The indexed links of Non-singular Morse-Smale flows on graph manifolds

TL;DR

This paper classifies the indexed links of non-singular Morse-Smale flows on graph manifolds, showing they can be generated from a fundamental link through finite operations, advancing understanding of flow topology.

Contribution

It provides a classification method for indexed links of Morse-Smale flows on graph manifolds, identifying a finite set of operations to generate all such links.

Findings

Indexed links can be obtained by finite operations on a fundamental link.

The fundamental link includes all singular Seifert fibers and certain regular fibers.

The classification applies to most graph manifolds.

Abstract

We classify the indexed links corresponding to the union of the closed orbits of non-singular Morse-Smale flows on most graph manifolds. We find that each of this kind of indexed links can be obtained by applying a finite steps of operations on a special indexed link, which consists of all of the singular Seifert fibers and some regular Seifert fibers with some precisely described conditions.

Equations

$$\text{centering}X(\theta ,x,y)=\frac{\partial }{\partial \theta }-\sum _{i=1}^k{x}_i\frac{\partial }{\partial x_i}+\sum _{j=1}^{2-k}{y}_j\frac{\partial }{\partial y_j}\text{@add@centering}$$

$$\begin{array}{rl}\text{@add@centering}\text{centering}W& =A{\cup }_{S^2}B=(A\setminus \mathrm{Int}{A}_1){\cup }_{\partial A_1\setminus S^2}{A}_1{\cup }_{S^2}{B}_1{\cup }_{\partial B_1\setminus S^2}(B\setminus \mathrm{Int}{B}_1)\\ & =(A\setminus \mathrm{Int}{A}_1){\cup }_{T_1\times \partial I}((T_1\times I)♯(T_2\times I)){\cup }_{T_2\times \partial I}(B\setminus \mathrm{Int}{B}_1).\end{array}$$

Full text

**Keywords: **non-singular Morse-Smale flows, graph manifolds

1. Introduction

1.1. Historic remarks and the aim of the paper

A Morse-Smale flow is a smooth flow whose chain recurrent set consists of finitely many hyperbolic closed orbits and fixed points, and it satisfies the transversality condition [13]. If a Morse-Smale flow has no fixed point, then we call it a non-singular Morse-Smale flow, abbreviated as an NMS flow.

From the viewpoint of dynamical systems, Morse-Smale flow is always regarded as a kind of simple system: there does not exist a homoclinic orbit in a Morse-Smale flow. Roughly speaking, such a system does not provide chaos. But from the viewpoint more close to topology, i.e. classifying Morse-Smale flows up to topological equivalence, it is quite complicated. Peixoto [10] began to systematically classify Morse-Smale flows on surfaces. Asimov ([1], [2]) did some significant works from the viewpoint close to topology. Similar to the well-known relationship between handle decompositions and gradient-like flows (Morse-Smale flows without closed orbits), he showed that there implies a combinatorial decomposition in an NMS flow, named a round handle decomposition (abbreviated as RH decomposition). Further he used RH decomposition to obtain several significant results about NMS flows on $n$-manifolds ($n\geq 4$), one of them ([1]) says that a closed $n$-manifold admits an NMS flow if and only if the Euler number of $M$ is zero. This result perfectly answered Question 1.1 in the cases $n\geq 4$. The recent progress about NMS flows we refer to [11], [3], [4], and [5].

Question 1.1**.**

Which closed $n$-manifolds admit NMS flows?

Note that by Poincare-Hopf Theorem, it is easy to know that the answer to this question in the case $n=2$ is similar to the cases $n\geq 4$: a closed surface $\Sigma$ admits an NMS flow if and only if $\Sigma$ is homeomorphic to either a torus or a Klein bottle. So, for Question 1.1, what left is the case $n=3$. Yano ([15]) studied the problem of the existence of NMS flows in a given homotopy class of non-singular vector fields on a closed $3$-manifold. Morgan ([9]) built three theorems to nearly describe the 3-manifolds admitting NMS flows. One of his theorems says that an irreducible closed $3$-manifold $M$ admits an NMS flow if and only if $M$ is a graph manifold. As a direct consequence, every hyperbolic closed $3$-manifold does not admit any NMS flow. Note that the Euler number of every closed $3$-manifold is zero, so the case $n=3$ is very special for Question 1.1. Therefore it is an interesting topic to more deeply understand NMS flows on $3$-manifolds. In particular, it is natural to ask:

Question 1.2**.**

For a given closed $3$-manifold $M$, how to describe the NMS flows on $M$?

The union of closed orbits $\Gamma$ of an NMS flow $\phi_{t}$ on $M$ is the set of finitely many pairwise disjoint embedded simple closed curves in $M$, i.e., a link in $M$. In [8], T. Kobayashi described the link type of $\Gamma$ when $M$ is an irreducible, simple, closed, orientable 3-manifold. We can label every closed orbit $\gamma$ of $\Gamma$ by an integer $k\in\{0,1,2\}$ corresponding to the dimension of the (strong) unstable manifold of $\gamma$, and the labeled link $\Gamma$ is called the indexed link of $\phi_{t}$. We call this integer $k$ the index of $\gamma$. The indexed link is a natural dynamical invariant of NMS flows, and is also closely related to knot theory. So, to understand the indexed links of NMS flows on $M$ is a suitable refinement of Question 1.2. In a broad sense, an indexed link in $M$ is a link in $M$ with index [math], $1$ or $2$ attached to each component. In [14], Wada built an algorithm to decide which indexed links can be realized as the union of closed orbits of NMS flows on $S^{3}$. This work can be regarded as a solid progress on Question 1.2 in the case $M\cong S^{3}$.

The main purpose of this paper is to generalize Wada’s description to most graph manifolds. That is to classify the indexed links of NMS flows on most graph manifolds, which can be regarded as a further progress to answer Question 1.2.

1.2. Some concepts and notations

To state our main results, we have to introduce some further concepts and notations.

1.2.1. General graph manifolds

We use the notation $M(\pm g,b;\frac{q_{1}}{p_{1}},\cdots,\frac{q_{k}}{p_{k}})$ to record a Seifert fibering of a Seifert manifold $M$, where $g$ is the genus of the base orbifold $B$, with sign $+$ if $B$ is orientable and $-$ if $B$ is nonorientable, and $b$ is the number of the boundary components of $B$. Here ‘genus’ for nonorientable surfaces means the number of $\mathbb{RP}^{2}$ connected summands. Moreover, $p_{i}$ and $q_{i}$ are coprime, and $p_{i}>0$ for $i=1,\cdots,k$. For some $i_{0}\in\{1,\cdots,k\}$, if $p_{i_{0}}>1$, then we say that there is a singular fiber of $M$ with slope-$\frac{q_{i_{0}}}{p_{i_{0}}}$. In particular, if every $p_{i}>1$, then $M$ contains $k$ singular fibers. The details we refer to Chapter 2 of Hatcher [7].

Let $T$ be an incompressible torus embedded in an irreducible Seifert manifold. It is a well-known fact that $T$ is isotopic to either a vertical torus, i.e., a union of regular fibers, or horizontal torus, i.e., transverse to all fibers. It is a classical result that most Seifert manifolds do not contain any horizontal torus, but admit a unique Seifert fibering up to isomorphism. The details we refer to Chapter 2 of Hatcher [7].

A compact irreducible orientable 3-manifold $W$ is a graph manifold if every JSJ piece of $W$ is a Seifert manifold. A closed graph manifold $W$ is called a general graph manifold if:

1. (1)

each Seifert piece $M_{i}$ of $W$ does not contain any horizontal torus and admits a unique Seifert fibering up to isomorphism; 2. (2)

the base orbifold of $M_{i}$ is orientable, and $M_{i}$ does not admit any singular fiber with slope-$\frac{q}{2}$ where $q$ is coprime to $2$; 3. (3)

$W$ is not homeomorphic to $M(0,0;\frac{q_{1}}{p_{1}},\frac{q_{2}}{p_{2}},\frac{q_{3}}{p_{3}})$ ($p_{1},p_{2},p_{3}>1$).

From its definition, general graph manifolds are the majority in the set of closed orientable graph manifolds. In this paper, we will only study the indexed links of NMS flows on general graph manifolds. Our results will show that the indexed links of NMS flows on this class of manifolds are strongly and cleanly related to the topology of the underlying graph manifolds.

1.2.2. Fat round handle decompositions

By using Lyapunov function, one can decompose and combine the an NMS flow $\phi_{t}$ on a $3$-manifold $M$ as follows. We can choose finitely many regular level sets of a Lyapunov function associated to $(M,\phi_{t})$ to decompose $(M,\phi_{t})$ into some canonical pieces, i.e., filtrating neighborhoods. A filtrating neighborhood is a compact $3$-manifold $N$ embedded into $M$ such that,

1. (1)

the boundary of $N$ is nonempty and $\phi_{t}$ is transverse to each boundary component of $N$; 2. (2)

the maximal invariant set of $\phi_{t}$ restricted to $N$ is a periodic orbit of $\phi_{t}$.

A filtrating neighborhood is called a fat round handle (abbreviated as FRH) by Morgan [9]. We say that a FRH $N$ is a $k$-FRH ($k=0,1,2$) if the index of the periodic orbit in $N$ is equal to $2-k$. A key fact is that there are only finitely many filtrating neighborhoods for NMS flows on irreducible manifolds, see [9], [14] or Section 2. In [9], the way that $(M,\phi_{t})$ is decomposed into finitely many FRHs is called a fat round handle decomposition (abbreviated as FRH decomposition). We refer to Section 2 for more details about FRH decompositions.

An NMS flow $\phi_{t}$ on a graph manifold $W$ is called an NMS flow related to incompressible torus decompositions if:

1. (1)

the indexed link $l$ of $\phi_{t}$ contains index-$1$ knots, and $l$ is knotted in $W$, i.e., every knot of $l$ cannot bound a disk in $W$; 2. (2)

there is a FRH decomposition of $\phi_{t}$ such that every boundary surface of each 1-FRH is either incompressible in $W$ or a boundary of some 0-FRH or 2-FRH in this decomposition.

The NMS flows related to incompressible torus decompositions will play a crucial role in this paper. One reason is that if $\phi_{t}$ is an NMS flow on a graph manifold $W$ related to incompressible torus decompositions, then we will prove that $\phi_{t}$ is transverse to the JSJ tori of $W$ (see Proposition 2.12).

1.3. Main results

Our first main result (Theorem 1.3) explains that the indexed links of NMS flows related to incompressible torus decompositions of general graph manifolds can be well understood:

Theorem 1.3**.**

Let $l$ be an indexed link in a general graph manifold $W$. Then $l$ is the indexed link of an NMS flow on $W$ related to incompressible torus decompositions if and only if there is a JSJ decomposition $W=M_{1}\cup\cdots\cup M_{s}$ with the JSJ tori set $\mathcal{T}$ such that:

1. (1)

$l\cap(\cup_{T\in\mathcal{T}}T)=\varnothing$* and $l$ contains both index-[math] knots and index-$2$ knots. If $T\in\mathcal{T}$ is separating in $W$, then each connected component of $W\smallsetminus T$ contains either index-[math] or index-$2$ knots of $l$.* 2. (2)

For each $i=1,\cdots,s$, $l_{i}=l\cap M_{i}$ consists of all the singular fibers and some regular fibers of $M_{i}$ such that every singular fiber knot is either index-[math]* or index-$2$.* 3. (3)

Suppose that $l_{i}$ contains $x_{i}$ index-$1$* knots and $z_{i}$ other knots, then $z_{i}+b_{i}=x_{i}-2g_{i}+2$, where $b_{i}$ is the number of boundary components of $M_{i}$, and $g_{i}$ is the genus of the base orbifold of $M_{i}$.*

The second main result (Theorem 1.4) shows that the indexed link of an NMS flow on a general graph manifold can be well understood by applying finite steps of combinatorial operations in Operation A. For the precise definition of Operation A we refer to Subsection 2.2. Note that Operation A is essentially consistent with Wada’s operations in [14]. Therefore, the indexed links of NMS flows on general graph manifolds can be well understood by using Theorem 1.3 and Theorem 1.4.

Theorem 1.4**.**

Let $W$ be a general graph manifold. An indexed link $l$ in $W$ is corresponding to the union of the closed orbits of an NMS flow $\phi_{t}$ on $W$ if and only if there is an NMS flow $\psi_{t}$ on $W$ related to incompressible torus decompositions such that $l$ can be obtained from the indexed link $l^{\prime}$ of $\psi_{t}$ by applying finitely many steps of operations in Operation A.

1.4. Further remarks

It is natural to expect a complete classification of NMS flows (up to topological equivalence) on a (general) graph $3$-manifold. But generally heteroclinic trajectories connecting saddle orbits will lead the discussion of NMS flows quite wild. As a first step to study this question, a reasonable object is to classify NMS flows without any heteroclinic trajectory. In [16], the second author of this paper devised a path for the $3$-manifold $S^{3}$ to discuss this issue and classified such NMS flows completely in orbits with a small number of periods (no more than $4$). But even on $S^{3}$, it still seems difficult to provide a complete classification for such NMS flows. Nonetheless, it remains an interesting further topic to consider similar problems on (general) graph manifolds.

