Positive flowspines and contact manifolds
Abstract.
We say that a contact structure on a closed, connected, oriented, smooth manifold is supported by a flowspine if it has a contact form whose Reeb flow is a flow of the flowspine. We then define a map from the set of positive flowspines to the set of contact manifolds up to contactomorphism by sending a positive flowspine to the supported contact manifold and show that this map is welldefined and surjective. We also determine the contact manifolds supported by positive flowspines with up to vertices. As an application, we introduce the complexity for contact manifolds and determine the contact manifolds with complexity up to .
Contents
1. Introduction
Triangulation is one of the basic tools to analyze topological manifolds and, in dimensional manifold theory, it is often used to understand geometric and combinatorial structures like ideal triangulations in hyperbolic geometry, normal surfaces for decomposition of manifolds and quantum invariants based on the symbols. The dual of a triangulation is a polyhedron with simple vertices. It also has many applications such as branched surfaces for studying laminations and foliations and effective calculus of various invariants. A spine is a dimensional polyhedron embedded in a closed manifold obtained from the manifold by removing an open ball and collapsing it from the boundary. For instance, the skeleton of the dual of a onevertex triangulation is a spine. There are several useful aspects of spines. One is the tabulation of manifolds using the complexity of almostsimple spines due to Matveev [41, 42]. Tabulation is a kind of classification and plays an important role as a treasury of concrete examples that lead us to deep understanding of research object, like the knot table in knot theory. Various studies on the Matveev complexity are ongoing as studies of those of lens spaces and certain hyperbolic manifolds [1, 15, 29, 30, 31, 32, 47, 54, 55].
Another aspect of spines can be found in the study of nonsingular flows in manifolds. For a given nonsingular flow, set a disk transverse to the flow and intersecting all orbits, and then float the boundary of the disk smoothly until it arrives in the disk itself. The object obtained in this way is a spine equipped with a special structure, called a flowspine. A flowspine was introduced by the first author in [26] and developed further in [27, 11]. This is very useful when one studies nonsingular flow combinatorially since any nonsingular flow has a flowspine [26]. A spine is described by a decorated trivalent graph on called the DSdiagram, that was introduced in a paper of Ikeda and Inoue [25]. The DS is an abbreviation of DehnSeifert. A flowspine is described by a DSdiagram with a specified simple loop called an cycle. Flowspines are also studied by Benedetti and Petronio in [2] in the context of branched standard spines.
A contact structure is a hyperplane distribution that is nonintegrable everywhere. In this paper, all 3manifolds are oriented, and contact structures are always assumed to be positive, that is, a contact form satisfies with respect to the orientation on the ambient 3manifold. Flexibility of contact structures allows us to study them combinatorially, especially in the dimensional case. Martinet and Lutz developed Dehn surgery technique for contact manifolds [40, 39], and Thurston and Winkelnkemper related contact structures to open book decompositions of manifolds [52]. Tightness and fillability were introduced and studied by Bennequin [4], Eliashberg [7, 8, 9], Gromov [21] and many other mathematicians. Great progress had been made by Giroux. There are two big directions in his works on dimensional contact topology: one is convex surface theory [17] and the other is the socalled Giroux correspondence [19]. Convex surface theory allows us to use cutandpaste methods for studying contact manifolds more flexibly. This technique is mainly used when one studies classification of tight contact structures on manifolds. The most famous application is the existence of a closed, orientable manifold that does not admit a tight contact structure due to Etnyre and Honda [14]. The Giroux correspondence gives us an easy way to “see” contact structures via open book decompositions, and it is especially useful when one wants to show Stein fillability of contact structures given in terms of open books. Branched surfaces are sometimes used in the study of contact structures, see for instance [3, 6].
The aim of this paper is to relate triangulation of manifolds with contact structures via flowspines, by regarding Reeb flows as flows of flowspines. This idea had appeared in the paper of Benedetti and Petronio [3]. They focused on the characteristic foliation on a branched standard spine embedded in a contact manifold and studied the contact structure using techniques in convex surface theory of Giroux. The point is that they did not use Reeb vector fields so much since convex surface theory uses rather contact vector fields. In this paper, we focus on Reeb flows more and define the correspondence between flowspines and contact structures as follows:
Definition.

