not the case if the splice diagram have more than two nodes, and the lack of understanding which splice diagram that in general arises as invariant of graph orbifolds is one of the reasons we have not been able to extend the main result to more than two nodes splice diagrams.

Splice Diagram Equations
We are in general going to be be interested in splice diagram satisfying the following stronger condition than the ideal condition.
Definition 4.1. A splice diagram Γ is said to satisfy the semigroup condition if for every node v and edge e at v. The edge weight lies in the following semigroup The semigroup is only interesting if one has no negative signs at nodes, so we will assume that splice diagrams satisfying the semigroup satisfy this. The semigroup condition is strictly stronger than the ideal condition for splice diagram with more than one node.
If M satisfy the semigroup condition, then given a node v and adjacent edge e, one can write the corresponding edge weight as where the α vw 's are non negative integers. We call the collection of α vw semigroup coefficients of d ve . It not hard to see that (1) is equivalent to where d v is the product of all edge weights adjacent to v.
From now on we will assume that Γ satisfy the semigroup condition, we then associate to each leaf w of Γ a variable z w , and have the following definitions.
Definition 4.2. Let v be a node of Γ and assume Γ has leaves w 1 , . . . , w n , then a v-weighting, or v-filtration, of [z w1 , . . . , z wn ] is to associate the weight l vwi to z wi . Definition 4.3. Let v be a node of Γ an e an adjacent edge, then an admissible monomial associated to v and e, is a monomial on the form M ve = w z αw w , where the product is taken over all leaves in Γ ve , and the α w 's is a choice of semigroup coefficients of d ve .
It is clear that each admissible monomial is v-weighted homogeneous, of total v-weight d v .
Definition 4.4. Let Γ with leaves w 1 , . . . , w n be a splice diagram satisfying the semigroup condition, then a set of splice diagram equations for Γ, is the following set of equations in the variables z w1 , . . . , z wn .
This defines n−2 equations in n variables, and the corresponding subscheme X(Γ) ⊂ n is called a splice diagram surface singularity.
We have the first important result concerning splice diagram equations, which is Theorem 2.6 of [NW05a].
Theorem 4.5. Let Γ be a splice diagram satisfy the semigroup condition, and let X = X(Γ) be an associated splice diagram surface singularity. Then X is a twodimensional complete intersection, with an isolated singularity at the origin. Example 5.1. Assume M is a graph manifold with following splice diagram

Plumbing and The Discriminant Group
P P P P P P P 3 n n n n n n n 4330 78 • 102 4755 n n n n n n n 5 P P P P P P P and assume that at the leaves named w 1 , w 2 and w 3 , we have orbifold curves of orbifold degrees 3, 2 and 5 respectively. Then M is going to have the following splice diagram If we want the reduced splice diagram of M , we just remove the leaf w 3 , and the the central vertex becomes a valence two vertex, and therefore has to be suppressed, and we get Let ∆(M ) be a plumbing diagram of M , then one can get a plumbing diagram ∆(M ) for M , by adding a arrow weighted with the orbifold degree at the corresponding vertex for each orbifold curve of M . By blowing up if necessary, we can assume that each vertex has at most one arrow attached. If v is a vertex of ∆(M ), then the orbifold degree o v of v is the orbifold degree of the arrow attached to v, if no arrow is attached to v then o v = 1. When ever we use that notation ∆(M ) and ∆(M ), we will assume that they are connected in this way.
Example 5.2. The graph orbifold M from Example 5.1 has the following plumbing Let E v ⊂ X be the curve where X is an analytic surface with ∂X = M corresponding to the vertex v ∈ ∆(M ) and E v the corresponding surface in X with ∂X = M . Then let and Now and are the same as -modules, but they have different intersection pairings defined by ∆(M ) and ∆(M ). Let A(M ) be the intersection matrix of in the basis given by E v . Then the intersection matrix for in the basis E v is given by the matrix A(M ) which is gotten from A(M ) by multiplying each column, which corresponds to a vertex v, with the orbifold degree o v of v as defined above.
One can construct Γ(M ) from ∆(M ) by suppressing all vertices of valence 2 in ∆(M ) to get the tree structure, and using the following propositions from [Ped10a] to get the decorations. In fact, each non-trivial element of D(M ) gives an element of (É/ ) n with at least two non-zero entries.