Readers may wonder how the main results of this paper can be generalized to all graph manifolds. In fact, there is no inherent difficulty, but similar results and proofs would be very complicated and subtle. For specificity and clarity, in this paper we focus on general graph manifolds.

1.5. Outline of the article

This paper is organized as follows. In Section 2, we introduce some definitions, and discuss the NMS flows on graph manifolds related to incompressible torus decompositions. In Section 3, we prove Theorem 1.4. In Section 4, we discuss the Lyapunov graphs of NMS flows. In Section 5, we prove Theorem 1.3.

2. preliminary

In this paper, when we talk about a flow on a manifold with boundary, we mean that the flow is transverse to the boundary of this manifold.

2.1. FRH decompositions

For a compact 3-manifold $N$, if there exists a non-singular flow on $N$ pointing inward on $\partial_{-}N$ ($\partial_{-}N$ is a specified union of components of $\partial N$) and outward on $\partial_{+}N=\partial N\smallsetminus\partial_{-}N$, then we use the notation $(N,\partial_{-}N)$ to denote it. It should be noted that we do not rule out the cases $\partial N=\varnothing$, $\partial_{-}N=\varnothing$ or $\partial_{+}N=\varnothing$.

Definition 2.1**.**

A round $k$-handle (abbreviated as $k$-RH) is the Whitney sum $E_{s}^{k}\oplus E_{u}^{2-k}$, where $E_{s}^{k}$ is a $k$-disk bundle over $S^{1}$, and $E_{u}^{2-k}$ is a $(2-k)$-disk bundle over $S^{1}$ for $k=0,1,2$.

Definition 2.2**.**

A 3-manifold $M$ is obtained from $(N,\partial_{-}N)$ by attaching a $k$-RH $R=E_{s}^{k}\oplus E_{u}^{2-k}$ if there is an embedding $\varphi:\partial E_{s}^{k}\oplus E_{u}^{2-k}\rightarrow\partial_{+}N$ such that $M\cong N\cup_{\varphi}R$ ($k=0,1,2$).

Remark 2.3**.**

We specify that $\partial_{-}R=\partial E_{s}^{k}\oplus E_{u}^{2-k}$, $\partial_{+}R=\overline{\partial R\smallsetminus\partial_{-}R}$, the closure of $\partial R\smallsetminus\partial_{-}R$, and $\partial_{+}M=\overline{(\partial_{+}N\smallsetminus\varphi(\partial_{-}R))\cup\partial_{+}R}$.

Definition 2.4**.**

A $3$-manifold $M$ admits a round handle decomposition (abbreviated as RH decomposition) $M=(\partial_{-}M\times I)\cup_{i=1}^{n}R_{i}$ if each $R_{i}$ denotes a RH attached to $(\partial_{-}M\times I)\cup_{j<i}R_{j}$, where $\partial_{-}M$ is a specified union of components of $\partial M$, $I=[0,1]$, and we specify that $\partial_{+}(\partial_{-}M\times I)=\partial_{-}M\times\{1\}$.

Suppose that $M$ is obtained from $(N,\partial_{-}N)$ by attaching a $1$-RH $h$, and that both $M$ and $N$ are orientable. Then $h=E_{s}\oplus E_{u}$, where $E_{s}$ and $E_{u}$ are either both trivial or both non-trivial. If $E_{s}$ and $E_{u}$ are both trivial, then we say that $h$ is untwisted. At this moment, $\partial_{-}h$ consists of two annuli (see Figure 1 (a)). If $E_{s}$ and $E_{u}$ are both non-trivial, then we say that $h$ is twisted. At this moment, $\partial_{-}h$ is one annulus (see Figure 1 (b)). We call the cores of the connected components of $\partial_{-}h$ attaching circles.

Let $A$ be a special union of components of $\partial_{+}N$ such that $\partial_{-}h$ is attached to $A$ and meets each component of $A$. We fatten up $A$ to get $A\times I$ such that $\partial_{-}h$ is attached to $A\times\{1\}$. The manifold $C(h)$ obtained from $A\times I$ by attaching $h$ is called a fat round $1$-handle (abbreviated as $1$-FRH), and we specify that $\partial_{-}(C(h))=A\times\{0\}$. Making $\partial_{-}(C(h))$ naturally attach to $\partial_{+}N$, we still get $M$. Namely, $M$ is obtained from $N$ by attaching a $1$-FRH $C(h)$. Therefore for a RH decomposition of $M$, we can get a new decomposition of $M$ by replacing each $1$-RH with the corresponding $1$-FRH, and call this new decomposition a fat round handle decomposition (abbreviated as FRH decomposition). In addition, we also call the [math]-RHs in the FRH decomposition [math]-FRHs, and call the $2$-RHs in the FRH decomposition $2$-FRHs.

Morgan [9] proved the following result that will be very useful in this paper.

Lemma 2.5** (Morgan [9]).**

Let $M$ be an irreducible orientable compact $3$-manifold so that either $\partial M=\varnothing$ or $\partial M$ is a disjoint union of tori. Let $M=(\partial_{-}M\times I)\cup_{i=1}^{n}R_{i}$ be a RH decomposition of $M$, and $M_{j}=(\partial_{-}M\times I)\cup_{i=1}^{j}R_{i}$ for $j=1,\cdots,n$. Then every $\partial M_{j}$ is a disjoint union of tori.

Suppose $C(h)\cong(A\times I)\cup_{\varphi}h$ is a $1$-FRH associated to a $1$-RH $h$. In [14], Wada classified the set of $1$-FRHs that can be embedded in $S^{3}$. He got $C(h)$ in $S^{3}$ is one of the types (a)-(e) in Lemma 2.6. Suppose that $M$ is the same as defined in Lemma 2.5. In Proposition 3.5 of [9], Morgan proved that the $1$-FRH $C(h)$ in $M$ is one of types (a), (b), (c) and (g) in Lemma 2.6 when there is an inessential attaching circle in $C(h)$. In fact, by using both the tools and the results due to Morgan [9] and Wada [14], it is not difficult to get all of the $1$-FRHs realized in irreducible $3$-manifolds. The following lemma describes the complete classification in question. Note that the proof is only a slight generalization of the tools in Morgan [9] and Wada [14], so we only give the sketch of a proof here.

Lemma 2.6**.**

Let $M$ be an irreducible orientable compact $3$-manifold so that either $\partial M=\varnothing$ or $\partial M$ is a disjoint union of tori. Let $C(h)$ be a $1$-FRH associated to a $1$-RH $h$ embedded in $M$, and $r$ be a core of $h$. Then $(C(h),r)$ is one of the following types:

1. (a)

$C(h)\cong(T_{1}\times I)\sharp(T_{2}\times I)$, that is, the connected sum of $T_{1}\times I$ and $T_{2}\times I$, where $T_{1}$ and $T_{2}$ are tori. Moreover, $\partial_{-}(C(h))=(T_{1}\times\left\{0\right\})\cup(T_{2}\times\left\{0\right\})$, and $r$ bounds a disk in $C(h)$. 2. (b)

$C(h)\cong(T^{2}\times I)\sharp(D^{2}\times S^{1})$, $\partial_{-}(C(h))=T^{2}\times\left\{0\right\}$ or $\partial_{-}(C(h))=(T^{2}\times\left\{0\right\})\cup(\partial D^{2}\times S^{1})$, and $r$ bounds a disk in $C(h)$. 3. (c)

$C(h)\cong V_{1}\sharp V_{2}$* where $V_{1}$ and $V_{2}$ are two solid tori, $\partial_{-}(C(h))=\partial V_{1}$, and $r$ bounds a disk in $C(h)$.* 4. (d)

$C(h)\cong F\times S^{1}$* where $F$ is a disk with two holes. Moreover, $\partial_{-}(C(h))$ is a (connected) component or a union of two components of $\partial C(h)$, and $r=\left\{*\right\}\times S^{1}$ for some point $*$ in ${\rm Int}F$, i.e. the interior of $F$.* 5. (e)

$C(h)\cong D^{2}\times S^{1}\smallsetminus{\rm Int}N$* where $N$ is a tubular neighborhood of the (2,1)-cable of $\left\{0\right\}\times S^{1}$ in $D^{2}\times S^{1}$, $\partial_{-}(C(h))=\partial N$, and $r=\left\{0\right\}\times S^{1}$.* 6. (f)

$C(h)\cong P\widetilde{\times}S^{1}$* where $P$ is a Möbius strip with one hole, $\partial_{-}(C(h))$ is a component of $\partial C(h)$, and $r=\left\{*\right\}\widetilde{\times}S^{1}$ for some point $*$ in ${\rm Int}P$.* 7. (g)

$C(h)\cong\mathbb{R}\mathbb{P}^{3}\sharp(T^{2}\times I)$, $\partial_{-}(C(h))=T^{2}\times\left\{0\right\}$, and a $(2,1)$-cable of $r$ bounds a disk in $C(h)$. Moreover, $C(h)\cong\mathbb{R}\mathbb{P}^{3}\sharp(T^{2}\times I)$ exists only if $M\cong\mathbb{R}\mathbb{P}^{3}$.

**Sketch of a proof : ** Suppose $C(h)\cong A\times I\cup_{\varphi}h$ is a $1$-FRH associated to $h$, where $\varphi:\partial_{-}h\rightarrow A\times\{1\}$. By Lemma 2.5, $A$ is a disjoint union of tori.

Case 1. $h$ is untwisted, then $\partial_{-}h$ consists of two annuli, which implies that $A$ is either a torus or a disjoint union of two tori. Let $c_{1}$ and $c_{2}$ be two attaching circles that are contained in different components of $\partial_{-}h$. We endow an orientation on $h$, then it induces the orientations on $c_{1}$ and $c_{2}$.

Case 1.1. $A$ is the disjoint union of two tori.

• •

If both $\varphi(c_{1})$ and $\varphi(c_{2})$ are essential in $A\times\{1\}$, then we get type (d).

• •

If one of $\varphi(c_{1})$ and $\varphi(c_{2})$ is essential in $A\times\{1\}$ and the other is inessential in $A\times\{1\}$, then we get type (b).

• •

If both $\varphi(c_{1})$ and $\varphi(c_{2})$ are inessential in $A\times\{1\}$, then either we get a manifold with a $S^{2}$ boundary component (see Figure 2 (a)) or we get type (a) (see Figure 2 (b)). By Lemma 2.5, the first case is impossible.

Case 1.2. $A$ is a torus.

• •

Suppose both $\varphi(c_{1})$ and $\varphi(c_{2})$ are essential in $A\times\{1\}$. If $\varphi(c_{1})$ and $\varphi(c_{2})$ are in opposite directions on $A\times\{1\}$, then we get type (f) (see Figure 2 (c)). Otherwise, we get type (d).

• •

If one of $\varphi(c_{1})$ and $\varphi(c_{2})$ is essential in $A\times\{1\}$ and the other is inessential in $A\times\{1\}$, then we get type (c).

• •

If both $\varphi(c_{1})$ and $\varphi(c_{2}$) are inessential in $A\times\{1\}$, then there are four cases as shown in Figure 3. The first manifold admits a $S^{2}$ boundary component, and both the third and fourth manifolds contain non-separating $2$-spheres. By Lemma 2.5 and the irreducibility of $M$, these three cases are impossible. The second manifold is of type (b).

Case 2. $h$ is twisted, then $\partial_{-}h$ is an annulus, which implies that $A$ is a torus. Let $c_{1}$ be an attaching circle.

• •

If $\varphi(c_{1})$ is essential in $A\times\{1\}$, then we get type (e).

• •

Otherwise, we get type (g).

$\Box$

Definition 2.7**.**

Let $C(h)$ be a $1$-FRH associated to a $1$-RH $h$ in a FRH decomposition of a $3$-manifold. We define $\widetilde{C}(h)$ to be the union of $C(h)$ and all of the [math]-RHs and $2$-RHs adjacent to $C(h)$ in this FRH decomposition.

The definition of $\widetilde{C}(h)$ is introduced by Morgan in [9] to describe the local situation of the $1$-FRH. Suppose that $\widetilde{C}(h)$ is embedded in a graph manifold $W$ and $\partial\widetilde{C}(h)\neq\varnothing$. In Proposition 3.5 of [9], Morgan discussed $\widetilde{C}(h)$ when each attaching circle is essential in $C(h)$, i.e., when $C(h)$ is one of types (d), (e) and (f). He got $\widetilde{C}(h)$ is one of types (4), (5) and (7) in Lemma 2.8, if there is an inessential attaching circle in $\widetilde{C}(h)$. Moreover, he found that in other cases $\widetilde{C}(h)$ is a Seifert manifold with a Seifert fibering extended from the attaching circles. In the following lemma, we describe all cases of $\widetilde{C}(h)$.

