The complexity of orientable graph manifolds


 We give an upper bound for the Matveev complexity of the whole class of closed connected orientable prime graph manifolds; this bound is sharp for all 14502 graph manifolds of the Recogniser catalogue (available at http://matlas.math.csu.ru/?page=search)



Introduction
Graph manifolds have been introduced and classified by Waldhausen in [13].They are defined as compact 3-manifolds that can be obtained by gluing Seifert fibre spaces along toric boundary components; so they can be described using a labeled digraph, as will be explained in the next section.
The notion of complexity for compact 3-dimensional manifolds has been introduced by Matveev in [10] (see also [11]) as a way to measure how "complicated" a manifold is.Indeed, for closed irreducible and P2 -irreducible manifolds the complexity coincides with the minimum number of tetrahedra needed to construct the manifold, with the only exceptions of S 3 , RP 3 and L(3, 1), all having complexity zero.Moreover, complexity is additive under connected sum and it is finite-to-one in the closed irreducible case.
The last property has been used in order to construct a census of manifolds according to increasing complexity: exact values of it are listed in the Recognizer catalogue 1 for the orientable case, up to complexity 12, and in the Regina catalogue 2 for the non-orientable case, up to complexity 11.
Upper bounds for the complexity of infinite families of 3-manifolds are given in [9] for lens spaces, in [8] for closed orientable Seifert fibre spaces and for orientable torus bundles over the circle, in [6] for orientable Sefert fibre space with boundary and in [2] for non-orientable compact Seifert fibre spaces.Very little is known for the complexity of graph manifolds: in [4] and [5] upper bounds are given only for the case of graph manifolds obtained by gluing along the boundary two or three Seifert fibre spaces with disk base space and at most two exceptional fibres.
The main goal of this paper is to furnish a potentially sharp upper bound for the complexity of all prime closed orientable graph manifolds different from Seifert fibre spaces and orientable torus bundles over the circle.It is worth noting that the upper bounds given in Theorems 1, 2 and 3 are sharp for all the 14502 manifolds of this type included in the Recognizer catalogue.
The organization of the paper is the following.In Section 2 we recall the definition of graph manifolds and their representation via decomposition graphs while in Section 3 we prove our main results.
We end this section by recalling the definition of complexity and skeletons.
A polyhedron P is said to be almost simple if the link of each point x ∈ P can be embedded into K 4 , the complete graph with four vertices.In particular, the polyhedron is called simple if the link is homeomorphic to either a circle, or a circle with a diameter, or K 4 .A true vertex of an (almost) simple polyhedron P is a point x ∈ P whose link is homeomorphic to K 4 .
A spine of a closed connected 3-manifold is a polyhedron P embedded in M such that M \ P ∼ = B3 , being B 3 an open 3-ball.The complexity c(M ) of M is the minimum number of true vertices among all almost simple spines of M .
We will construct a spine for a given graph manifold by gluing skeletons for its Seifert pieces.We recall their definitions from [7] and [8].
Consider a compact connected 3-manifold M whose boundary either is empty or consists of tori.A skeleton of M is a sub-polyhedron P of M such that (i) Given two manifolds M 1 and M 2 as above with non-empty boundary, let P i be a skeleton of M i for i = 1, 2. Take two components T 1 ⊆ ∂M 1 and T 2 ⊆ ∂M 2 such that P i ∩ T i = θ i and consider an homeomorphism ϕ : (T 1 , θ 1 ) → (T 2 , θ 2 ).Then P 1 ∪ ϕ P 2 is a skeleton for M 1 ∪ ϕ M 2 : we call this operation, as well as the manifold M 1 ∪ ϕ M 2 , an assembling of M 1 and M 2 .