A flow is said to be carried by a flowspine if is a flow of .

A contact structure on is said to be supported by a flowspine if there exists a contact form on such that and its Reeb flow is carried by .
To our aim, we need to introduce a notion of positivity for flowspines. Each region of a flowspine is canonically oriented since it is transverse to the flow. This structure is called a branching. A branched simple polyhedron has two kinds of vertices: the vertex on the left in Figure 1 is said to be of type and the one on the right is of type. We define a flowspine to be positive if it has at least one vertex and all vertices are of type^{1}^{1}1 In Section 3.3, two flowspines and are obtained from a normal pair . In [26], is called a positive flowspine and is a negative one. The positivity for flowspines introduced in this paper is different from the one for introduced in [26].. An essential reason for concerning this condition is that we cannot expect one of the existence and the uniqueness of a contact manifold for a given flowspine without this condition, see Theorem 4.3. Another reason, which is more technical, is that a form called a reference form, which plays a key role in the proof of Theorem 1.1, cannot be defined unless we restrict flowspines to positive ones.
With these observations, we restrict our attention to positive flowspines. Our main theorems are the following.
Theorem 1.1.
For any positive flowspine of a closed, oriented manifold , there exists a unique contact structure supported by up to isotopy.
Recall that two contact structures and are said to be isotopic if there exists a oneparameter family of contact forms connecting two contact forms and whose kernel are and , respectively. In particular, and are contactomorphic by Gray’s stability [20].
Theorem 1.2.
For any closed, oriented contact manifold , there exists a positive flowspine of that supports .
These theorems show the existence of the surjection
for any closed, connected, oriented, smooth manifold and the surjection
(1.1) 
To get the onetoone correspondence, we need to find suitable moves of positive flowspines. Moves of flowspines are known in [27, 2, 11]. However moves of positive flowspines have never been studied. Such moves are very rare since most known moves for flowspines yield type vertices. Theorem 1.2 implies that any flowspine can be deformed to a positive one by regular moves, see Corollary 8.3.
To prove Theorem 1.1, we first give a reference form on the manifold with respect to the flowspine explicitly and then consider a contact form on of the form , where is a form defined by extending a form on with to the complement and is a sufficiently large positive real number. This is analogous to the contact form for open books used by Thurston and Winkelnkemper and then in the Giroux correspondence, where is the monodromy diffeomorphism and is the parameter of pages of the open book. The discussion in the open book case is easier because is closed. In the case of flowspine, although is not closed, it satisfies , that is, it gives a confoliation [10]. We use the positivity of flowspines to get this property. Then Theorem 1.1 follows by the same strategy as the Giroux correspondence though the argument is much more complicated. Theorem 1.2 is proved by giving a positive flowspine explicitly. In the proof, we first use the existence of an open book for a given contact manifold, due to Giroux, and then make a positive flowspine from the open book. The proof is done by showing that the contact structure supported by this flowspine is isotopic to the original one.
The second surjection (1.1) obtained from our main theorems allows us to define the complexity for contact manifolds as the Matveev complexity for usual manifolds. That is, we define the complexity of a contact manifold to be the minimum number of vertices of positive flowspines supporting , where is a closed, connected, oriented manifold and is a contact structure on . In the final part of this paper we determine the supported contact structures of several positive flowspines. To determine them, we read off the Seifert fibrations from their DSdiagrams and use branched cover technique and specific Dehn surgeries called coil surgeries. For example, by considering branched covers, we may determine that the complexity of the quaternion space with canonical contact structure is and that of the Poincaré homology sphere with the canonical contact structure is . Here the canonical contact structure means the contact structure of the link of a complex hypersurface singularity given by its complex tangency, which is uniquely determined by [5]. In particular, we determine the supported contact structures for all positive flowspines with up to vertices and obtain the following tabulation.
Theorem 1.3.
The contact manifolds with complexity up to are determined as follows:

if and only if ;

if and only if or ;

if and only if , , or the quaternion space with the canonical contact structure.
Here denote the standard contact structure on and a tight contact structure.
All the contact manifolds with complexity up to are tight. In some sense, this means that tight contact manifolds have smaller complexities than overtwisted ones.
This paper is organized as follows: In Section 2, we shortly recall known results in dimensional contact topology that will be used in this paper. In Section 3, we introduce flowspines and DSdiagrams. New observation starts from Section 4, where we introduce the admissibility condition and the notion of positivity for flowspines. In Section 5, we introduce the reference form for a positive flowspine that plays an important role in the proof of Theorem 1.1. The proof of the existence of the supported contact structure in Theorem 1.1 is given in Section 6 and the proof of its uniqueness is given in Section 7. Section 8 is devoted to the proof of Theorem 1.2. In Section 9, we introduce several techniques to determine the supported contact structures for given positive flowspines and prove Theorem 1.3. We close the section with a few questions concerning dynamics of flows carried by positive flowspines. The list of positive flowspines with up to vertices is given in the appendix.
The authors wish to express their gratitude to Riccardo Benedetti for many insightful comments. The second author would like to thank Shin Handa and Atsushi Ichikawa for useful discussions in the early stages of research. The second author is supported by JSPS KAKENHI Grant Numbers JP19K03499, JP17H06128 and Keio University Academic Development Funds for Individual Research. The third author is supported by JSPS KAKENHI Grant Numbers JP15H03620, JP17K05254, JP17H06463, and JST CREST Grant Number JPMJCR17J4. The fourth author is supported by JSPS KAKENHI Grant Number JP19K21019.
2. Contact manifolds
Throughout this paper, for a polyhedral space , represents the interior of , represents the boundary of , and represents a closed regular neighborhood of a subspace of in , where is equipped with the natural PL structure if is a smooth manifold. The set is the interior of in .
In this section, we briefly recall notions and known results in dimensional contact topology that will be used in this paper. The reader may find general explanation, for instance, in [13, 16, 44].
Let be a closed, oriented, smooth manifold. A contact structure on is the plane field on given by the kernel of a form on satisfying everywhere. The form is called a contact form. If everywhere on then the contact structure given by is called a positive contact structure and the form is called a positive contact form. The pair of a closed, oriented, smooth manifold and a contact structure on is called a contact manifold and denoted by . In this paper, by a contact structure we mean a positive one.
There are two ways of classification of contact manifolds: up to isotopy and up to contactomorphism. Two contact structures and on are said to be isotopic if there exists a oneparameter family of contact forms , , such that and . Two contact manifolds and are said to be contactomorphic if there exists a diffeomorphism such that . The map is called a contactomorphism. If then we also say that and are contactomorphic. The Gray theorem states that if two contact structures are isotopic then they are contactomorphic [20].
The contact structure on given as the kernel of the form is called the standard contact structure on , where are the standard coordinates of and is the unit sphere in . We also say that is the plane field given by the complex tangency at each point of in . This contact structure satisfies the following important property, called “tightness”, which was shown by Bennequin [4]. A contact structure on is said to be tight if there does not exist a disk embedded in such that is everywhere tangent to and the framing of along coincides with that of . Otherwise is said to be overtwisted and the disk is called an overtwisted disk. Note that the tightness is an invariant of contact manifolds up to contactomorphism. Due to Eliashberg, it is shown that the classification of overtwisted contact structures on up to isotopy is equivalent to the classification of homotopy classes of plane fields on [7]. Therefore, in dimensional contact topology, determining tightness of contact manifolds and classifying tight contact manifolds become crucial problems.
It is shown by Eliashberg [9] that , and admit the unique tight contact structure. The tight contact structures on the lens space are classified by Giroux [18] and Honda [22]. The number of isotopy classes of tight contact structures of is determined by and . For convenience, we write the numbers for first few lens spaces in Table 1 below. The involution of gives a contactomorphism between the contact structures corresponding to the euler classes and , see [12, 22]. The numbers of contactomorphism classes are also written in the table.
manifold  