Lemma 2.8**.**

Let $C(h)$ be a $1$-FRH of type (d), (e) or (f) associated to a $1$-RH $h$. Suppose that $\widetilde{C}(h)$ is embedded in a graph manifold $W$ and $\partial\widetilde{C}(h)\neq\varnothing$. Then $\widetilde{C}(h)$ is one of the following types:

1. (1)

$\widetilde{C}(h)\cong C(h)$; 2. (2)

$\widetilde{C}(h)\cong M(0,2;\frac{q}{p})$; 3. (3)

$\widetilde{C}(h)\cong M(0,1;\frac{q_{1}}{p_{1}},\frac{q_{2}}{p_{2}})$, where $p_{1},p_{2}>1$; 4. (4)

$\widetilde{C}(h)\cong S^{1}\times D^{2}$; 5. (5)

$\widetilde{C}(h)\cong(S^{1}\times D^{2})\sharp(S^{1}\times D^{2})$; 6. (6)

$\widetilde{C}(h)\cong M(-1,1;\frac{q}{p})$; 7. (7)

$\widetilde{C}(h)\cong W\sharp(S^{1}\times D^{2})$.

In particular, $\widetilde{C}(h)$ of type (7) occurs only when $W$ is homeomorphic to a lens space.

**Proof : **If there is no [math]-RH or $2$-RH adjacent to $C(h)$, then $\widetilde{C}(h)\cong C(h)$. In the following of the proof, we consider the case that $C(h)$ is adjacent to some [math]-RHs or $2$-RHs.

Case 1. $C(h)$ is of type (d).

The three boundary components of $C(h)$ are symmetric in topology, so we only need to consider two cases: attaching one solid torus to $C(h)$ or attaching two solid tori to $C(h)$. Let $T_{i}$ be the boundary component of $C(h)$ for $i=1,2,3$. There is a circle bundle $\pi:F\times S^{1}\cong C(h)\rightarrow F$, where $F$ is a pair-of-pants, i.e., a disk with two holes. We choose a cross section $s:F\rightarrow F\times S^{1}$, and endow $C(h)$ with an orientation, then it induces an orientation on $\partial C(h)$. Let $d_{i}=s\cap T_{i}$, and $l_{i}$ be a fiber in $T_{i}$.

Case 1.1. Attach one solid torus to $C(h)$.

We may assume that a solid torus $R_{1}$ is attached to $C(h)$ along $T_{1}$. let $\widetilde{m_{1}}$ be a meridian of $\partial R_{1}$. Suppose that $\widetilde{C}(h)\cong C(h)\cup_{\psi}R_{1}$, where $\psi:\partial R_{1}\to T_{1}$ is a diffeomorphism such that $\psi(\widetilde{m_{1}})=p_{1}d_{1}+q_{1}l_{1}$ ($p_{1}\geq 0$). If $p_{1}=0$, then $\widetilde{C}(h)\cong(S^{1}\times D^{2})\sharp(S^{1}\times D^{2})$. Otherwise, $\widetilde{C}(h)\cong M(0,2;\frac{q_{1}}{p_{1}})$. In particular, $\widetilde{C}(h)\cong T^{2}\times I$, if $p=1$.

Case 1.2. Attach two solid tori to $C(h)$.

We may assume that two solid tori $R_{1}$ and $R_{2}$ are attached to $C(h)$ along $T_{1}$ and $T_{2}$ respectively. Let $\widetilde{m_{j}}$ be a meridian of $\partial R_{j}$ ($j=1,2$). Suppose that $\widetilde{C}(h)\cong R_{1}\cup_{\psi_{1}}C(h)\cup_{\psi_{2}}R_{2}$, where $\psi_{j}:\partial R_{j}\to T_{j}$ is a diffeomorphism such that $\psi_{j}(\widetilde{m_{j}})=p_{j}d_{j}+q_{j}l_{j}$ ($p_{j}\geq 0$).

If $p_{1}=p_{2}=0$, then there is a non-separating $S^{2}$ in $W$, which contradicts to the irreducibility of $W$. If one of $p_{1}$ and $p_{2}$ is [math] and the other is not, then we may assume that $p_{1}=0$ and $p_{2}\neq 0$. Then $C(h)\cup_{\psi_{1}}R_{1}\cong V_{1}\sharp V_{2}$, where $V_{1}$ and $V_{2}$ are two solid tori, and $\partial V_{2}=T_{2}$. Thus $\widetilde{C}(h)\cong V_{1}\sharp\left(V_{2}\cup_{\psi_{2}}R_{2}\right)$. If $W$ isn’t homeomorphic to any lens space, then $V_{2}\cup_{\psi_{2}}R_{2}\cong S^{3}$, which implies that $\widetilde{C}(h)\cong S^{1}\times D^{2}$. Otherwise, $V_{2}\cup_{\psi_{2}}R_{2}\cong W$ or $S^{3}$, then $\widetilde{C}(h)\cong W\sharp(S^{1}\times D^{2})$ or $S^{1}\times D^{2}$.

Suppose that $p_{1}\cdot p_{2}\neq 0$. If $p_{1}=1$ or $p_{2}=1$ and we assume that $p_{1}=1$ without confusion, then $C(h)\cup_{\psi_{1}}R_{1}=T^{2}\times I$, which implies that $\widetilde{C}(h)\cong S^{1}\times D^{2}$. If $p_{1},p_{2}\geq 2$, then $\widetilde{C}(h)\cong M(0,1;\frac{q_{1}}{p_{1}},\frac{q_{2}}{p_{2}})$.

Case 2. $C(h)$ is of type (e), then $C(h)\cong M(0,2;\frac{1}{2})$.

Removing a small open tubular neighborhood $N$ of the singular fiber of $C(h)$, we get a manifold $F\times S^{1}$ with a circle bundle $\pi:F\times S^{1}\rightarrow F$, where $F$ is a pair-of-pants. Similar to Case 1, it is easy to prove that $\widetilde{C}(h)$ is one of types (3), (4) and (7). In particular, if $\widetilde{C}(h)$ is of type (7), then $W\cong\mathbb{R}\mathbb{P}^{3}$. If $\widetilde{C}(h)$ is a Seifert manifold, then $\widetilde{C}(h)$ admits a Seifert fibering such that $\widetilde{C}(h)$ contains a singular fiber with slope-$\frac{q}{2}$, where $q$ is coprime to $2$.

Case 3. $C(h)$ is of type (f), then $C(h)\cong M(-1,2;)$.

Similarly, it is easy to prove that $\widetilde{C}(h)$ is of type (6).

$\Box$

Remark 2.9**.**

Suppose that $\widetilde{C}(h)$ is a Seifert manifold with boundary embedded in a graph manifold, then:

1. (1)

if $C(h)$ is of type (e), then $\widetilde{C}(h)$ admits a Seifert fibering such that $\widetilde{C}(h)$ contains a singular fiber with slope-$\frac{q}{2}$, where $q$ is coprime to $2$; 2. (2)

if $C(h)$ is of type (f), then $\widetilde{C}(h)$ admits a Seifert fibering such that the base orbifold of $\widetilde{C}(h)$ is non-orientable.

Form now on, we discuss the relationship between NMS flows and RH decompositions. Let $X$ be the vector field on $N=I\times D^{k}\times D^{2-k}$ ($k=0,1,2$) given by

[TABLE]

where $x_{i}$ and $y_{j}$ denote the standard coordinate functions on $\mathbb{R}^{k}$ and $\mathbb{R}^{2-k}$ respectively, and $\frac{\partial}{\partial\theta}$ is a vertor field along the direction of $I$.

Let $R$ be a $k$-RH. If $R$ is a twisted $1$-RH, then we can get an NMS flow on $R$ induced by $X$ and the attaching map $p:N\to N/(0,x,y)\sim(1,-x,-y)$, such that this flow is tangent to $\partial_{-}R\cap\partial_{+}R$, transverse inwardly to ${\rm Int}(\partial_{-}R)$ and transverse outwardly to ${\rm Int}(\partial_{+}R)$. Otherwise, we can also get an NMS flow on $R$ induced by $X$ and the attaching map $p^{\prime}:N\to N/(0,x,y)\sim(1,x,y)$, such that this flow is tangent to $\partial_{-}R\cap\partial_{+}R$, transverse inwardly to ${\rm Int}(\partial_{-}R)$ and transverse outwardly to ${\rm Int}(\partial_{+}R)$. We call the above NMS flow the natural NMS flow on $R$. Obviously, the natural NMS flow on $R$ admits only one closed orbit, which is a core of $R$.

Let $\phi_{t}$ be an NMS flow on an orientable $3$-manifold $M$. Suppose that $\phi_{t}$ is transverse inwardly to $\partial_{-}M$, a union of some connected components of $\partial M$, and transverse outwardly to $\partial M\smallsetminus\partial_{-}M$. Let $n$ be the number of the closed orbits of $\phi_{t}$. We say that a RH decomposition $M=(\partial_{-}M\times I)\cup_{i=1}^{n}R_{i}$ of $M$ is a RH decomposition of $\phi_{t}$ if:

1. (1)

for each $i$, $\phi_{t}|_{R_{i}}$ is topologically equivalent to the natural NMS flow of $R_{i}$; 2. (2)

$\phi_{t}$ is tangent to $\partial_{-}R_{i}\cap\partial_{+}R_{i}$, transverse inwardly to ${\rm Int}(\partial_{-}R_{i})$ and transverse outwardly to ${\rm Int}(\partial_{+}R_{i})$.

For a RH decomposition of $\phi_{t}$, each closed orbit of $\phi_{t}$ is a core of a RH obviously. We call a FRH decomposition associated to a RH decomposition of $\phi_{t}$ a FRH decomposition of $\phi_{t}$.

Let $M=(\partial_{-}M\times I)\cup_{i=1}^{n}C_{i}$ be a FRH decomposition of $\phi_{t}$, which is associated to a RH decomposition $M=(\partial_{-}M\times I)\cup_{i=1}^{n}R_{i}$ of $\phi_{t}$. Let $M_{j}=(\partial_{-}M\times I)\cup_{i=1}^{j}R_{i}$ for $j=1,\cdots,n$. It is easy to observe that $\phi_{t}$ is transverse to $\partial M_{j}$, then $\phi_{t}$ is transverse to the boundary of each $C_{i}$. Thus, every $C_{i}$ is a filtrating neighborhood for $\phi_{t}$. Conversely, it is a fact that a filtrating neighborhood is a FRH. By No-Cycle Theorem [13], we can get a gluing order of the filtrating neighborhoods for $\phi_{t}$, and then we construct a FRH decomposition of $\phi_{t}$. Therefore, essentially, a FRH decomposition of $\phi_{t}$ is a method to decompose $(M,\phi_{t})$ into some filtrating neighborhoods.

Due to Theorem N of Asimov [1] and Proposition in page 43 of Morgan [9], RH decompositions and NMS flows are closely related:

Theorem 2.10** (Asimov [1] and Morgan [9]).**

If an orientable $3$-manifold $M$ admits a RH decomposition, then there exists an NMS flow $\phi_{t}$ on $M$ such that this RH decomposition is a RH decomposition of $\phi_{t}$. Conversely, if $M$ admits an NMS flow $\phi_{t}$, then there exists a RH decomposition of $\phi_{t}$.

2.2. Indexed links

From now on, we define some operations to discuss the indexed link of an NMS flow on a graph manifold $W$. Let $k$ be an indexed knot in $W$. We denote by $N(k,W)$ the regular neighborhood of $k$ in $W$, and denote by ${\rm Ind}(k)$ the index of $k$.

Let $l^{\prime}$ be a link in $W$, and $l^{\prime\prime}$ be a link in $S^{3}$. Let $N^{\prime}$ (resp. $N^{\prime\prime}$) be a small open tubular neighborhood of $l^{\prime}$ (resp. $l^{\prime\prime}$) in $W$ (resp. $S^{3}$). Delete small open $3$-balls $B^{\prime}$ and $B^{\prime\prime}$ from $W\smallsetminus N^{\prime}$ and $S^{3}\smallsetminus N^{\prime\prime}$ respectively, and we obtain a new $W$ by gluing $W\smallsetminus B^{\prime}$ and $S^{3}\smallsetminus B^{\prime\prime}$ along their $2$-sphere boundaries. We denote by $l^{\prime}\cdot l^{\prime\prime}$ this new link in $W$.

Let $k_{1}$ be a knot in $W$, and $k_{2}$ be a knot in $S^{3}$. We choose an open $3$-ball $B_{1}$ (resp. $B_{2}$) in $W$ (resp. $S^{3}$) such that $B_{1}\cap k_{1}$ (resp. $B_{2}\cap k_{2}$) can be embedded in a $2$-sphere. By gluing $W\smallsetminus B_{1}$ and $S^{3}\smallsetminus B_{2}$ along their $2$-sphere boundaries, we can get a new knot in $W$, called the connected sum $k_{1}\sharp k_{2}$ (see Figure 4).

Both $l^{\prime}$ and $l^{\prime\prime}$ are defined as before, and let $k^{\prime}$ (resp. $k^{\prime\prime}$) be a knot of $l^{\prime}$ (resp. $l^{\prime\prime}$). The connected sum $l^{\prime}\sharp l^{\prime\prime}$ associated to $k^{\prime}\sharp k^{\prime\prime}$ is a new link in $W$ obtained from $l^{\prime}$ and $l^{\prime\prime}$ by doing connected sum $k^{\prime}\sharp k^{\prime\prime}$. Obviously, $l^{\prime}\cdot l^{\prime\prime}$ and $l^{\prime}_{1}\sharp l^{\prime}_{2}$ associated to $k^{\prime}\sharp k^{\prime\prime}$ can be realized by doing connected sum $W\sharp S^{3}$.