Graph manifolds and their combinatorial description
Let us start by fixing some notations for Seifert fibre spaces.We will consider only orientable compact connected Seifert fibre spaces with nonempty boundary, described as S = (g, d, (p 1 , q 1 ), . . ., (p r , q r ), b) where: g ∈ Z coincides with the genus of the base space if it is orientable and with the opposite if it is non-orientable, d > 0 is the number of boundary components of S, (p j , q j ) are lexicographically ordered pairs of coprime integers such that 0 < q j < p j for j = 1, . . ., r describing the type of the exceptional fibres of S and b ∈ Z can be considered as a (non-exceptional) fibre of type (1, b).
Up to fibre-preserving homeomorphism, we can assume (see [3]) that the Seifert pieces appearing in a graph manifold belong to the set S of the orientable compact connected Seifert fibre spaces with non-empty boundary that are different from fibred solid tori and from the spaces S 1 × S 1 × I and N ×S 1 (i.e., the orientable circle bundle over the Moebius strip N , which will be considered with the alternative Seifert fibre structure (0, 1, (2, 1), (2, 1), b)).
Any Seifert fibre space S i = (g i , d i , (p 1 , q 1 ), . . ., (p r i , q r i ), b i ) ∈ S, with base space B i = p i (S i ), is equipped with a fixed global orientation and coordinate systems on the toric boundary components, as follows.
Let Si be the closed Seifert fibre space Si = (g i , 0, (p 1 , q 1 ), . . ., (p r i , q r i ), b i ), with base space Bi = pi ( Si ).Moreover, let si : B i → S i be the section naturally associated to Si , up to isotopy, where B i = pi ( S i ) and S i is obtained from Si by removing r i + 1 open fibred solid tori which are regular neighborhoods of the exceptional fibres of Si and of the fibre of type (1, b).Then S i is obtained from Si by removing d i open fibred solid torus which are regular neighborhoods of d i regular fibres of type (1, 0) of Si .Therefore, p i = pi| S i and ∂(B i ) ⊂ B i .The section si induces a coordinate system (i.e., a positive basis of the first homology group) (µ k , λ k ) on the k-th toric boundary component T k of S i such that: µ k is the image under si of a boundary component c k of B i and λ k is a fibre of S i .Moreover, µ k and λ k are oriented in such a way that their intersection index is 1, with the orientation of T k induced by the one of S i .When B i is orientable, we require that any µ k is oriented according to the orientation of c k induced by a fixed orientation on B i .
Consider a finite connected digraph G = (V, E, ι), where V = {v i | i ∈ I} is the set of vertices and E = {e j | j ∈ J} is the set of oriented edges of G, with incidence structure ι : E → V × V given by ι(e j ) = (v i j , v i j ).Let H = 0 1 1 0 , then associate: • to each vertex v i ∈ V having degree d i a Seifert fibre space S i = (g i , d i , (p 1 , q 1 ), . . ., (p r i , q r i ), b i ) ∈ S (i.e., the degree of v i is equal to the number of components of ∂S i ); • to each edge e j ∈ E a matrix A j = α j β j γ j δ j ∈ GL − 2 (Z) such that β j = 0 and 0 ≤ j α j , j δ j < |β j |, where j = β j /|β j |.We call normalized a matrix of GL − 2 (Z) satisfying these conditions.Moreover: (i) A j = ±H when either S i j or S i j is the space (0, 1, (2, 1), (2, 1), −1); The graph manifold M associated to the above data is obtained by gluing, for each edge e j ∈ E with ι(e j ) = (v i j , v i j ), a toric boundary component of S i j with one of S i j , using the homeomorphism represented by A j with respect to the fixed coordinate systems on the tori. 4Clearly, M is a closed, orientable and connected 3-manifold.On the contrary each prime closed orientable connected graph manifold different from a Seifert fibre space and an orientable torus bundle over the circle can be obtained in this way (see [3, §11]).We will call G a decomposition graph of M and given a graph G, we denote with M G the graph manifolds associated to G. Note that all decomposition graphs are non-trivial (i.e., |E| ≥ 1) since the Seifert fibre spaces in S have non-empty boundary.
Remark 1.There is no restriction in assuming that all the matrices associated to the edges of a decomposition graph are normalized: this is because of the following two operations that do not change the resulting graph manifold (see [3, §11] and [12]): 1) replacement of the matrix A j with A j = A j • 1 0 k 1 and of the parameter b i j of the Seifert space S i j with the parameter b i j + k; 2) replacement of the matrix A j with A j = 1 0 k 1 • A j and of the parameter b i j of the Seifert space S i j with b i j − k.