We will embed (É/ ) n into n via the following map

Congruence Condition for Graph Orbifolds
The plumbing diagram ∆ we use will be assumed to be quasi-minimal, this means that all weights on strings of ∆ have weights less that −1, unless the string consist of a single vertex with weight −1.
To any string We associate 1/0 to the empty string. We need the following standard facts about this relation ship which proofs are not hard and can be found many places.
Lemma 6.1. Reversing a string with continued fraction n/p gives one with continued fraction n/p ′ with pp ′ ≡ 1( mod n). Moreover the following relations holds: • , and the continued fraction in the last case is p ′ /n ′ with n ′ = (pp ′ − 1)/n.
For each n/p ∈ [1, ∞], there is a unique quasi minimal string, and in this case the continued fraction associated to the reverse direction is n/p ′ where p ′ is the unique number satisfying p ′ ≤ n and pp ′ ≡ 1( mod n). Proof. This follows from elementary linear algebra.
Lemma 6.4. For any v, v ′ ∈ vert(∆(M )), we have that Proof. This follows from Cramer's rule, i.e. if M ij is the i, j'th minor of A(M ) and M ij is the i, j'th minor of A(M ), then M ij is M ij with each column l multiplied by the corresponding orbifold degree o l , hence det(M ij ) = l =j o l det(M ij ) and Proof. e v · e v ′ = −l vv ′ d by Lemma 6.4 in [NW05a], but by the definition of M it follows that Proposition 6.6. If v is a node in Γ(M ) and w is a adjacent leaf and the continued fraction associated to the string in ∆(M ) (and ∆( M )) is given by (n w /o w )/p, where n w is the weight of the leaf and o w is the orbifold degree of the leaf. Let p ′ be the smallest positive integer such that p ′ p = 1 mod (n w /o w ). Let {n i } k i=1 be the other weights adjacent to v and let N = k i=1 n i . Then Proof. By using 6.3 and the formula for e w · e w given by proposition 6.6 in [NW05a] we get Corollary 6.7. The class of e w ′ , where w ′ is a leaf, transforms the monomial z αw w by multiplication by the root of unity We are now able to give formulas for checking the congruence condition.
Proposition 6.8. Let ∆ be an orbifold plumbing diagram which splice diagram Γ satisfy the semigroup condition. Then the orbifold congruence condition is equivalent to the following: for every node v and adjacent edge e, there is an admissible monomial M ve = z αw w where w is a leaf in Γ ve , so that for every leaf w ′ of Γ ve , Proof. The proof follows exactly as the prof given for the manifold case in Proposition 6.8 in [NW05a].
We will now look closer to how the congruence condition looks in the two node case. Let Γ be the splice diagram and let the orbifold degree corresponding to n ij be o ij . let ∆ be a plumbing diagram where we have made sure by blowing up that all the vertices v with o v = 1 have valence one. Then ∆ is going to look like Let p ′ ij be the unique positive integer satisfying p ij p ′ ij ≡ 1 mod (n ij /o ij ) and p ′ ij < (n ij /o ij ). Let N i = j n ij . Then the equations for the congruence condition becomes n li be the choice of admissible monomials. This equality is of course equivalent to where we use that nd = r 0 r 1 − N 0 N 1 by the edge determinant equation (Corollary 3.3 in [Ped10a]. Since n lj /o lj is an integer, the equation becomes equivalent to the following α lj o lj p ′ lj ≡ −n mod (n lj /o lj ).
Using the definition of p ′ ij gives us the following set of equations one need to check for the congruence to be satisfied α lj o lj ≡ −np lj mod (n lj /o lj ). Let M be the underlying manifold of M , then the splice diagram Γ = Γ(M ) is equal to Γ except that given an edge e ∈ Γ at a node v, then the edge weight d ve in Γ is given by d ve = dve w∈Γve ow where d ve is the edge weight in Γ. If {α w } is a set on M reducible semigroup coefficients then {α w }, where α w = α w /o w , is a set of semigroup coefficients for M since Proposition 7.2. Assume that {α w } satisfy the orbifold congruence condition for ∆(M ) then {α w } satisfy the congruence condition for ∆(M ).