The terminologies of knot theory refer to [12]. For a given indexed link $l_{1}$ in a graph manifold $W$, we define seven operations as follows.

Operation A: Choose an indexed link $l_{2}$ of an NMS flow on $S^{3}$.

1. I.

To make $l_{1}\cdot l_{2}\cdot u$, where $u$ is an unknot in $S^{3}$ with index 1. 2. II.

To make $l_{1}\cdot(l_{2}\smallsetminus k_{2})\cdot u$, where ${k}_{2}$ is a knot of ${l}_{2}$ with index [math] or $2$. 3. III.

To make $(l_{1}\smallsetminus k_{1})\cdot l_{2}\cdot u$, where ${k}_{1}$ is a knot of ${l}_{1}$ with index [math] or $2$. 4. IV.

To make $(l_{1}\smallsetminus k_{1})\cdot(l_{2}\smallsetminus k_{2})\cdot u$, where ${k}_{i}$ is a knot of ${l}_{i}$ ($i=1,2$), and ${\rm Ind}({k}_{1})=2-{\rm Ind}({k}_{2})=0$ or $2$. 5. V.

To make $(l_{1}\sharp l_{2})\cup m$. The connected sum $l_{1}\sharp l_{2}$ is associated to $k_{1}\sharp k_{2}$, where $k_{i}$ is a knot of $l_{i}$ with index [math] or $2$ ($i=1,2$). ${\rm Ind}(k_{1}\sharp k_{2})={\rm Ind}(k_{1})$ or ${\rm Ind}(k_{2})$, and $m$ is a meridian of $k_{1}\sharp k_{2}$ with index $1$. 6. VI.

Choose a knot $k_{1}$ of $l_{1}$ with index [math] or $2$, and replace $N(k_{1},W)$ by $S^{1}\times D^{2}$ with three indexed circles in it: $S^{1}\times\left\{0\right\}$, $k_{2}$ and $k_{3}$. Here, $k_{2}$ and $k_{3}$ are two parallel $\left(p,q\right)$-cables of $S^{1}\times\left\{0\right\}$ (see Figure 5). The indices of $S^{1}\times\left\{0\right\}$ and $k_{2}$ are either [math] or $2$, and one of them is equal to ${\rm Ind}(k_{1})$. ${\rm Ind}(k_{3})=1$. 7. VII.

Choose a knot $k_{1}$ of $l_{1}$ with index [math] or $2$, and replace $N(k_{1},W)$ by $S^{1}\times D^{2}$ with two indexed knots in it: $S^{1}\times\left\{0\right\}$ and a $\left(2,q\right)$-cable knot $k_{2}$ of $S^{1}\times\left\{0\right\}$ (see Figure 6). ${\rm Ind}(S^{1}\times\left\{0\right\})=1$, and ${\rm Ind}(k_{2})={\rm Ind}(k_{1})$.

In [14], Wada classified the indexed links of NMS flows on $S^{3}$. Therefore, Operation A that we define is clear. In fact, Operation A is consistent with Wada’s operations in [14], except that Wada’s operations require that both $l_{1}$ and $l_{2}$ are two given indexed links in $S^{3}$.

2.3. NMS flows related to incompressible torus decompositions

In Section 1, we introduced the concept of NMS flows related to incompressible torus decompositions. Recall that an NMS flow $\phi_{t}$ on a graph manifold $W$ is called an NMS flow related to incompressible torus decompositions if:

1. (1)

the indexed link $l$ of $\phi_{t}$ contains index-$1$ knots, and $l$ is knotted in $W$, i.e., every knot of $l$ cannot bound a disk in $W$; 2. (2)

there is a FRH decomposition of $\phi_{t}$ such that every boundary surface of each 1-FRH is either incompressible in $W$ or a boundary of some 0-FRH or 2-FRH in this decomposition.

Note that the [math]-FRH is the same as the [math]-RH, and the $2$-FRH is the same as the $2$-RH.

We say that an NMS flow $\phi_{t}$ on a graph manifold $W$ is related to an incompressible torus decomposition $W=(\partial_{-}W\times I)\cup_{i}C_{i}$ if $\phi_{t}$ is an NMS flow related to incompressible torus decompositions and $W=(\partial_{-}W\times I)\cup_{i}C_{i}$ is a FRH decomposition of $\phi_{t}$ satisfying the condition (2). At this time, we call the decomposition $W=(\partial_{-}W\times I)\cup_{i}C_{i}$ an incompressible torus decomposition of $\phi_{t}$.

Lemma 2.11**.**

Suppose that $\phi_{t}$ is an NMS flow on a graph manifold $W$ with saddle closed orbits, and $\phi_{t}$ admits at least two saddle closed orbits when $\partial W=\varnothing$. Then $\phi_{t}$ is related to an incompressible torus decomposition $W=(\partial_{-}W\times I)\cup_{j=1}^{n}\widetilde{C}(h_{j})$ if and only if this decomposition is a FRH decomposition of $\phi_{t}$ such that:

1. (1)

each $C(h_{j})$ is one of types (d), (e) and (f) in Lemma 2.6; 2. (2)

every boundary component of each $\widetilde{C}(h_{j})$ is incompressible in $\widetilde{C}(h_{j})$.

**Proof : **Suppose that $W=(\partial_{-}W\times I)\cup_{j=1}^{n}\widetilde{C}(h_{j})$ is a FRH decomposition of $\phi_{t}$. By the definition of $\phi_{t}$, we have $n\geq 2$ when $\partial W=\varnothing$. Then the boundary of each $\widetilde{C}(h_{j})$ is nonempty. Let $l$ be the indexed link of $\phi_{t}$, then the intersection of $l$ and each RH of the RH decomposition associated to the decomposition $W=(\partial_{-}W\times I)\cup_{j=1}^{n}\widetilde{C}(h_{j})$ is a core of this RH.

Necessity. Suppose that $\phi_{t}$ is related to an incompressible torus decomposition $W=(\partial_{-}W\times I)\cup_{j=1}^{n}\widetilde{C}(h_{j})$. Since $l$ is knotted in $W$ and $l\cap h_{j}$ is a core of $h_{j}$, each $C(h_{j})$ cannot be of type (a), (b) or (c) by Lemma 2.6. If there is a $C(h_{j_{0}})$ is of type (g), then by the irreducibility of $W$, $W\cong\mathbb{RP}^{3}$. Moreover, it is easy to prove that each boundary component of $C(h_{j_{0}})$ is compressible in $W$. By the definition of incompressible torus decompositions, each boundary component of $C(h_{j_{0}})$ must bound either a [math]-RH or $2$-RH, which contradicts that $n\geq 2$. Therefore, each $C(h_{j})$ is one of types (d), (e) and (f) in Lemma 2.6. Let $T$ be a boundary component of $\widetilde{C}(h_{j})$, then $T$ is not bound any [math]-RH or $2$-RH. By the definition of incompressible torus decompositions, $T$ is incompressible in $W$, then $T$ is incompressible in $\widetilde{C}(h_{j})$.

Sufficiency. Suppose that the FRH decomposition $W=(\partial_{-}W\times I)\cup_{j=1}^{n}\widetilde{C}(h_{j})$ of $\phi_{t}$ satisfies conditions (1) and (2). Let $T^{\prime}$ be a boundary component of a $1$-FRH in this decomposition that does not bound a 0-RH or 2-RH, then $T^{\prime}$ is a boundary component of $\widetilde{C}(h_{j^{\prime}})$ for some $j^{\prime}\in\{1,\cdots,n\}$. By the condition (2), $T^{\prime}$ is incompressible in $\widetilde{C}(h_{j^{\prime}})$. Since $W=(\partial_{-}W\times I)\cup_{j=1}^{n}\widetilde{C}(h_{j})$ and every boundary component of each $\widetilde{C}(h_{j})$ is incompressible in $\widetilde{C}(h_{j})$, $T^{\prime}$ is incompressible in $W$.

Let $l_{j}=l\cap\widetilde{C}(h_{j})$ for $j=1,\cdots,n$. Note that the intersection of $l_{j}$ and each RH of the RH decomposition associated to $\widetilde{C}(h_{j})$ is a core of this RH. According to the proof of Lemma 2.8, $\widetilde{C}(h_{j})$ is a Seifert manifold, and $l_{j}$ consists of some fibers of a Seifert fibering of $\widetilde{C}(h_{j})$. Then $l_{j}$ is knotted in $\widetilde{C}(h_{j})$. If there is a knot $k$ of $l$ bounding a disk in $W$, then $k$ is a knot of some $l_{j^{\prime\prime}}$ for $j^{\prime\prime}\in\{1,\cdots,n\}$. Since every boundary component of each $\widetilde{C}(h_{j})$ is incompressible in $\widetilde{C}(h_{j})$, $k$ must bound a disk in $\widetilde{C}(h_{j^{\prime\prime}})$, which contradicts that $l_{j^{\prime\prime}}$ is knotted in $\widetilde{C}(h_{j^{\prime\prime}})$. Therefore, $l$ is knotted in $W$. Then $\phi_{t}$ is related to an incompressible torus decomposition $W=(\partial_{-}W\times I)\cup_{j=1}^{n}\widetilde{C}(h_{j})$.

$\Box$

Proposition 2.12**.**

Let $\phi_{t}$ be an NMS flow on a graph manifold $W$ related to incompressible torus decompositions. Then there is a JSJ decomposition $W=M_{1}\cup\cdots\cup M_{s}$ with the JSJ tori set $\mathcal{T}$ such that:

1. (1)

$\phi_{t}$* is transverse to each $T\in\mathcal{T}$;* 2. (2)

$\phi_{t}|_{M_{i}}$* is an NMS flow on $M_{i}$ related to incompressible torus decompositions.*

**Proof : **If $W$ is a Seifert manifold, then the conclusion in this case is correct. From now on, we suppose that $W$ is not a Seifert manifold.

Since $\phi_{t}$ admits saddle closed oribits, any FRH decomposition of $\phi_{t}$ must contain $1$-FRHs. Let $W=(\partial_{-}W\times I)\cup_{j=1}^{n}\widetilde{C}(h_{j})$ be an incompressible torus decomposition of $\phi_{t}$. Since the indexed link of $\phi_{t}$ is knotted in $W$, every $C(h_{j})$ cannot be of type (a), (b) or (c). Since $W$ is not a Seifert manifold, $C(h_{j})$ cannot be of type (g). Thus each $C(h_{j})$ is one of types (d), (e) and (f).

If $n=1$ and $\partial W\neq\varnothing$, then $W=(\partial_{-}W\times I)\cup\widetilde{C}(h_{1})\cong\widetilde{C}(h_{1})$. By Lemma 2.8 and the irreducibility of $W$, $W$ is a Seifert manifold. If $n=1$ and $\partial W=\varnothing$, then similar to Lemma 2.8, we can prove that $W$ is homeomorphic to $M(0,0;\frac{q_{1}}{p_{1}},\frac{q_{2}}{p_{2}},\frac{q_{3}}{p_{3}})$ or $M(-1,0;\frac{q^{\prime}_{1}}{p^{\prime}_{1}},\frac{q^{\prime}_{2}}{p^{\prime}_{2}})$. Therefore, when $n=1$, $W$ is a Seifert manifold, which contradicts to the supposition. Then $n\geq 2$, which implies that the boundary of every $\widetilde{C}(h_{j})$ is nonempty. By the definition of $\phi_{t}$, the boundary components of each $\widetilde{C}(h_{j})$ are incompressible in $W$. By Lemma 2.8, each $\widetilde{C}(h_{j})$ is a Seifert manifold. Then there is a JSJ decomposition $W=M_{1}\cup\cdots\cup M_{s}$ with the JSJ tori set $\mathcal{T}$ such that for any $T\in\mathcal{T}$, there is $k\in\{1,\cdots,n\}$ such that $T$ is a boundary component of $\widetilde{C}(h_{k})$. Then $\phi_{t}$ is transverse to $T$. Moreover, the indexed link of $\phi_{t}|_{M_{i}}$ is knotted in $M_{i}$ and must contain index-$1$ knots. It is easy to observe that $\phi_{t}|_{M_{i}}$ is an NMS flow related to incompressible torus decompositions. Proposition 2.12 is proved.

$\Box$

3. Proof of Theorem 1.4

In Section 1, we introduced the concept of general graph manifolds. Recall that a closed graph manifold $W$ is called a general graph manifold if:

1. (1)

each Seifert piece $M_{i}$ of $W$ does not contain any horizontal torus and admits a unique Seifert fibering up to isomorphism; 2. (2)

the base orbifold of $M_{i}$ is orientable, and $M_{i}$ does not admit any singular fiber with slope-$\frac{q}{2}$ where $q$ is coprime to $2$; 3. (3)

$W$ is not homeomorphic to $M(0,0;\frac{q_{1}}{p_{1}},\frac{q_{2}}{p_{2}},\frac{q_{3}}{p_{3}})$ ($p_{1},p_{2},p_{3}>1$).