Indeed, given a matrix
β where x denotes the floor of x.Then the matrix ing properties hold:

Theta graphs and Farey triangulation
Consider the upper half-plane model of the hyperbolic plane H 2 and let F be the ideal Farey triangulation (see [1]).The vertices of F coincide with the points of Q ∪ {∞} ⊂ R ∪ {∞} = ∂H 2 and the edges of F are geodesics in H 2 with endpoints the pairs a/b, c/d such that ad f be the triangle of the Farey triangulation with vertices a/b, c/d, e/f ∈ Q ∪ {∞} and set Let T 2 be a torus, it is a well-known fact that the vertex set of F is in bijection with the set S(T ) of slopes (i.e., isotopy classes of non-contractible simple closed curves) on T 2 via a/b ↔ aµ + bλ, where (µ, λ) is a fixed basis of H 1 (T 2 ).This bijection induces a bijection between the set of triangles of the Farey triangulation and the set Θ(T 2 ) of non-trivial theta-graphs on T , up to isotopy.Indeed, given θ ∈ Θ(T ), consider the three slopes l 1 , l 2 , l 3 on T 2 formed by the pairs of edges of θ.The triangle associated to θ is ∆ l 1 ,l 2 ,l 3 .
Note that this bijection is well defined since the intersection index of l i and l j , with i = j, is always ±1.
The graph F * dual to F is a tree and will be called Farey tree.Given two triangles ∆ and ∆ in F the distance d(∆, ∆ ) between them is the number of edges of the unique simple path joining the vertices v ∆ and v ∆ corresponding to ∆ and ∆ in F * , respectively.Given two theta graphs θ, θ it is possible to pass from one to the other by a sequence of flip moves (see Figure 1): the distance on the set of triangles of the Farey triangulation induces a distance on Θ(T 2 ) such that d(θ, θ ) is the minimal number of flips necessary to pass from θ to θ (see [8]).We define the complexity c A of a matrix A ∈ GL 2 (Z) as We end this section by stating a lemma about the complexity of normalized matrices.Let S : is the expansion of the positive rational number a/b as a continued fraction, with a 1 , . . ., a k > 0.
Proof.The first statement is straightforward since Indeed, since αδ − βγ = −1, we have: So, where we suppose δ − γ = 0, otherwise the last inequality is straightforward.
. Referring to Figure 2, where for convenience we use the Poincaré disk model of H 2 , all the triangles of T β/δ different from A∆ − (and A∆ + ) are contained in the two hyperbolic half-planes depicted in gray.As a consequence we have min{d(∆, We end this section by introducing a notation to distinguish edges corresponding to ±H from the others.Given a decomposition graph (V, E, ι)