Proof. Let v ∈ Γ be a node and w ′ ∈ Γ ve be a leaf, then .
Where we use that {α w } satisfy the orbifold congruence condition to get from the third line to the fourth.
Associate to each leaf w ∈ Γ a variable z w , for each node v ∈ Γ let {α vw } be a reducible choice of semigroup coefficients satisfying the orbifold congruence condition, and let α vw = αvw ow . Let Z ve = w∈Γve z αvw w and Z ve = w∈Γve z αvw w . Let t be the number of leaves of Γ and Let V be the subvariety of t defined by the equations where for all v the maximal minors of the ((δ v −2)×δ v )-matrix (a vie ) have maximal rank. Likewise let V be the subvariety of t defined by the equations Let X be a singularity which has resolution ∆(M ) and hence has link M . Since the equations for V satisfy the congruence conditions for ∆(M ), V defines the universal abelian cover of X branched over the origin by the work of Neumann and Wahl [NW05a]. Let π : V → X denote the covering map, then deg(π) = |H 1 (M )| = A priori π is just a continues map, we next turn to prove that π is an abelian orbifold cover.
Let K w ⊂ M be the singular fiber corresponding to the leaf w ∈ Γ. Let S be the singular set of M , then S = w:ow>1 K w . We want to determine π −1 (S). Proof. We both need to show that the exist open set U ⊂ Ê 3 and D ⊂ Ê 2 and a branched abelian coverπ : U → D × S 1 and a homomorphism ψ : /a → /α such that π is equivariant with respect to ψ and the following diagram commutes The vertical maps are the ones given from the orbifold structures of π −1 (N Kw ) and N Kw , i.e. the upper maps are quotient maps and lower maps homeomorphisms. We can choose D to be a disk, and by choosing it small enough (i.e. choosing N Kw small enough) we get that π −1 (N Kw ) is a disjoint union of solid torus neighborhoods V K of an singular fiber K, hence we can choose U to be a disjoint unions of D × S 1 . We now only need to see that the diagrams commute for each of these component.
Let t n = e 2πi/n , then /α acts on K is a solid torus neighborhood of a singular fiber of degree a ′ . Let α ′ = α/o w and β ′ = β/o w then the orbifold structure on N ′ K is given by Since π is an abelian orbifold cover there exist an homomorphism ψ ′ : /a ′ → /α ′ , and an abelian cover π ′ such that the following diagram commutes a ′ x, t a ′ s). Now since π is an abelian cover, it follows that α ′ = λa ′ for some λ and hence, ψ ′ (t ′ a ) = t λ α ′ . Now restricting F to V K one sees that F is a branched cover and if V K = D × S 1 then F (x, s) = (x ow , s) ∈ V ′ K = D × S 1 . Now the Seifert fibered structure on V ′ K is define by the curve of slope b ′ /a ′ , and it lifts to the curve of slope o w b ′ /a ′ and hence a = a ′ and b = o w b ′ and hence v a ′ ≡ o w v a mod a ′ .
We can now defineπ and ψ.π(x, s) = π ′ (x ow , s) and ψ(t a ) = t owλ α . So first we need to check thatπ is equivariant with respect to ψ So what is left is just checking that the diagram commutes. Start by taking (x, s) ∈ D × S 1 , then the one composition is takingπ(x, s) and send it to the class in (D × S 1 )/( /α ), and the other composition is sending (x, s) into the class in (D × S 1 )/( /a ) and then take π. If we denote the class in (D × S 1 )/( /k ) where the action is given by the integers k, l by [x, s] (k,l) , then we need to see that π([x, s] (a,b) ) = [π(x, s)] (α,β) . Now π([x, s] (a,b) ) = f (π(F ([x, s] (a,b) ))) by definition. Since the Seifert fibered structure on V K is given by pulling back the Seifert fibered structure on V ,β) , and the diagram commutes.
Last we need to calculate the degree of π| U . First the degree of π| VK : Combining the above results gives us the following theorem Proof. The above show that π : L(V ) → M ′ is an orbifold abelian cover of degree |H orb 1 (M )|, and hence the universal abelian cover of, M ′ , combining this with the second main theorem of [Ped10a] gives the result.