isotopy  
contactom. 
Next we introduce the Reeb vector field. Let be a contact form on . A vector field on determined by the conditions and is called the Reeb vector field of on . Such a vector field is uniquely determined by and we denote it by . The nonsingular flow on a manifold generated by a Reeb vector field is called a Reeb flow.
The Reeb vector field plays important roles in many studies in contact geometry and topology. In dimensional contact topology, it is used to give a correspondence between contact structures and open book decompositions of manifolds. Let be an oriented, compact surface with boundary and be a diffeomorphism such that is the identity map on . If a closed, oriented manifold is orientationpreservingly homeomorphic to the quotient space obtained from by the identification for and for each and any , then we say that it is an open book decomposition of . The image of in by the quotient map is called the binding, which equips the orientation as the boundary of . The image of the surface in is called a page. We denote the open book by .
A contact structure on is said to be supported by an open book if there exists a contact form on such that and the Reeb vector field of satisfies that

is tangent to and the orientation on induced from coincides with the direction of , and

is positively transverse to for any .
Any open book has a supported contact structure and this is used by Thurston and Winkelnkemper to prove that any closed, oriented, smooth manifold admits a contact structure [52]. Note that the existence of a contact structure for any manifold was first proved by Martinet [40]. Giroux then showed that the contact structure supported by a given open book is unique up to isotopy [19]. He also proved the following theorem, that will be used in the proof of Theorem 1.2.
Theorem 2.1 (Giroux, cf. [13]).
For any contact manifold , there exists an open book decomposition of that supports .
These results of Giroux give the surjection
for any closed, oriented, smooth manifold and the surjection
Taking the quotient of the set of open books of by an operation for pages of open books, called stabilizations, we can think onetoone correspondence between open books and contact manifolds, that is called the Giroux correspondence.
3. Flowspines and DSdiagrams
3.1. Branched polyhedron
A compact topological space is called a simple polyhedron, or a quasistandard polyhedron, if every point of has a regular neighborhood homeomorphic to one of the three local models shown in Figure 2. A point whose regular neighborhood is shaped on the model (iii) is called a true vertex of (or vertex for short), and we denote the set of true vertices of by . The set of points whose regular neighborhoods are shaped on the models (ii) and (iii) is called the singular set of , and we denote it by . Each connected component of is called a region of and each connected component of is called an edge of . A simple polyhedron is said to be special, or standard, if each region of is an open disk and each edge of is an open arc. Throughout this paper, we assume that all regions are orientable.
A branching of a simple polyhedron is an assignment of orientations to regions of such that the three orientations on each edge of induced by the three adjacent regions do not agree. We note that even though each region of a simple polyhedron is orientable, does not necessarily admit a branching. See [26, 2, 33, 46] for general properties of branched polyhedra.
3.2. Flowspines and DSdiagrams
A polyhedron is called a spine of a closed, connected, oriented manifold if it is embedded in and with removing an open ball collapses onto . If a spine is simple then it is called a simple spine.
If a spine admits a branching, then it allows us to smoothen in the ambient manifold as in the local models shown in Figure 3. A point of whose regular neighborhood is shaped on the model (iii) is called a vertex of type and that on the model (iv) is a vertex of type.
Definition.
Let be a closed, connected, oriented manifold.

A branched simple spine of is called a flowspine of if it is a flowspine of for some nonsingular flow on . The flow is said to be carried by .
Let be the unit ball in equipped with the righthanded orientation. Consider a homeomorphism from to the interior of that takes the biggest horizontal open disk in Figure 4 to the horizontal open disk bounded by the equator of . Now we take the geometric completions of these open balls. Then the inverse map induces a continuous map from to that maps the boundary of to the flowspine . The preimage of the singular set of by the map constitutes a trivalent graph on containing the equator . The map restricted to , denoted by , is called the identification map.
Let be the regions of that are oriented so that the flow is positively transverse to these regions. Let and denote the upper and lower hemispheres of , respectively. The identification map satisfies the following properties:

The preimage of each region by consists of two connected regions and bounded by each of which is homeomorphic to and which are contained in and , respectively.