A complexity upper bound
In this section we provide an upper bound for the complexity of graph manifolds.Before stating the general result (Theorem 3), we deal with two special classes of graph manifolds: the first one is the class of manifolds having a decomposition graph in which all edges are associated to matrices different from ±H, that is J = ∅ (Theorem 1), while the second one concerns manifolds whose decomposition graph admits a spanning tree containing all the edges associated to matrices of type ±H (Theorem 2).
In all cases, the upper bound is obtained by assembling skeletons for the Seifert pieces in order to obtain a spine of the graph manifold.The construction of the skeleton for the Seifert pieces is essentially the one described in [2], specialized to our case (i.e., orientable Seifert manifolds) and adapted to take care of the fact that the boundary components of the Seifert pieces will be glued together to obtain a closed graph manifold.For the sake of the reader we recall how to construct a skeleton for S = (g, d, (p 1 , q 1 ), . . ., (p r , q r ), b) ∈ S. The construction and the number of true vertices of the resulting skeletons depend on some choices: we will discuss in the proof of theorems how to fix them in order to minimize the number of true vertices of the spine.
where Φ k (resp.Φ 0 ) is a closed solid torus having the k-th exceptional fibre (resp.a regular fibre) as core.Let p 0 : S 0 → B 0 be the projection map and set s = d + r.Note that if g ≥ 0 (resp.g < 0) then B = p 0 (S ) is a disk with 2g + s orientable (resp.−g non-orientable and s orientable) handles attached.We set h = 2g if g ≥ 0 and h = −g if g < 0. We are going to construct a skeleton P S for S, starting from a skeleton P S of S ∪ Φ 0 .
Let D = p 0 (Φ 0 ) and let A 0 be the union of the disjoint arcs properly embedded in B depicted by thick lines in Figure 3. Then A 0 is non-empty and is composed by h edges with both endpoints in ∂D and s edges with an endpoint in ∂D and the other in another component of ∂B .By construction B \ (A 0 ∪ ∂B ) is homeomorphic to an open disk and the number of points of A 0 belonging to ∂D is at least three, since the conditions on the class S ensure that d + r + 2h > 2.