Let $W$ be a general graph manifold, then $W$ cannot be homeomorphic to $M(0,0;\frac{q_{1}}{p_{1}},\frac{q_{2}}{p_{2}},\frac{q_{3}}{p_{3}})$. Now, we will prove it. Suppose that $W$ is homeomorphic to $M(0,0;\frac{q_{1}}{p_{1}},\frac{q_{2}}{p_{2}},\frac{q_{3}}{p_{3}})$. Without loss of generality, we assume that $p_{1},p_{2},p_{3}\geq 1$. According to the condition (3) in the definition of general graph manifolds, the case that $p_{1},p_{2},p_{3}>1$ is impossible. For other cases, $W$ contains at most two singular fibers, then $W$ admits some different Seifert fiberings up to isomorphism (see Chapter 2 in Hatcher [7]), which contradicts the first condition in the definition of general graph manifolds. Therefore, $W$ cannot be homeomorphic to $M(0,0;\frac{q_{1}}{p_{1}},\frac{q_{2}}{p_{2}},\frac{q_{3}}{p_{3}})$.

In Subsection 2.2, we defined Operation A in the indexed link level. From now on, we prove Theorem 1.4. For the convenience of the reader, we outline the idea of the proof here.

Proof ideas:

Necessity. Let $l$ be the indexed link of an NMS flow $\phi_{t}$ on $W$.

1. (1)

By Theorem 2.10, we can get a FRH decomposition of $\phi_{t}$, which does not contain any $1$-FRH of type (g) in Lemma 2.6. 2. (2)

Note that a $1$-FRH of type (a), (b) or (c) is a connected sum of two $3$-manifolds. By the irreducibility of $W$, we can get a FRH decomposition of some NMS flow $\phi_{t}^{0}$ on $W$, which does not contain any $1$-FRH of type (a), (b), (c) or (g). Moreover, $l$ is obtained from the indexed link $l^{\prime}_{0}$ of $\phi_{t}^{0}$ by applying finitely many steps of Operations I-IV in Operation A. 3. (3)

By replcing some solid tori with [math]-RHs or $2$-RHs, we can construct an NMS flow $\psi_{t}$ on $W$ related to incompressible torus decompositions such that $l^{\prime}_{0}$ is obtained from the indexed link $l^{\prime}$ of $\psi_{t}$ by applying finitely many steps of Operations V-VII in Operation A.

Sufficiency. Let $l_{1}$ be the indexed link of an NMS flow $\phi_{t}^{1}$ on $W$, and $l_{0}$ be an indexed link in $W$.

1. (1)

If $l_{0}$ is obtained from $l_{1}$ by applying one of Operations I-V in Operation A, then by doing connected sum $W\sharp S^{3}$, we can construct an NMS flow on $W$ with indexed link $l_{0}$. 2. (2)

If $l_{0}$ is obtained from $l_{1}$ by applying Operation VI or VII in Operation A, then by changing the flow on a small neighborhood of an attracting or repelling closed orbit of $\phi_{t}^{1}$, we can construct a new NMS flow on $W$ with indexed link $l_{0}$.

Now we prove Theorem 1.4.

**Proof of Theorem 1.4 : ** Necessity. Let $l$ be the indexed link of an NMS flow $\phi_{t}$ on $W$. By Theorem 2.10, we can obtain a FRH decomposition $W=\cup_{i}C_{i}$ of $\phi_{t}$, then the intersection of $l$ and each RH of the RH decomposition associated to the decomposition $W=\cup_{i}C_{i}$ is a core of this RH.

If the decomposition $W=\cup_{i}C_{i}$ does not contain any $1$-FRH, then $W$ is homeomorphic to one of $T^{2}\times I$, $S^{1}\times D^{2}$, $S^{3}$ and a lens space, which contradicts to the definition of general graph manifolds. Thus, this decomposition must contain $1$-FRHs. In addition, since $W$ cannot be homeomorphic to $\mathbb{RP}^{3}$, this decomposition cannot contain any $1$-FRH of type (g) in Lemma 2.6. Let $C_{i_{0}}$ be a $1$-FRH associated to a $1$-RH $h$ in this FRH decomposition.

Suppose that $C_{i_{0}}\cong(T_{1}\times I)\sharp(T_{2}\times I)$, where $T_{1}$ and $T_{2}$ are two tori. Then a $2$-sphere $S^{2}$ associated to the connected sum separates both of $C_{i_{0}}$ and $W$. Suppose that $C_{i_{0}}=A_{1}\cup_{S^{2}}B_{1}$ and $W=A\cup_{S^{2}}B$, where $A_{1}\cup_{S^{2}}B^{3}\cong T_{1}\times I$, $B_{1}\cup_{S^{2}}B^{3}\cong T_{2}\times I$, $A_{1}\subseteq A$, and $B_{1}\subseteq B$. Then

[TABLE]

By the irreducibility of $W$, either $(A\smallsetminus{\rm Int}A_{1})\cup_{T_{1}\times\partial I}(T_{1}\times I)\cong S^{3}$ or $(T_{2}\times I)\cup_{T_{2}\times\partial I}(B\smallsetminus{\rm Int}B_{1})\cong S^{3}$. Without loss of generality, we assume that $(T_{2}\times I)\cup_{T_{2}\times\partial I}(B\smallsetminus{\rm Int}B_{1})\cong S^{3}$, then $(A\smallsetminus{\rm Int}A_{1})\cup_{T_{1}\times\partial I}(T_{1}\times I)\cong W$. Since both $A\smallsetminus{\rm Int}A_{1}$ and $B\smallsetminus{\rm Int}B_{1}$ admit FRH decompositions induced by $W=\cup_{i}C_{i}$, we get a new FRH decomposition of $W$ which contains fewer $1$-FRHs of type (a), and get a FRH decomposition of $S^{3}$. By Theorem 2.10, we can construct an NMS flow on $W$ (resp. $S^{3}$) such that the above FRH decomposition of $W$ (resp. ${S}^{3}$) is a FRH decomposition of this flow. Let $l^{\prime}_{1}$ (resp. $l^{\prime}_{2}$) be the indexed link of the above NMS flow on $W$ (resp. $S^{3}$). It is easy to observe that $l=l^{\prime}_{1}\cdot l^{\prime}_{2}\cdot u$, where $u$ is a core of $h$. Namely, $l$ is obtained from $l^{\prime}_{1}$ and $l^{\prime}_{2}$ by applying Operation I in Operation A.

For this new FRH decomposition of $W$ and $l^{\prime}_{1}$, we continue to discuss its $1$-FRHs of type (a) in the same way. After finitely many steps, we end up with a FRH decomposition of an NMS flow on $W$, which does not contain any $1$-FRH of type (a). Let $l_{0}$ be the indexed link of this flow, then $l$ is obtained from $l_{0}$ by applying finitely many steps of Operation I in Operation A. Similarly, it is easy to observe that discussing the $1$-FRH of type (b) corresponds to Operation II or III, and discussing the $1$-FRH of type (c) corresponds to Operations IV. Therefore, we can finally obtain a FRH decomposition $W=\cup_{w}C^{\prime}_{w}$ of some NMS flow $\phi_{t}^{0}$ on $W$, which does not contain any $1$-FRH of type (a), (b), (c) or (g). Let $l^{\prime}_{0}$ be the indexed link of $\phi_{t}^{0}$, then $l$ is obtained from $l^{\prime}_{0}$ by applying finitely many steps of Operations I-IV in Operation A.

Claim 3.1**.**

There is an NMS flow $\psi_{t}$ on $W$ related to incompressible torus decompositions such that $l^{\prime}_{0}$ is obtained from the indexed link $l^{\prime}$ of $\psi_{t}$ by applying finitely many steps of Operations V-VII in Operation A.

**Proof : **If the decomposition $W=\cup_{w}C^{\prime}_{w}$ does not contain any $1$-FRH, then $W$ is homeomorphic to one of $T^{2}\times I$, $S^{1}\times D^{2}$, $S^{3}$ and a lens space. If the decomposition $W=\cup_{w}C^{\prime}_{w}$ contains only one $1$-FRH, then similar to Lemma 2.8, we can prove that $W$ is homeomorphic to $M(0,0;\frac{q_{1}}{p_{1}},\frac{q_{2}}{p_{2}},\frac{q_{3}}{p_{3}})$ or $M(-1,0;\frac{q^{\prime}_{1}}{p^{\prime}_{1}},\frac{q^{\prime}_{2}}{p^{\prime}_{2}})$. The above two cases contradict to the definition of $W$. Therefore the decomposition $W=\cup_{w}C^{\prime}_{w}$ contains at least two $1$-FRHs.

Let $W=\cup_{w}C^{\prime}_{w}=\cup_{j=1}^{n}\widetilde{C}(h_{j})$ ($n\geq 2$), where $C(h_{j})$ is a $1$-FRH of type (d), (e) or (f) associated to a $1$-RH $h_{j}$. Then the boundary of each $\widetilde{C}(h_{j})$ is nonempty. Before proving, we introduce the following definitions.

Let $W=\cup_{v}C^{\prime\prime}_{v}$ be a FRH decomposition of $W$ containing $1$-FRHs. Let $W=\cup_{v}C^{\prime\prime}_{v}=\cup_{v^{\prime}}\widetilde{C}(h^{\prime}_{v^{\prime}})$, and $T$ be a boundary component of some $\widetilde{C}(h^{\prime}_{v^{\prime}})$. If $T$ is compressible in $W$, then we call $T$ a compressible torus boundary component of this FRH decomposition.

Let $y$ be the number of the compressible torus boundary components of the FRH decomposition $W=\cup_{w}C^{\prime}_{w}$. If $y=0$, then every boundary component of $\widetilde{C}(h_{j})$ is incompressible in $\widetilde{C}(h_{j})$ for every $j=1,\cdots,n$. By Lemma 2.11, $\phi_{t}^{0}$ is related to an incompressible torus decomposition $W=\cup_{w}C^{\prime}_{w}$. If $y>0$, then there is some $j_{0}\in\left\{1,\cdots,n\right\}$ such that $\widetilde{C}(h_{j_{0}})$ supports compressible boundary components in $\widetilde{C}(h_{j_{0}})$. Since $W$ is not homeomorphic to any lens space, by Lemma 2.8, $\widetilde{C}(h_{j_{0}})$ is homeomorphic to $V_{1}\sharp V_{2}$ or $V_{1}$, where $V_{1}$ and $V_{2}$ are two solid tori. Let $m=l^{\prime}_{0}\cap h_{j_{0}}$, then $m$ is a core of $h_{j_{0}}$ with index $1$.

Case 1. Suppose that $\widetilde{C}(h_{j_{0}})\cong V_{1}\sharp V_{2}$.

Similar to the discussion of the $1$-FRH of type (a), we can get a FRH decomposition of an NMS flow on $W$ such that the number of compressible torus boundary components of this FRH decomposition is less than $y$. In addition, we get a FRH decomposition of an NMS flow on $S^{3}$. In fact, one of $V_{1}$ and $V_{2}$ is a RH in this decomposition of $W$, and the other is a RH in this decomposition of $S^{3}$. Without loss of generality, we assume that $V_{1}$ is a RH in this decomposition of $W$. Let ${l}_{1}$ and ${l}_{2}$ be the indexed links of the above NMS flows on $W$ and ${S}^{3}$ respectively. Let $k_{1}=l_{1}\cap V_{1}$ and $k_{2}=l_{2}\cap V_{2}$, then $k_{1}$ is a core of $V_{1}$ and $k_{2}$ is a core of $V_{2}$. Moreover, each of the indices of $k_{1}$ and $k_{2}$ is either [math] or $2$.

Due to the proof of Lemma 2.8, $C(h_{j_{0}})\cong F\times S^{1}$, and $m=\{\ast\}\times S^{1}$, where $F$ is a pair-of-pants and $\ast\in{\rm Int}F$. Moreover, $\widetilde{C}(h_{j_{0}})=C(h_{j_{0}})\cup R_{0}$, where $R_{0}$ is a [math]-RH or $2$-RH. Let $\partial_{0}$, $\partial_{1}$ and $\partial_{2}$ be the boundary components of $F$. Suppose that $\partial_{0}\subset\partial R_{0}$, $\partial_{1}\subset\partial V_{1}$ and $\partial_{2}\subset\partial V_{2}$. Then for each $p\in\partial_{0}$, $\{p\}\times S^{1}$ bounds a meridian disk of $R_{0}$. Let $c(R_{0})=l^{\prime}_{0}\cap R_{0}$, then it is a core of $R_{0}$.