The orientation of (respectively, ) induced from that of coincides with (respectively, is opposite to) the orientation of through the map .
The manifold is restored from by identifying the pairs of regions and for .
Definition.
The trivalent graph on equipped with the identification map obtained from a flowspine of as above is called the DSdiagram of . The equator in is called the cycle of . The cycle is oriented as the boundary of with the orientation induced from that of .
Remark 3.1.

To be precise, the DSdiagram defined above is a DSdiagram with an cycle. If a simple spine is given then we may obtain its DSdiagram without cycles in the same manner. There is a formal definition of cycles for DSdiagrams, see [27]. Note that a DSdiagram of a simple spine may have several possible positions of cycles. In this paper, by a DSdiagram we mean a DSdiagram with a fixed cycle.

Conversely, if a DSdiagram with an cycle is given, we may obtain a closed manifold from the diagram by using the identification map . The image in is a flowspine of .
For convenience, taking the stereographic projection of from the south pole, we describe the DSdiagram on the real plane so that is the unit circle, is the inside of and is the outside. The real plane is oriented such that it coincides with that of as the boundary of . The image lies on (respectively, lies on ) and its orientation induced from that of coincides with (respectively, is opposite to) the orientation of . For simplicity, we denote , and by , and , respectively.
Example 1.
Consider the diagram described on the left in Figure 5. The manifold is obtained from by identifying and for so that the labeled edges along their boundary coincide. Let be the simple spine obtained as the image of by the identification map . The edges with label are the preimage of the edge of by and the vertices with label are the preimage of the vertex of . The righttop is the union of and the region , which is obtained from the union of , and neighborhoods of the edges and the vertices by identifying them by . The polyhedron is obtained from this branched polyhedron by attaching the region along the boundary. Remark that the branched polyhedron on the righttop in Figure 5 is abstract, not an object embedded in . The polyhedron embedded in is described on the rightbottom. This polyhedron can be obtained from the ball by collapsing from the boundary. Hence it is a spine of . Furthermore, setting a flow in positively transverse to and extending it to canonically as in Figure 4, we may see that is a flowspine of . It will be shown in Lemma 9.1 that the flow tangent to the Seifert fibration whose regular orbit is a torus knot is carried by . Thus, it supports the standard contact structure on , which will be stated in Corollary 9.2. This flowspine is called the positive abalone. The flowspine obtained from the mirror of the DSdiagram on the left in Figure 5 is called the negative abalone. The abalone in Figure 5 is named “positive” since it supports the standard contact structure on . The underlying abstract simple polyhedra of the positive and negative abalones are the same. This polyhedron first appeared in [23]. See also [24].
3.3. Normal pairs
In this subsection, we introduce the flowspine from a normal pair. This is the original definition of a flowspine given in [26]. Let be a nonsingular flow on . A compact surface embedded in with boundary is called a compact local section of if it is included in an open surface that is transverse to everywhere. We define two functions and on as
where means that the flow brings the point to the point after the time . We set (respectively, ) if for any (respectively, ). If (respectively, ) then we define (respectively, ) by (respectively, ).
Definition.
The pair of a nonsingular flow on and its compact local section is called a normal pair on if