∂D ∂B
Moreover, if b = 0, the manifold S is obtained from S 0 by removing |b| disjoint trivially fibred solid torus Φ 1 , . . ., Φ |b| each being a fiber-neighborhood of regular fibres φ 1 , . . .φ |b| , and by attaching back a solid torus D 2 × S 1 via a homeomorphism ψ l : ∂(D 2 × S 1 ) → ∂Φ l such that ψ l (∂D 2 × { * }) is a curve of type (1, sign(b)) on ∂Φ l , with respect to the basis fixed on ∂Φ l (see Section 2), for l = 1, . . ., |b|.It is convenient to take the fibre φ l corresponding to an internal point Q l of A 0 and suppose that p 0 (Φ l ) is a "small" disk intersecting the component δ l of A 0 containing Q l in an interval and being disjoint from ∂B and from the other components of A 0 (see Figure 4).In this way δ l \ int(p 0 (Φ l )) is the disjoint union of two arcs δ l and δ l .Let , then the polyhedron ) is a nonsimple polyhedron since, as represented in the central picture of Figure 5, the set int(s (A)) is a collection of quadruple lines in the polyhedron (the link of each point is homeomorphic to a graph with two vertices and four edges connecting them), and a similar phenomenon occurs for s (∂D \ A).In order to make the polyhedron P simple we perform "small" shifts by moving in parallel the disk s (D) along the fibration and the components of p −1 (A) as depicted in left and right pictures of Figure 5.As shown by the pictures, the shift of any component of p −1 (A) may be performed in two different ways that are not usually equivalent in term of complexity of the final spine.On the contrary, the two possible parallel shifts for s (D) are equivalent as it is evident from Figure 6, which represents the torus T 0 = ∂Φ 0 = p −1 (∂D).It is convenient to think the shifts of p −1 (A) as performed on the components of A. Moreover, the shifts on δ l and δ l can be chosen independently (see Figure 4).
)) be the polyhedron obtained from P after the shifts, where D and W are the results of the shifts of s (D) and p −1 (A), respectively.It is easy to see that P ∪ ∂S is simple and P intersects each component of ∂S different from p −1 (∂D) = ∂Φ 0 in a non-trivial theta graph.Since S ∪ Φ 0 \ (P ∪ ∂S ) is the disjoint union of |b| + 2 open balls, in order to obtain a skeleton P S for S ∪ Φ 0 it is enough to remove a suitable open 2-cell from the torus T 0 = p −1 (∂D) = ∂Φ 0 ⊂ P and one on each torus T l = ∂Φ l , for l = 1, . . ., |b|, connecting in this way the balls.In all cases Γ m = T m ∩ (s (B ) ∪ D ∪ W ) is a graph cellularly embedded in T m whose vertices are true vertices of P : we will remove the region R m of T m \ Γ m having in the boundary the highest number of vertices of Γ m , for m = 0, 1, . . ., |b|.Referring to Figure 6, the graph Γ 0 is composed by two horizontal parallel loops ξ = ∂(s (D)) and ξ = ∂D , and an arc with both endpoints on ξ for each boundary point of A belonging to ∂D.Changing the shift of a component of A has the same effect as performing a symmetry along ξ of the correspondent arc(s).Clearly, if the shift is applied to a component of A which is the cocore of a handle, both arcs corresponding to the endpoints change as just described.Moreover, given an orientable handle having the cocore with both endpoints in ∂D, the two arcs (which are not consecutive in Γ 0 , as suggested by the dots in the pictures) corresponding to these endpoints are as in the picture or in the mirrored ones with respect to ξ.If the handle is non-orientable, the rightmost arc in each picture has to be symmetrized with respect to the loop ξ.A region of T 0 \ Γ 0 has 5 vertices when the arcs belonging to its boundary are parallel and either 4 or 6 vertices otherwise.
When b = 0, the graph Γ l , for l = 1, . . ., |b|, is depicted in Figure 7 (resp.Figure 8) for a fibre of type (1, 1) (resp.(1, −1)), just labeled with + (resp.−) inside the disk.If we take for δ l and δ l the shifts induced by the one of δ l , then we can choose as region R l the gray one containing in its boundary all the true vertices of P belonging to ∂Φ l except one (see the first two pictures).On the contrary, if one of the two shifts is changed as in the third draw of Figures 7 and 8, then R l can be chosen containing in its boundary all true vertices of P belonging to ∂Φ l .We remark that changing the shift of a component of A changes the intersection between the corresponding element of W and ∂S (which is a non-trivial theta graph) by a flip move (see bottom and top face of the block of Figure 9).
In order to construct a skeleton for Φ k , for k = 1, . . ., r, consider the skeleton P F depicted in Figure 9: it is a skeleton for T 2 × [0, 1] with one true vertex and such that θ 0 = P F ∩ T 2 × {0} (the graph in the upper face) and θ 1 = P F ∩ T 2 × {1} (the graph in the bottom face) are two theta graphs differing for a flip move.Denote with θ p j /q j the closest theta graph to θ + between those containing the slope p j /q j .The skeleton X k for Φ k is obtained by assembling several flip skeletons connecting the theta graph P S ∩ Φ k to a theta graph which is one step closer to θ + than θ p j /q j ,with respect to the distance on Θ(T 2 ) (see [6]).The number of the required flips is either S(p j , q j ) − 2 or S(p j , q j ) − 1 depending on the shift chosen for the corresponding component of A used in the construction of the skeleton of S ∪ Φ 0 .We call the shift of a component regular in the first case and singular in the second one (see Figure 10, where the shifted arcs are denoted by dotted lines).
The skeleton P S of S is obtained by assembling P S with X k , via the identity, for k = 1, . . ., r.