So to prove that the the splice diagram always define the universal abelian cover, one just have to show that given M with Γ(M ) satisfying the semigroup condition, there always exist a M ′ with a reducible set of admissible monomials satisfying the orbifold congruence condition such that Γ(M ′ ) = Γ(M ). We will in the next section show this is always true in the case of a splice diagram with only two nodes, by constructing such a M ′ from any two node splice diagram satisfying the semigroup condition.
8. Algorithm for construction an orbifold with a given two node splice diagram In this section we will make an algorithm which given any two node splice diagram Γ satisfying the ideal generator condition gives a graph orbifold M with Γ(M ) = Γ. We will construct M by giving a plumbing diagram ∆ such that ∆ = ∆(M ). We will not give a complete plumbing diagram, but specify the orbifold degrees, the weight at the nodes, and the continued fraction associated to the strings. From this data one can the obtain the complete plumbing diagram if needed. Let the splice diagram look like the following • . Let N j = i n ji , and let D be the edge determinant of the central edge. The plumbing diagram will be given by . . .

•
(o 0k 0 ) s s g g g g g g g • (o 1k 1 ) ) ) T T T T T T .
So we need to specify o ji , p ji and b j , from the information given by Γ.
• First chose integer α ji such that r 1−j = i α ji Nj nji , these integer exist since Γ satisfy the ideal generator condition. If Γ furthermore satisfy the semigroup condition, then the α ji 's can be chosen to be non negative, and the choice of α ji 's is a choice of semigroup coefficients for r 1−j .
• Let λ ji be the smallest integer, such that λ ji n ji ≥ ε j α ji if D > 0, and λ ji n ji ≥ −ε j α ji if D < 0 • Let o ji = gcd(n ji , α ji ). Notice that gcd(n ji /o ji , p ji ) = 1 so these choices gives a well-defined plumbing.
Proposition 8.1. Let M be the graph orbifold given by ∆ with the above choices, then Γ(M ) = Γ and |H orb 1 (M )| = D(e). Proof. Since the weight to the leaves in Γ(M ) is (n ji /o ji )o ji , it is the same weight as in Γ. Next we start by considering the case that D > 0, then the unnormalized edge determinant equation (Lemma 3.2 in [Ped10a]) implies that det(M ) > 0, so the only thing to check is thatr j , the weights associated to the central string in Γ(M ), is ε 1−j r j .
The case whit D < 0 is similar, but now det(M ) < 0 andr j = −ε 1−j r j . The last statement follows from the edge determinant equation (Corollary 3.3 in [Ped10a]), since the fiber intersection number of e is 1, or by using the above calculation to calculate det(∆).
Notice that M do depend on the choice of α ji 's.
Corollary 8.2. Let Γ be a splice diagram satisfying the semigroup condition. Then if M is a graph orbifold given by the above algorithm, then M satisfy the orbifold congruence condition.
Proof. We need to check that the equations α lj o lj ≡ −np ji mod (n ji /o ji ) given by (10) are satisfied. Now n = 1 so by definition Which implies that − α lj o lj ≡ p ji mod (n ji /o ji ), and hence the congruence condition is satisfied.
The edge determinant equation and the above mentioned condition implies that |H orb 1 (M )| = 30 if M is a graph orbifold realizing Γ ′ . Then using this and the edge determinant equation we can make a splice diagram for M v1e1 since we know that |H orb 1 (M v1e1 )| = 90. We can continue doing this until we get that there exists a graph manifold M ′ with |H 1 (M ′ )| = 60 and with the following splice diagram. • , we now this has to be manifold, since all the singular fibers has come from the process of creating M ve 's, and hence does not have orbifold curves. This means M ′ is a Seifert fibered manifold with Seifert invariants (1, −b), (6, β 1 )(21, β 2 ), (10, β 3 ), a simple calculation shows that such a Seifert fibered manifold can not have first homology group of order 60.
The failure of the last example is not as easy as the first to specify in a nice condition, so do at the moment not have a good idea on a set of necessary conditions for a splice diagram to be realized by graph orbifold. Even without this, it might still be possible to used Theorem 7.5 to prove it for more general graph orbifolds that just the once having two node splice diagram.
An other interesting question is, what are splice diagram singularities coming from diagrams as above.