is a disk,

any orbit of intersects ,

if and then and intersect at transversely, and

if and then .
The conditions (iii) and (iv) are satisfied by setting in general position.
For a normal pair , let and be the subsets of defined as
Roughly speaking, the subsets and are obtained from by floating by the flow and , respectively. Therefore, they are simple polyhedra. Floating , instead of , smoothly, we can easily see that these polyhedra are flowspines.
An important fact is that any nonsingular flow has a normal pair and hence the following theorem holds.
Theorem 3.2 ([26]).
Any pair of a closed, connected, oriented, smooth manifold and a nonsingular flow on admits a flowspine.
The existence of a normal pair of a given flow after a slight deformation is mentioned in [26, Remark (1) in page 509]. We may easily remove the necessity of a slight deformation of the flow as follows. For each point in , we choose a pair of a flowbox of and a disk intersecting all orbits of in the flowbox transversely. Since is compact, we may choose a finite set of these flowboxes so that the union of the disks in these flowboxes intersects all orbits of . We then isotope the positions of these disks with keeping the same property so that they are disjoint. Then the disk of a normal pair can be obtained by connecting these disks by bands transverse to as done in the proof of [26, Theorem 1.1]. The flowspine in Theorem 3.2 is obtained from the normal pair by [26, Theorem 1.2].
The disk of a normal pair is restored from a flowspine by cleaving it along the singular set as shown in Figure 6. The flowspine in the figure is for this normal pair . Assign the orientation to induced from that of , which is the orientation indicated in the figure, and observe the function from to . This function is leftcontinuous at the point in the figure if the vertex is of type and it is rightcontinuous if the vertex is of type.
3.4. Regular moves
In this subsection, we introduce two kinds of moves of branched simple polyhedra.
Definition.
These moves are enough to trace deformation of nonsingular flows in manifolds. Two nonsingular flows and in a closed, orientable manifold is said to be homotopic if there exists a smooth deformation from to in the set of nonsingular flows.
Theorem 3.3 ([27]).
Let and be flowspines of a closed, orientable manifold . Let and be flows on carried by and , respectively. Suppose that and are homotopic. Then is obtained from by applying first and second regular moves successively.
Remark 3.4.
We may apply the moves from the left to the right in Figures 7 and 8 with fixing the flow carried by the flowspine. On the other hand, we may need to homotope the flow when we apply the moves from the right to the left. Sometimes, such homotopy move cannot be obtained in the set of Reeb flows, see the proof of Theorem 4.3 below.
4. Admissibility and positivity
4.1. Admissibility condition
Let be a closed, connected, oriented manifold. Let be a branched simple spine of and be the regions of . Assign orientations to these regions so that they satisfy the branching condition. Let be the metric completion of with the path metric inherited from a Riemannian metric on . Let be the natural extension of the inclusion . Assign an orientation to each edge of in an arbitrary way.
Definition.
A branched simple spine is said to be admissible if there exists an assignment of real numbers to the edges , respectively, of such that for any
(4.1) 
where is an open interval or a circle on such that is a homeomorphism, and if the orientation of coincides with that of induced from the orientation of and otherwise.
Example 2.