The case of decomposition graphs with J = ∅
Before stating our first result, we need to discuss how to fix the choices in the construction of the skeleton P S previously described, when the Seifert fibre space S = (g, d, (p 1 , q 1 ), . . ., (p r , q r ), b) is a piece of a graph manifold having gluing matrices all different from ±H.According to the notations introduced at the end of Section 2, since J = ∅, then d = d + + d − .Remark 2. Let θ + and θ − be the theta graphs corresponding, respectively, to ∆ + and ∆ − in the Farey triangulation.The intersection of each boundary   11: they project to components of A 0 corresponding to handles having the cocore with both endpoints in ∂D, exceptional fibres and d − boundary components.In this way: (i) we can remove a cell from T 0 containing 6 true vertices (ii) we can remove from T l a region R l containing in its boundary all true vertices of P belonging to ∂Φ l (as in the third draw of Figure 8) for each l = 1, . . ., |b| and (iii) we can take all regular shifts in the skeletons X k corresponding to the exceptional fibres for k = 1, . . ., r.If b = 0 we do not have to remove any tubular neighborhood Φ l of regular fibres but still (i) and (iii) hold.An analogue situation happens if b ≥ 1, but in this case in order to satisfy (i), (ii) and (iii) the fibres of type (1, 1) correspond just to handles having the cocore with both endpoints ∂D, eand d + boundary components (see Figure 12).As a result, when m ≤ b ≤ M the skeleton  Let f m,M : Z → N be the function defined by for m, M ∈ Z, m < M , m ≤ 1 and M ≥ −1 (see the graph in Figure 13).
Theorem 1.Let M be a graph manifold associated to a decomposition graph Proof.Let T = (V, E T ) be a spanning tree of G, set J T = {j ∈ J | e j ∈ E T } and denote with M T the graph manifold (with boundary if T = G) having decomposition graph T .We will construct a skeleton P M T for M T by assembling, via A j , the skeletons P S i for S i with the skeletons P A j for the thickened torus T 2 × I, for each i ∈ I and j ∈ J T .Let θ j be the theta graph corresponding to A j ∆ − .We construct the skeleton P A j by assembling flip blocks (see Figure 9) in such a way that (i) P A j ∩ T 2 × {0} = θ j and (ii) P A j ∩ T 2 × {1} = θ + .By Lemma 1 we have c A j = d(A j ∆ − , ∆ + ) = S(β j /δ j ) − 1, so the number of flip blocks required to construct P A j , as well as the number of true vertices of P A j , is S(β j /δ j ) − 1.
By Remark 2, we can construct the skeleton P S i with a number of true vertices equal to 3( For each e j ∈ E\E T , the gluing corresponding to A j is a self-gluing of M T , that identifies two boundary components.Denote with M T ∪e j the resulting manifold (having decomposition graph (V, E T ∪ {e j })).If we take a skeleton P A j for T 2 ×I as described above, the polyhedron P M T ∪ A j P A j is not a skeleton for the resulting manifold M T ∪e j since M T ∪e j \ (P M T ∪ A j P A j ∪ ∂M T ∪e j ) is an open solid torus; so we have to add to P M T ∪ A j P A j the torus T 2 × {1}.Moreover, we construct P A j in such a way that (i) P A j ∩ T 2 × {0} = θ j and (ii) P A j ∩ T 2 × {1} = θ , where θ in the Farey tree is one step closer to θ + than θ j .Clearly the graphs θ and θ + differ for a flip, since they correspond to adjacent triangles ans so they have two intersection points (see Figure 14).As a consequence, the graph θ + ∪ θ has 6 vertices of degree greater than 2, each corresponding to a true vertex in the skeleton P M T ∪ A j P A j ∪ T 2 × {1}.Moreover the skeleton P A j consists of S(β j /δ j ) − 2 flip blocks, so the new skeleton has 5 + S(β j /δ j ) − 1 true vertices more than P M T .By repeating this construction for each e j ∈ E \ E T and observing that |E \ E T | = |E| − |V | + 1, we get the statement.
Figure 14: The two intersections of two theta graphs differing for a flip move.