Let $\alpha$ be an essential arc in $F$ such that its end points $p_{1}$ and $p_{2}$ are contained in $\partial_{0}$ (see Figure 7 (a)), then $\{p_{1}\}\times S^{1}$ (resp. $\{p_{2}\}\times S^{1}$) bounds a meridian disk $D_{1}$ (resp. $D_{2}$) of $R_{0}$. We get $S^{2}\cong D_{1}\cup(\alpha\times S^{1})\cup D_{2}$ which is a separating $2$-sphere associated to the connected sum of $V_{1}\sharp V_{2}$. It is easy to observe that $c(R_{0})\simeq k_{1}\sharp k_{2}$ whose index is equal to either ${\rm Ind}(k_{1})$ or ${\rm Ind}(k_{2})$, and $m$ is a meridian of $k_{1}\sharp k_{2}$ (see Figure 7 (b)). Thus, $l^{\prime}_{0}$ is obtained from $l_{1}$ and $l_{2}$ by applying Operation V in Operation A.

Case 2. Suppose that $\widetilde{C}(h_{j_{0}})\cong V_{1}$.

According to $\partial_{-}(\widetilde{C}(h_{j_{0}}))=\varnothing$ or $T^{2}$, we replace $\widetilde{C}(h_{j_{0}})$ with a [math]-RH or $2$-RH $R$. Then we get a FRH decomposition of an NMS flow on $W$, such that the number of compressible torus boundary components of this FRH decomposition is less than $y$. Let $l_{1}$ be the indexed link of the above NMS flow, and $k=l_{1}\cap R$, then $k$ is a core of $R$ with index $2$ or [math].

Suppose that $C(h_{j_{0}})$ is of type (d), then $\widetilde{C}(h_{j_{0}})=R_{1}\cup C(h_{j_{0}})\cup R_{2}$, where $R_{1}$ and $R_{2}$ are RHs which are not $1$-RHs. Let $N^{\prime}=R_{1}\cup C(h_{j_{0}})$ and $N^{\prime\prime}=C(h_{j_{0}})\cup R_{2}$. By the proof of Lemma 2.8, one of the following holds:

• •

One of $N^{\prime}$ and $N^{\prime\prime}$ is the same to $\widetilde{C}(h)$ of type (5) in Lemma 2.8, and the other is not.

• •

Both $N^{\prime}$ and $N^{\prime\prime}$ are Seifert manifolds, and either $N^{\prime}$ or $N^{\prime\prime}$ is homeomorphic to $T^{2}\times I$.

For the first case, similar to Case 1, we can get an indexed link $l_{2}$ of an NMS flow on $S^{3}$ such that $l^{\prime}_{0}$ is obtained from $l_{1}$ and $l_{2}$ by applying Operation V. For the second case, we may assume that $N^{\prime}\cong T^{2}\times I$. Then $\widetilde{C}(h_{j_{0}})$ contains three indexed knots of $l^{\prime}_{0}$: $k_{1}$, $k_{2}$ and $m$, where $k_{i}$ is a core of $R_{i}$ for $i=1,2$. Moreover, $k_{1}$ and $m$ are two parallel $\left(p,q\right)$-cables of $k_{2}$, and $k_{2}$ is a core of the solid torus $\widetilde{C}(h_{j_{0}})$. The indices of $k_{1}$ and $k_{2}$ are either [math] or $2$, and one of them is equal to ${\rm Ind}(k)$. Namely, $l^{\prime}_{0}$ is obtained from $l_{1}$ by applying Operation VI.

Suppose that $C(h_{j_{0}})$ is of type (e), then we can get $F\times S^{1}$ by removing a small open tubular neighborhood of $m$ from $C(h_{j_{0}})$, where $F$ is a pair-of-pants. Similar to the above discussion, it is easy to observe that $l^{\prime}_{0}$ is obtained from $l_{1}$ by applying Operation VII.

In both Case 1 and Case 2, we get a FRH decomposition of an NMS flow on $W$ such that the number of compressible torus boundary components of this FRH decomposition is less than $y$. For this FRH decomposition, we continue to discuss its compressible torus boundary components in the same way. After finitely many steps, we end up with a FRH decomposition of some NMS flow $\psi_{t}$ on $W$ such that there is no compressible torus boundary component of this decomposition. By Lemma 2.11, $\psi_{t}$ is an NMS flow related to incompressible torus decompositions. Let $l^{\prime}$ be the indexed link of $\psi_{t}$, then $l^{\prime}_{0}$ is obtained from $l^{\prime}$ by applying finitely many steps of Operations V-VII in Operation A. Claim 3.1 is proved.

$\Box$

By Claim 3.1, the necessity of Theorem 1.4 is proved.

Sufficiency. Let $l_{1}$ and $l_{2}$ be two indexed links of two NMS flows on $W$ and $S^{3}$ respectively. Let $W=\cup_{j=1}^{s}C_{j}$ and ${S}^{3}=\cup_{j=1}^{w}C^{\prime}_{j}$ be two FRH decompositions of the above flows respectively. Let $T_{1}$ (resp. $T_{2}$) be a boundary component of $C_{1}$ (resp. $C^{\prime}_{1}$), and let $l_{0}$ be an indexed link in $W$.

If $l_{0}$ is obtained from ${l}_{1}$ and ${l}_{2}$ by applying Operation I in Operation A, we repalce ${C}_{1}$ by ${C}_{1}\cup(T_{1}\times I)$ and repalce $C^{\prime}_{1}$ by $C^{\prime}_{1}\cup(T_{2}\times I)$. Then

$W\cong W\sharp{S}^{3}\cong\left(T_{1}\times I\right)\sharp\left(T_{2}\times I\right)\cup\left(\cup_{j=1}^{s}C_{j}\right)\cup\left(\cup_{j=1}^{w}C^{\prime}_{j}\right)$.

By considering $(T_{1}\times I)\sharp(T_{2}\times I)$ as a $1$-FRH of type (a) in Lemma 2.6, we get a new FRH decomposition of $W$. By Theorem 2.10, we can construct an NMS flow on $W$ whose indexed link is $l_{0}$.

Similarly, if $l_{0}$ is obtained by Operation II or III, then by using the $1$-FRH of type (b), we can construct an NMS flow on $W$ whose indexed link is $l_{0}$. If $l_{0}$ is obtained by Operation IV, then by using the $1$-FRH of type (c), we can construct an NMS flow on $W$ whose indexed link is $l_{0}$. In addition, if $l_{0}$ is obtained by Operation V, then by using $\widetilde{C}(h)$ of type (5) in Lemma 2.8, we can construct an NMS flow on $W$ whose indexed link is $l_{0}$. If $l_{0}$ is obtained by Operation VI or VII, then by using $\widetilde{C}(h)$ of type (4) in Lemma 2.8, we can construct an NMS flow on $W$ whose indexed link is $l_{0}$.

Let $\psi_{t}$ be an NMS flow on $W$ related to incompressible torus decompositions, and $l^{\prime}$ be the indexed link of $\psi_{t}$. Suppose that an indexed link $l$ in $W$ can be obtained from $l^{\prime}$ by applying finitely many steps of operations in Operation A. According to the above discussion, there is an NMS flow $\phi_{t}$ on $W$ such that $l$ is the indexed link of $\phi_{t}$. Sufficiency is proved, and therefore the proof of Theorem 1.4 is completed

$\Box$

Let $l_{1}$ be the indexed link of an NMS flow $\phi_{t}$ on a general graph manifold $W$. Let $N\cong T^{2}\times I$ be a submanifold of $W$ such that $\phi_{t}|_{N}$ is topologically equivalent to the flow induced by $\frac{\partial}{\partial x}$ along the $I$ direction. Let $V$ be a RH neighborhood of an attracting or repelling closed orbit of $\phi_{t}$. According to the proof of Theorem 1.4, Operations I and II are corresponding to change the NMS flow $\phi_{t}|_{N}$, and Operations III-VII are corresponding to change the NMS flow $\phi_{t}|_{V}$ for some $N,V\subset W$.

4. Lyapunov graphs

The Lyapunov graph was firstly used by Franks ([6]) to classify nonsingular Smale flows on $S^{3}$. For a smooth flow $\phi_{t}:M\rightarrow M$ with a Lyapunov function $f:M\rightarrow\mathbb{R}$, a Lyapunov graph is an oriented graph by identifying each connected component of $f^{-1}(c)$ to a point for each $c\in\mathbb{R}$, and each edge is oriented by the flow direction. In fact, we can also define the Lyapunov graph in an abstractive way (see Franks [6]):

Definition 4.1**.**

An abstract Lyapunov graph is a finite, connected, oriented graph $L$ which satisfies the following conditions:

1. (1)

$L$ possesses no oriented cycles; 2. (2)

each vertex of $L$ is labeled with a chain recurrent flow on a compact space.

By No-Cycle Theorem (see Smale [13]), a Lyapunov graph of an NMS flow possesses no oriented cycles. Thus a Lyapunov graph of an NMS flow is an abstract Lyapunov graph.

Definition 4.2**.**

An abstract Lyapunov graph $L$ is called a tree if its first Betti number $\beta_{1}(L)$ is equal to [math].

Let $L$ be an abstract Lyapunov graph. If an endpoint of an edge in $L$ does not connect any vertex, then we call this endpoint a boundary component of $L$. Let $v$ be a vertex of $L$, and let $e_{v}^{+}$ (resp. $e_{v}^{-}$) be the number of incoming (resp. outgoing) edges connecting $v$. If $e_{v}^{-}\cdot e_{v}^{+}\neq 0$, then we call $v$ a saddle vertex. If $e_{v}^{+}=0$ (resp. $e_{v}^{-}=0$), we call $v$ a source (resp. sink) vertex. In addition, if $v$ connects only one edge, then we call $v$ an end. Obviously, each end is either a source vertex or a sink vertex.

By cutting $L$ along the midpoint of each edge connecting two vertices, we obtain some subgraphs of $L$, each of which contains only one vertex. We call these subgraphs of $L$ the star neighborhoods of vertices (see Figure 8). In particular, if $L$ contains only one vertex $v$, then we say that $L$ is the star neighborhood of $v$.

Suppose that $L$ be a Lyapunov graph of an NMS flow $\phi_{t}$ on a $3$-manifold $M$. Then $L$ naturally corresponds to a FRH decomposition of $\phi_{t}$, as follows: by a Lyapunov function associated to $L$, the star neighborhood of each vertex in $L$ corresponds to a filtrating neighborhood which is a FRH. By No-Cycle Theorem [13], we can get a gluing order of the filtrating neighborhoods for $\phi_{t}$, and then we construct a FRH decomposition $M=(\partial_{-}M\times I)\cup_{i=1}^{w}C_{i}$ of $\phi_{t}$. Conversely, according to the numbers of the connected components of $\partial_{-}C_{i}$ and $\partial_{+}C_{i}$, we obtain a Lyapunov graph of $\phi_{t}|_{C_{i}}$. Based on the gluing rules of the FRHs of the FRH decomposition $M=(\partial_{-}M\times I)\cup_{i=1}^{w}C_{i}$, we get a Lyapunov graph $L$ of $\phi_{t}$.

Proposition 4.3**.**

Let $L$ be a Lyapunov graph of an NMS flow $\phi_{t}$ on $3$-manifolds, then each of the source and sink vertices of $L$ connects exactly one edge.

**Proof : **Let $v$ be a sink (resp. source) vertex of $L$. Note that the star neighborhood of $v$ in $L$ corresponds to a filtrating neighborhood of an attracting (resp. repelling) closed orbit of $\phi_{t}$. Since any filtrating neighborhood of an attracting (resp. repelling) closed orbit of $\phi_{t}$ is a solid torus, $v$ connects only one edge in $L$.

$\Box$

Let $L$ be an abstract Lyapunov graph with saddle vertices such that each of source and sink vertices connects only one edge. Then the union of ends is the union of sink vertices and source vertices.

By cutting $L$ along the midpoints of the edges connecting two saddle vertices, we obtain some subgraphs of $L$, each of which contains one saddle vertex. We call these subgraphs of $L$ the subgraphs associated to saddle vertices (see Figure 9). In particular, if $L$ contains only one saddle vertex $v$, then we say that $L$ is the subgraph associated to $v$.

Let $L^{\prime}$ and $L^{\prime\prime}$ be two finite oriented (possibly disconnected) graph. If $L^{\prime\prime}$ is attached to $L^{\prime}$ by an injective map $\varphi:\partial_{-}L^{\prime\prime}\rightarrow\partial_{+}L^{\prime}$, then we write the resulting graph as $L^{\prime}+L^{\prime\prime}$, where $\partial_{-}L^{\prime\prime}$ denotes the union of incoming boundary components of $L^{\prime\prime}$, and $\partial_{+}L^{\prime}$ denotes the union of outgoing boundary components of $L^{\prime}$.

Lemma 4.4**.**

Let $L$ be an abstract Lyapunov graph with saddle vertices such that each of source and sink vertices connects only one edge. Then $L$ admits a decomposition $L=(\partial_{-}L\times I)+L_{1}+\cdots+L_{n}$, where $L_{1},\cdots,L_{n}$ are the subgraphs of $L$ associated to saddle vertices.

**Proof : **Let $w=\beta_{1}(L)$. By induction on the number of the saddle vertices, we can prove that $L$ admits a decomposition when $w=0$. Assume that $w\geq 1$, and that an abstract Lyapunov graph with saddle vertices admits a decomposition if each of the source and sink vertices connects only one edge and its first Betti number is equal to $w-1$.