Let be the positive abalone in Figure 5 embedded in . Suppose that there exists a contact structure supported by , that is, there exists a contact form on whose Reeb flow is carried by . The abalone is obtained from the branched polyhedron described on the righttop in the figure by attaching the disk corresponding to the region . Since the Reeb flow is transverse to the regions and , the integrated values and should be positive, where the orientation of is induced from that of for each . From the figure, we see that and . Setting and , we have the inequalities and . Therefore, the existence of an assignment of real numbers satisfying these inequalities to the edges is a necessary condition for to have a contact form whose Reeb flow is positively transverse to . This is the admissibility condition. Since satisfies the two inequalities, the positive abalone is admissible.
There are many flowspines that satisfy the admissibility condition. Below, we show that any branched special spine of a rational homology sphere satisfies the condition.
Let be the rank of . Take a maximal tree of and assign to the edges on . Let be the edges on not contained in and set
for and . In this setting, is admissible if and only if the set
is nonempty.
Let be the matrix with the entries . If the spine is special then is a regular matrix.
Lemma 4.1.
Let be a branched special spine of . Then if and only if is a rational homology sphere.
Proof.
The manifold has a cell decomposition whose cells are the vertices of , cells correspond to the edges of , cells correspond to the open disks and cell corresponds to the open ball . Consider the chain complex of this decomposition:
where is the chain of dimensional cells. The map is the zero map because each cell appears twice on the boundary of the cell with opposite orientations. The map is given by the matrix . Since has no torsion, if and only if . By Poincaré duality and the universal coefficient theorem, if and only if for . Thus the assertion follows. ∎
Proposition 4.2.
A branched special spine of a rational homology sphere is admissible.
Proof.
The set is the union of the open half spaces in given by for . Since we are considering a rational homology sphere, we have by Lemma 4.1. Therefore the vectors , , are linearly independent and hence is nonempty. ∎
Example 3.
Let be an oriented sphere with two disjoint oriented disks and in and let be the branched polyhedron obtained from by identifying and by an orientationpreserving diffeomorphism. Then is a flowspine of without vertices (cf. [11, Remark 1.2]). The flow described in Figure 9 is a flow carried by . This flowspine satisfies and hence it is not admissible. In particular, there does not exist a contact form on whose Reeb flow is positively transverse to .
Example 4.
Let be the flowspine given by the DSdiagram in Figure 10. This polyhedron is special and has vertices. From the cell decomposition of , we may easily see that the closed manifold of satisfies . Since there is no closed, orientable, irreducible manifold with and with complexity up to in the table of manifolds [42], the manifold is . Since is not a rational homology sphere, it satisfies by Lemma 4.1. However we have , and hence it is admissible.
4.2. Necessity of positivity
As mentioned in the introduction, we define the positivity of a flowspine as follows.
Definition.
A flowspine is said to be positive if it has at least one vertex and all vertices are of type.
Although there is a flowspine that is not positive but carries a Reeb flow, we need to restrict our discussion to positive flowspines. The reason is that either the existence or the uniqueness of a contact manifold for a given flowspine does not hold at least in some case as we will prove in the following theorem. Recall that a contact structure on is said to be supported by a flowspine if there exists a contact form such that and its Reeb flow is carried by .
Theorem 4.3.
Suppose that admits a tight contact structure. Then one of the following holds:

There exists a flowspine of that does not support any contact structure.

There exists a flowspine of supporting two contact structures that are not contactomorphic.
Proof.
Let and be contact structures on that belong to the same homotopy class of plane fields. Let and be Reeb flows generated by Reeb vector fields of and , respectively. These flows are homotopic. Let and be flowspines that carry and , respectively.
Suppose that is tight, and let be an overtwisted contact structure on obtained from by applying a Lutz fulltwist [39]. Note that a Lutz fulltwist does not change the homotopy class of . By Theorem 3.3, there exists a sequence of flowspines such that is obtained from by some regular move for . Now assume that both of the statements (1) and (2) in the assertion do not hold. Then each supports a unique contact structure up to contactomorphism. Since the regular moves from the left to the right in Figures 7 and 8 can be applied with fixing the flows, the contact structures supported by and are contactomorphic. Hence and are contactomorphic and this is a contradiction. ∎
Remark 4.4.
The positive abalone in Figure 5 is a positive flowspine. The flowspine in Example 3 carries a flow transverse to for any . Such a flow cannot be a Reeb flow. This flowspine is not positive by definition. The flowspine in Example 4 is positive. The flow carried by this flowspine is not homotopic to the flow carried by the flowspine in Example 3. This can be verified by using the ReidemeisterTuraev torsion, see [34, 35].
4.3. Admissibility of positive flowspines
In this subsection, we prove the following proposition.
Proposition 4.5.
Any positive flowspine satisfies the admissibility condition.
Let be a positive flowspine. In the DSdiagram of , the preimage of each edge of by the identification map consists of three copies of , one lies on the cycle, another one lies in the inside of the cycle and the last one lies outside. The diagram in a neighborhood of an edge on the cycle is one of the four cases described on the top in Figure 11, the corresponding part of the spine is described in the middle, and the diagram in a neighborhood of the corresponding edge in the inside of the cycle is as shown on the bottom. The cycles in the top and bottom figures are described by dashed lines.
We put the labels and to the two sides of each edge of type as shown on the bottom in Figure 11 and we put the labels in the same manner.
Lemma 4.6.
A region in the inside of the cycle in a DSdiagram and not adjacent to the cycle has only label .
Proof.
Let be a region in the assertion. From the bottom figures in Figure 11, we see that does not contain edges of type and . Hence has only labels and . We observe how the orientations on the edges change when we travel along . We say that the orientation of an edge is “consistent” if it coincides with that of and “opposite” otherwise. From the bottom figures in Figure 11, we see that the orientation changes from “consistent” to “opposite” after passing through the sides and and there is no case where the orientation changes from “opposite” to “consistent”. Hence cannot have labels and . Moreover, since the orientations of the edges near the sides and are “opposite” and “consistent”, respectively, can have either only label or only label . Suppose that it has only label . Then, from the middle figure in Figure 11, we see that corresponds to the cycle. However, in the same figure, we see that there is an edge of that is not contained in . Since is the image of the cycle by , we have a contradiction. ∎
Now we prove the proposition.
Proof of Proposition 4.5.
We set the assignment of a real number to each edge of the singular set of a positive flowspine as
By Lemma 4.6, if a region lies inside the cycle and is not adjacent to the cycle then .
Let be a region of the DSdiagram of lying inside the cycle and adjacent to the cycle. The boundary of consists of an alternative sequence of paths on the cycle and paths lying inside the cycle, see Figure 12. The orientation of each edge on the path is consistent with the orientation of . The starting edge of the path is of type if the label on the region is and of type if it is . Their orientations are consistent with that of . We follow from the starting edge and check if the orientations of the edges are consistent with or opposite to that of . From the four figures on the bottom in Figure 11, we see that the orientation changes from “consistent” to “opposite” when passes through an edge with label , , or , and there is no edge where the orientation changes from “opposite” to “consistent”. This means that the path divides into two paths and so that the orientations of all edges on are consistent with that of and those of all edges of are opposite.
We are going to observe which types of edges can appear on each of paths and . By the four figures on the top in Figure 11, the sequence of types of the edges of is either or , where . The sum of the assignment to the edges on is
By the four figures on the bottom in Figure 11, the sequence of types of the edges of is either or , where , and the sum of the assignment to the edges on is
By the same figures, the sequence of types of the edges of is either or , where , and the sum of the assignment to the edges on is
The minimum values of , and are , and , respectively. Hence the sum of real numbers on the edges on is positive unless the types of , and are , and , respectively, for all .
Suppose that the boundary of consists of a sequence of edges
(4.2) 
where is an edge of type and is an edge of type . For each , we set a relation between two edges and as . For each region inside the cycle whose boundary is given in the above form, we give the relation to the edges of type in the same manner.
Now assume that there exists a sequence of edges of of type that gives a cyclic order as . For each , the relation implies that there exists a region whose boundary contains the sequence , where and are edges of type . However, this is impossible since the edges appear on the cycle as shown in Figure 13 and we cannot set for . This means that is a partial order for the edges of of type .
Let be the set of edges of of type , equipped with the partial order . We modify the assignment to these edges as , where ’s are sufficiently small positive real numbers such that if . Then the sum of the real numbers assigned to the edges in (4.2) becomes positive. This shows that the modified assignment satisfies the admissibility condition. ∎
5. Reference forms of positive flowspines
Let be a closed, connected, oriented, smooth manifold and be a positive flowspine of . In this section, we define a form on associated with such that using the decomposition (5.1).
5.1. Decomposition and gluing maps
For a positive flowspine of , set and let be the connected component of contained in the region of . Let be a neighborhood of in , choose a neighborhood of in sufficiently thin with respect to and set . Then decomposes as
(5.1) 
where is the connected component of the closure of containing , which is diffeomorphic to , and is the closure of , which is a ball. By identifying the ball in Section 3.2 with , we draw the cycle on and regard as , where is the unit disk, so that a tubular neighborhood of the cycle on corresponds to .
To define the gluing maps of these pieces, we rechoose and slightly larger. For , we denote the gluing map of to by , which is a diffeomorphism to the image, where and . Also, we denote the gluing map of to by , which is a diffeomorphism