3.2
The case of decomposition graphs with a tree containing the elements of J In this subsection we deal with the class of graph manifolds having a decomposition graph admitting a tree containing all the edges associated to the matrices ±H.Up to complexity 12, almost the 99% of graph manifolds belong to this class (they are exactly 14346 over 14502).
Before stating our result we introduce some notations.If J = ∅ let Ψ = {ψ : J → {+, −}} and set: Theorem 2. Let M be a graph manifold associated to a decomposition graph G = (V, E, ι).If there exists a spanning tree T = (V, E T ) of G such that e j ∈ E T for any j ∈ J , then where Proof.As in the proof of Theorem 1, we start by constructing a skeleton P M T for the graph manifold (with boundary) M T having decomposition graph T .By Lemma 1 we have 0 = c ±H = d(±H∆ − , ∆ − ) = d(±H∆ + , ∆ + ), so whenever j ∈ J , no flip block is required in P A j and we can assemble directly P S i j with P S i j .So, if T j and T j denote, respectively, the boundary components of S i j and S i j that are glued by A j , we require that either (i) P S i j ∩ T j = θ + and P S i j ∩ T j = θ + or (ii) P S i j ∩ T j = θ − and P S i j ∩ T j = θ − .
In order to take care of these two possibilities we use a "coloring" ψ : J → {+, −} of the edges of E .If ψ(j) = − (resp.ψ(j) = +) then the shifts in the construction of P S i j and P S i j are chosen so that (i) holds (resp.(ii) holds).Following the construction of Remark 2, with Moreover, when j ∈ J we have A j = ±H and so c A j = 0.As a consequence, the number of true vertices of the skeleton of M T is j∈J (S(β j /δ j ) − 1) + i∈I (3(d . Since all matrices associated to edges e j / ∈ E T are different from ±H, starting from the skeleton of M T we can construct a spine for M as described in the proof of Theorem 1.This concludes the proof.

The general case
For the general case we need to consider a certain set of spanning trees of G. Let T G = {T = (V, E T ) G | T is a spanning tree of G} and for We are now ready to state the general case.where Given ψ ∈ Ψ T , we construct a skeleton P M T for M T as described in the proof of Theorem 2. Let j ∈ J \ J T then A j = ±H and glues together two toric boundary components T j ⊂ ∂S i j and T j ⊂ ∂S i j of M T .Therefore, as in the proof of Theorem 1, to construct a spine for M , we must add to P M T the torus T 2 × {1} = T j of P ±H .Let (P S i j ∩ T j , P S i j ∩ T j ) = (θ j , θ j ).If ψ (j) = −+ (resp.ψ (j) = +−) then we choose the shifts in S i j and S i j so that (±Hθ j , θ j ) = (θ + , θ − ) (resp.(±Hθ j , θ j ) = (θ − , θ + )).Let U = 1 0 1 1 .If ψ (j) = ++ (resp.ψ (j) = +) we take (θ j , θ j ) = (θ − , θ − ) and replace ±H with ±HU −1 (resp.±U H) so that, according to Remark 1, b i j (resp.b i j ) is replaced with b i j − 1 (resp.b i j − 1).Finally, if ψ (j) = −− (resp.ψ (j) = −) we take (θ j , θ j ) = (θ + , θ + ) and replace ±H with ±HU (resp.±U −1 H) so that, according to Remark 1, b i j (resp.b i j ) is replaced with b i j + 1 (resp.b i j + 1).

Figure 1 :
Figure 1: Two theta graphs connected by a flip move.

Figure 5 :
Figure 5: The two possible shifts on a component of p −1 (A).

Figure 7 :
Figure 7: The graph Γ l , with b = 1, embedded in the torus T l = ∂Φ r l with different choices of the shifts for δ l and δ l .

Figure 8 :
Figure 8: The graph Γ l , with b = embedded in the torus T l = ∂Φ r l with different choices of the shifts for δ l and δ l .

Figure 9 :
Figure 9: A skeleton for T × [0, 1] connecting two theta graphs differing by a flip move.

Figure 10 :
Figure 10: Regular shift (on the left) and singular shift (on the right).
and (iii) hold and there are exactly m − b (resp.b − M ) torus in which we remove a region R l containing in its boundary all the true vertices of P belonging to ∂Φ l except one (see the first two pictures of Figure 7 and 8).Finally, if b < m = 1 (resp.b > M = −1) then (i) does not hold so we remove a region from T 0 containing 5 true vertices, there are exactly m − b − 1 (resp.b − M − 1) torus in which we remove a region R l containing in its boundary all the true vertices of P belonging to ∂Φ l except one and (iii) holds.So if b < m (resp.b > M ) then the number of true vertices of P S increases by m − b (resp.b − M ).