By cutting $L$ along the midpoint of an oriented edge $e$ in a circle of $L$, we get a graph $L^{\prime}$ such that $\beta_{1}(L^{\prime})=w-1$ (see Figure 10). By the assumption of induction, $L^{\prime}$ admits a decomposition $L^{\prime}=(\partial_{-}L^{\prime}\times I)+L_{1}+\cdots+L_{n}$ where $L_{1},\cdots,L_{n}$ are the subgraphs of $L^{\prime}$ associated to saddle vertices. In fact, $L_{1},\cdots,L_{n}$ are also the subgraphs of $L$ associated to saddle vertices.

Let $v_{1}$ and $v_{2}$ be two saddle vertices of $L$ such that $v_{1}$ reaches $v_{2}$ through the oriented edge $e$ in $L$. Since $L$ does not contain oriented circles, $v_{2}$ cannot reach $v_{1}$ through an oriented path in $L^{\prime}$. Suppose that $L_{i}$ and $L_{j}$ are the subgraphs of $L$ associated to $v_{1}$ and $v_{2}$ respectively. If $j>i$, then $L$ admits a decomposition $L=(\partial_{-}L\times I)+L_{1}+\cdots+L_{n}$. Otherwise, let $S$ be the union of the subgraphs of $L$ associated to saddle vertices that can be reached by $v_{2}$ through oriented paths in $L^{\prime}$. We move $L_{j}$ and the graphs in $S$ to the end of the decomposition of $L^{\prime}$ perservng their order in the decomposition of $L^{\prime}$, and replace $\partial_{-}L^{\prime}\times I$ with $\partial_{-}L\times I$. Then we get a decomposition of $L$. Lemma 4.4 is proved.

$\Box$

Definition 4.5**.**

Let $\mathcal{S}=\{L\}$ be a set of finite connected graphs without boundary such that:

1. (1)

each vertex of $L$ connects either one edge or three edges; 2. (2)

the edges of $L$ connecting ends are oriented, and $L$ contains both source ends and sink ends; 3. (3)

if $p$ is a separating point of $L$, then each connected graph obtained by cutting $L$ along $p$ must contain ends.

Lemma 4.6**.**

Let $L$ be a graph in $\mathcal{S}$. Then other edges of $L$ can be oriented such that the resulting graph is an abstract Lyapunov graph with exactly one edge attached to each source and each sink vertex.

**Proof : **Suppose $L$ contains $x$ edges ($x\geq 1$). It is easy to observe that the conclusion of the lemma is correct when $x=1$. Now we assume that $x\geq 2$, and that the graphs in $\mathcal{S}$ with less than $x$ edges satisfy the conclusion of the lemma. We choose a path $\gamma$ with no self intersection from a source end $v$ to a sink end $v^{\prime}$, and endow $\gamma$ with an orientation from $v$ to $v^{\prime}$. By deleting $v$, $v^{\prime}$ and all edges of $\gamma$, we get some subgraphs $L_{1},\cdots,L_{m}$. Let $n$ be the number of the vertices of $L_{1}\cap\gamma$.

Suppose that $n=1$ and let $v_{0}=L_{1}\cap\gamma$, then the point in the edge $e$ of $L_{1}$ connecting $v_{0}$ is separating in $L$. By the condition (3) in the definition of $\mathcal{S}$, the ends of $L_{1}$ consist of $v_{0}$ and at least one end of $L$. Thus we can always endow an orientation on $e$ such that $L_{1}$ contains both source ends and sink ends. If $p$ is a separating point in $L_{1}$, then $p$ is separating in $L$. Suppose that $L_{1}\smallsetminus\{p\}=L^{\prime}_{1}\sqcup L^{\prime\prime}_{1}$, where $v_{0}\in L^{\prime}_{1}$. Then $L^{\prime\prime}_{1}$ is a component of $L\smallsetminus\{p\}$. Due to the definition of $\mathcal{S}$, $L^{\prime\prime}_{1}$ must contain ends. Therefore, $L_{1}\in\mathcal{S}$.

Suppose that $n\geq 2$, then let $v_{1},\cdots,v_{n}$ be the vertices in $L_{1}\cap\gamma$. For every $i,j$, we define $v_{i}\prec v_{j}$ if $v_{i}$ points to $v_{j}$ in $\gamma$. Obviously, $\{v_{1},\cdots,v_{n}\}$ is a totally ordered set associated to $\prec$. We may assume that $v_{1}\prec\cdots\prec v_{n}$. In $L_{1}$, we endow an orientation on the edges connecting $v_{1},\cdots,v_{n}$ such that $v_{1}$ is a source end of $L_{1}$ and $v_{2},\cdots,v_{n}$ are $n-1$ sink ends of $L_{1}$. If $C$ is a cycle in $L_{1}\cup\gamma$ but $C\nsubseteq L_{1}$, then $C$ must be non-cycle, i.e., not oriented under the orientations on these edges.

Let $p$ be a separating point in $L_{1}$. If $p$ is separating in $L$, then similar to the case that $n=1$, we can prove that each component of $L_{1}\smallsetminus\{p\}$ contains ends. If $p$ is inseparating in $L$, then there is a path $\gamma^{\prime}$ in $L_{1}$ with no self intersection from $v_{i^{\prime}}$ to $v_{j^{\prime}}$ for $i^{\prime},j^{\prime}\in\{1,\cdots,n\}$ such that $p\in\gamma^{\prime}$. Then one component of $L_{1}\smallsetminus\{p\}$ contains $v_{i^{\prime}}$, and the other component of $L_{1}\smallsetminus\{p\}$ contains $v_{j^{\prime}}$. Namely, each component of $L_{1}\smallsetminus\{p\}$ contains ends. Therefore, $L_{1}\in\mathcal{S}$.

Obviously, the number of the edges in $L_{1}$ is less than $x$. By the inductive assumption, other edges of $L_{1}$ can be oriented such that the resulting graph is an abstract Lyapunov graph with exactly one edge attached to each source and each sink vertex. Similar discussions also work for $L_{2},\cdots,L_{m}$, and therefore $L$ can be oriented such that the resulting graph is an abstract Lyapunov graph with exactly one edge attached to each source and each sink vertex.

$\Box$

5. Proof of Theorem 1.3

In this section, we will prove Theorem 1.3. Before that, we will introduce the definition of JSJ graphs of graph manifolds to describe the JSJ decompositions of graph manifolds.

Let $W$ be a graph manifold, and $W=M_{1}\cup\cdots\cup M_{s}$ be a JSJ decomposition associated to the JSJ tori set $\mathcal{T}$. Let $v_{1},\cdots,v_{s}$ be $s$ vertices in $\mathbb{R}^{3}$. If $T\in\mathcal{T}$ is adjacent to $M_{i}$ and $M_{j}$, then we connect an edge ending at $v_{i}$ and $v_{j}$, where $i,j\in\{1,\cdots,s\}$ and it is possible that $i=j$. According to the JSJ decomposition $W=M_{1}\cup\cdots\cup M_{s}$, we get a connected graph $G$, called a JSJ graph of $W$.

In fact, we construct a one-to-one correspondence between the JSJ decomposition of $W$ and $G$ such that:

1. (1)

for each $i$, $M_{i}$ is corresponding to $v_{i}$; 2. (2)

if $T\in\mathcal{T}$ is adjacent to $M_{i}$ and $M_{j}$, then $T$ is corresponding to an edge of $G$ ending at $v_{i}$ and $v_{j}$.

In addition, it is easy to find that a torus $T\in\mathcal{T}$ is separating in $W$ if and only if the corresponding edge of $G$ is separating in $G$.

5.1. Proof of the necessity of Theorem 1.3

In order to prove the necessity of Theorem 1.3, we need to discuss in depth the NMS flow $\phi_{t}$ on a Seifert piece of a general graph manifold, where $\phi_{t}$ is related to incompressible torus decompositions.

Proposition 5.1**.**

Let $M$ be a Seifert piece of a general graph manifold $W$ with a genus $g$ base orbifold, and $\phi_{t}$ be an NMS flow on $M$. Suppose that $\phi_{t}$ is related to an incompressible torus decomposition $(\partial_{-}M\times I)\cup_{j=1}^{n}\widetilde{C}(h_{j})$, then:

1. (1)

For every $j$, $C(h_{j})$ is of type (d) in Lemma 2.6, and $\widetilde{C}(h_{j})$ is a Seifert manifold with a unique Seifert fibering up to isomorphism. 2. (2)

$M$* is obtained from $\left\{\widetilde{C}(h_{j})|j=1,\cdots,n\right\}$ by some gluing homeomorphisms that preserve the corresponding regular Seifert fibers.* 3. (3)

Let $L$ be the Lyapunov graph of $\phi_{t}$ corresponding to this decomposition, then $\beta_{1}(L)=g$.

**Proof : **Since $M$ cannot be homeomorphic to $\mathbb{RP}^{3}$, the decomposition $(\partial_{-}M\times I)\cup_{j=1}^{n}\widetilde{C}(h_{j})$ cannot contain any $1$-FRH of type (g) in Lemma 2.6. Since the indexed link of $\phi_{t}$ is knotted in $M$, every $C(h_{j})$ cannot be of type (a), (b) or (c).

Suppose that $n=1$, then $M\cong\widetilde{C}(h_{1})$, so $\widetilde{C}(h_{1})$ is a Seifert manifold with the unique Seifert fibering up to isomorphism. If $\partial M=\varnothing$, then $W\cong M$. Similar to Lemma 2.8, we can prove that $W$ is homeomorphic to $M(0,0;\frac{q_{1}}{p_{1}},\frac{q_{2}}{p_{2}},\frac{q_{3}}{p_{3}})$ or $M(-1,0;\frac{q^{\prime}_{1}}{p^{\prime}_{1}},\frac{q^{\prime}_{2}}{p^{\prime}_{2}})$, which contradicts to the definition of $W$. If $\partial M\neq\varnothing$ and $C(h_{1})$ is of type (e) or (f), then by Remark 2.9, either $M$ contains singular fibers with slope-$\frac{q}{2}$ where $q$ is coprime to $2$, or the base orbifold of $M$ is non-orientable. This also contradicts to the definition of $W$. Therefore, $C(h_{1})$ must be of type (d), i.e., a circle bundle over a pair-of-pants.

Suppose that $n>1$, then for each $j=1,\cdots,n$, $\partial(\widetilde{C}(h_{j}))\neq\varnothing$ and each boundary component of $\widetilde{C}(h_{j})$ is incompressible in $M$. By Lemma 2.8, $\widetilde{C}(h_{j})$ is a Seifert manifold. It is a well-known fact that any incompressible torus in an irreducible Seifert manifold is isotopic to either a vertical torus or horizontal torus (see Hatcher [7]). Since $M$ does not contain any horizonal torus, each boundary component of $\widetilde{C}(h_{j})$ is isotopic to a vertical torus of $M$. Then $\widetilde{C}(h_{j})$ admits a Seifert fibering induced by the Seifert fibering of $M$. Therefore, $M$ is obtained from $\widetilde{C}(h_{1}),\cdots,\widetilde{C}(h_{n})$ due to the gluing homeomorphisms that preserve the corresponding regular Seifert fibers.

By Lemma 2.8, $\widetilde{C}(h_{j})$ admits a unique Seifert fibering up to isomorphism, unless $\widetilde{C}(h_{j})$ is homeomorphic to $M(0,1;\frac{1}{2},\frac{1}{2})=M(-1,1;)$ (see Chapter 2 in Hatcher [7]). If $\widetilde{C}(h_{j})$ is homeomorphic to $M(0,1;\frac{1}{2},\frac{1}{2})=M(-1,1;)$, then either $M$ contains singular fibers with slope-$\frac{q}{2}$ where $q$ is coprime to $2$, or the base orbifold of $M$ is non-orientable. This contradicts the definition of $W$. Therefore, $\widetilde{C}(h_{j})$ admits a unique Seifert fibering up to isomorphism. Similar to the case that $n=1$, we can prove that $C(h_{j})$ must be of type (d). The conclusions (1) and (2) of Proposition 5.1 is proved.

According to the proof of Lemma 2.8, the boundaries of the [math]-RHs and the $2$-RHs in $\widetilde{C}(h_{j})$ are vertical tori of $\widetilde{C}(h_{j})$, and the singular fibers of $\widetilde{C}(h_{j})$ are carried by [math]-RHs or $2$-RHs in $\widetilde{C}(h_{j})$. Then by removing the interiors of all 0-RHs or 2-RHs of this decomposition, we get a circle bundle $\pi:M^{\prime}\rightarrow B^{\prime}$. Obviously, $B^{\prime}$ is a surface obtained from the base orbifold $B$ of $M$ by removing some open disks and the small open neighborhoods of all singular points. Thus, the genus of $B^{\prime}$ is equal to $g$. Let $L^{\prime}$ be the graph obtained from $L$ by deleting all ends. From now on, we prove that $\beta_{1}(L^{\prime})$ is equal to the genus of $B^{\prime}$.

Let $p_{1},\cdots,p_{s}$ be a set of maximal inseparating points of $L^{\prime}$ ($s=\beta_{1}(L^{\prime})$), then they correspond to a set of inseparating circles $c_{1},\cdots,c_{s}$ of $B^{\prime}$. Let $\Sigma$ be the surface obtained by cutting $B^{\prime}$ along these circles, then $\Sigma$ is connected obviously. It is easy to observe that the star neighborhood of each saddle vertex in $L^{\prime}$ is corresponding to a pair-of-pants in $B^{\prime}$, then $\Sigma$ is planar. Therefore, the genus of $B^{\prime}$ is equal to $\beta_{1}(L^{\prime})$, which implies that $\beta_{1}(L)=g$. The proof of Proposition 5.1 is completed.

$\Box$

**Proof of the necessity of Theorem 1.3 : **

Let $\phi_{t}$ be an NMS flow on $W$ related to incompressible torus decompositions, and $l$ be the indexed link of $\phi_{t}$.

Since $W$ is a closed manifold, any NMS flow on $W$ must contain both attracting closed orbits and repelling closed orbits. Namely, $l$ must contain both index-[math] knots and index-$2$ knots. By Proposition 2.12, there is a JSJ decomposition $W=M_{1}\cup\cdots\cup M_{s}$ with the JSJ tori set $\mathcal{T}$ such that $\phi_{t}$ is transverse to each $T\in\mathcal{T}$. Then $l\cap T=\varnothing$. If $T^{\prime}\in\mathcal{T}$ is separating in $W$, then we suppose that $W\smallsetminus T^{\prime}=W^{\prime}\sqcup W^{\prime\prime}$. Since $\phi_{t}$ is transverse to $T^{\prime}$ and $\partial\overline{W^{\prime}}$ is connected, $\phi_{t}|_{\overline{W^{\prime}}}$ must contain attracting closed orbits or repelling closed orbits. Then $l\cap W^{\prime}=l\cap\overline{W^{\prime}}$ must contain either index-[math] or index-$2$ knots. Similarly, $l\cap W^{\prime\prime}$ must also contain either index-[math] or index-$2$ knots. Then the conclusion (1) is satisfied.

For each $i=1,\cdots,s$, let $l_{i}=l\cap M_{i}$, and we suppose that $l_{i}$ contains $x_{i}$ index-$1$ knots and $z_{i}$ other knots. Proposition 2.12 shows that $\phi_{t}|_{M_{i}}$ is an NMS flow related to incompressible torus decompositions, then $x_{i}\geq 1$. Let $M_{i}=(\partial_{-}M_{i}\times I)\cup_{j=1}^{x_{i}}\widetilde{C}(h_{j}^{i})$ be an incompressible torus decomposition of $\phi_{t}|_{M_{i}}$.

By Proposition 5.1, each $C(h_{j}^{i})$ is of type (d) in Lemma 2.6, $\widetilde{C}(h_{j}^{i})$ is a Seifert manifold with a unique Seifert fibering up to isomorphism, and $M_{i}$ is obtained from $\left\{\widetilde{C}(h_{j}^{i})|j=1,\cdots,x_{i}\right\}$ by the gluing homeomorphisms that preserve the corresponding regular Seifert fibers. Note that the intersection of $l_{i}$ and each RH of the RH decomposition associated to $\widetilde{C}(h_{j}^{i})$ is a core of this RH. According to the proof of Lemma 2.8, $l_{i}\cap\widetilde{C}(h_{j}^{i})$ consists of all the singular fibers and some regular fibers of $\widetilde{C}(h_{j}^{i})$, such that each singular fiber knot is either index-[math] or index-$2$. Thus $l_{i}$ consists of all the singular fibers and some regular fibers of $M_{i}$, such that each singular fiber knot is either index-[math] or index-$2$. Therefore, the conclusion (2) is satisfied.

Let $b_{i}$ be the number of boundary components of $M_{i}$, and $g_{i}$ be the genus of the base orbifold of $M_{i}$. Let $L_{i}$ be the Lyapunov graph of $\phi_{t}|_{M_{i}}$, which corresponds to the decomposition $M_{i}=(\partial_{-}M_{i}\times I)\cup_{j=1}^{x_{i}}\widetilde{C}(h_{j}^{i})$. Then $L_{i}$ contains $x_{i}$ saddle vertices, $z_{i}$ ends, and $b_{i}$ boundary components. By Proposition 5.1, $\beta_{1}(L_{i})=g_{i}$, and each $C(h_{j}^{i})$ is of type (d) in Lemma 2.6. Thus each saddle vertex of $L_{i}$ connects three edges. By Proposition 4.3, each of the source and sink vertices connects only one edge.

Let $p^{i}_{1},\cdots,p^{i}_{g_{i}}$ be a set of maximal inseparating points of $L_{i}$. By cutting $L_{i}$ along these points, we can get a tree $L^{\prime}_{i}$ that contains $x_{i}$ saddle vertices, $z_{i}$ ends and $b_{i}+2g_{i}$ boundary components. By induction on the number of saddle vertices in $L^{\prime}_{i}$, it is easy to prove that $z_{i}+b_{i}+2g_{i}=x_{i}+2$. Namely, $z_{i}+b_{i}=x_{i}-2g_{i}+2$. Then the conclusion (3) is satisfied, and therefore the necessity of Theorem 1.3 is proved.

$\Box$

5.2. Constructions

In this Subsection, we will construct an NMS flow on a general graph manifold from an indexed link in this manifold. It is worth noting that all our definitions and symbols are compatible.

Let $l$ be an indexed link in a general graph manifold $W$ such that there is a JSJ decomposition $W=M_{1}\cup\cdots\cup M_{s}$ with the JSJ tori set $\mathcal{T}$ satisfies the following conditions:

1. (1)

$l\cap(\cup_{T\in\mathcal{T}}T)=\varnothing$ and $l$ contains both index-[math] knots and index-$2$ knots. If $T\in\mathcal{T}$ is separating in $W$, then each connected component of $W\smallsetminus T$ contains either index-[math] or index-$2$ knots of $l$. 2. (2)

For each $i=1,\cdots,s$, $l_{i}=l\cap M_{i}$ consists of all the singular fibers and some regular fibers of $M_{i}$ such that every singular fiber knot is either index-[math] or index-$2$. 3. (3)

Suppose that $l_{i}$ contains $x_{i}$ index-$1$ knots and $z_{i}$ other knots, then $z_{i}+b_{i}=x_{i}-2g_{i}+2$, where $b_{i}$ is the number of boundary components of $M_{i}$, and $g_{i}$ is the genus of the base orbifold of $M_{i}$.

Claim 5.2**.**

Let $m_{i}$ be the number of singular fibers of $M_{i}$, then we have the following inequalities:

1. (1)

$\Sigma_{i=1}^{s}z_{i}\geq 2$, $b_{i}+z_{i}\geq 2$, $m_{i}\leq z_{i}$, and $x_{i}\geq 1$. 2. (2)

If $b_{i}=0$ or $g_{i}\geq 1$, then $x_{i}\geq 2$. 3. (3)

If $g_{i}=b_{i}=0$, then $m_{i}\geq 4$. 4. (4)

If $g_{i}=0$ and $b_{i}=1$, then $m_{i}\geq 2$.

**Proof : **Since $l$ contains both index-[math] knots and index-$2$ knots, $\Sigma_{i=1}^{s}z_{i}\geq 2$. If $b_{i}=0$, then $W\cong M_{i}$ and $z_{i}\geq 2$. If $b_{i}=1$, then $\partial M_{i}$ corresponds to a separating JSJ torus $T\in\mathcal{T}$. By the condition (1) of the definition of $l$, each connected component of $W\smallsetminus T$ contains either index-[math] or index-$2$ knots of $l$, and $l\cap T=\varnothing$. Then $z_{i}\geq 1$, which implies that $b_{i}+z_{i}\geq 2$. Thus, we always have $b_{i}+z_{i}\geq 2$. Since every singular fiber knot is either index-[math] or index-$2$, we have $m_{i}\leq z_{i}$.

Suppose that $g_{i}=0$, then $b_{i}+z_{i}=x_{i}+2$. If $b_{i}=0$, then $M_{i}\cong W$. Since $W$ is a general graph manifold, $W$ cannot be homeomorphic to $M(0,0;\frac{q_{1}}{p_{1}},\frac{q_{2}}{p_{2}},\frac{q_{3}}{p_{3}})$. Then $z_{i}\geq m_{i}\geq 4$, which implies that $x_{i}\geq 2$. If $b_{i}=1$ and $m_{i}\leq 1$, then $M_{i}\cong S^{1}\times D^{2}$, which cannot be a Seifert piece of a closed graph manifold. Thus when $b_{i}=1$, we have $z_{i}\geq m_{i}\geq 2$, which implies that $x_{i}\geq 1$. If $b_{i}=2$ and $z_{i}=0$, then $M_{i}\cong T^{2}\times I$. This means that $W$ can only admit one Seifert piece, which contradicts that $\Sigma_{i=1}^{s}z_{i}\geq 2$. Thus when $b_{i}=2$, we have $z_{i}\geq 1$, then $x_{i}\geq 1$. When $b_{i}>2$, by the equation $b_{i}+z_{i}=x_{i}+2$, we have $x_{i}\geq 1$.

When $g_{i}\geq 1$, we have $2\leq b_{i}+z_{i}=x_{i}-2g_{i}+2\leq x_{i}$. Therefore, we always have $x_{i}\geq 1$. In particular, if $b_{i}=0$ or $g_{i}\geq 1$, then $x_{i}\geq 2$.

$\Box$

Let $G$ be the JSJ graph of $W$ associated to the JSJ decomposition $W=M_{1}\cup\cdots\cup M_{s}$. Let $v_{i}$ be the vertex of $G$ corresponding to $M_{i}$, and $\mbox{Star}(v_{i})$ be a small neighborhood of $v_{i}$ in $G$. If $\mathcal{T}$ contains $r\geq 1$ tori, then $G$ contains $r$ edges, and we denote the edges of $G$ by $e_{1},\cdots,e_{r}$. We denote by $\mathcal{E}$ the set of the edges of $G$ corresponding to separating JSJ tori. We label the edge in $\mbox{Star}(v_{i})$ by $e_{j}$ if this edge lies in $e_{j}$.

Lemma 5.3**.**

There is a connected graph $L$ without boundary and a surjective projection $\pi:L\to G$ such that:

1. (1)

For each edge $e_{j}$ and each vertex $v_{i}$ of $G$, $\pi^{-1}(e_{j})$ is an edge of $L$, and $L_{i}=\pi^{-1}(\mbox{Star}(v_{i}))$ is a connected graph with $b_{i}$ boundary components such that each vertex connects either one edge or three edges. 2. (2)

$L_{i}$* contains $z_{i}$ ends and $x_{i}$ vertices connecting three edges, and $\beta_{1}(L_{i})=g_{i}$. Suppose that $L_{i}$ contains $y_{i}$ vertices adjacent to two ends, then $2y_{i}\leq m_{i}$.* 3. (3)

If $v_{i}$ is in a circle of $G$, then the midpoint of the edge in $L_{i}$ is inseparating in $L$ unless this edge connects ends or intersects $\cup_{e\in\mathcal{E}}\pi^{-1}(e)$.

**Proof : **For each $i=1,\cdots,s$, we suppose that $L_{i}$ is a connected graph with $b_{i}$ boundary components. Since $W$ is a closed graph manifold, $\mbox{Star}(v_{i})$ contains $b_{i}$ edges. Then there is a one-to-one correspondence between the set of the edges in $\mbox{Star}(v_{i})$ and the set of the edges in $L_{i}$ containing boundary components. We label these edges in $L_{i}$ by the labels on the corresponding edges in $\mbox{Star}(v_{i})$.

Suppose that $g_{i}=0$, then $b_{i}+z_{i}=x_{i}+2$. If $b_{i}=0$, then by Claim 5.2, we have $z_{i}\geq m_{i}\geq 4$, and $L_{i}$ can be constructed as Figure 11 (a). If $b_{i}=1$, then by Claim 5.2, we have $z_{i}\geq m_{i}\geq 2$, and $L_{i}$ can be constructed as Figure 11 (b). If $b_{i}\geq 2$, then $L_{i}$ can be constructed as Figure 11 (c).

Suppose that $g_{i}\geq 1$. Since $2\leq b_{i}+z_{i}=x_{i}-2g_{i}+2$, we can construct $L_{i}$ as shown in Figure 12. It is easy to observe that the above $L_{i}$ that we constructed satisfy the condition (2). Moreover, each vertex of $L_{i}$ connects either one edge or three edges.

By gluing $L_{1},\cdots,L_{s}$ along the endpoints of the edges labeled by the same letters, we can get a connected graph $L$ without boundary. Let $L^{\prime}_{i}$ be the connected graph obtained from $L_{i}$ by removing the edges containing boundary components. By Claim 5.2, $x_{i}\geq 1$, then $L^{\prime}_{i}\neq\varnothing$. By mapping each graph in $\{L^{\prime}_{1},\cdots,L^{\prime}_{s}\}$ into a vertex, we can get $G$ from $L$. Thus we can construct a surjective projection $\pi:L\to G$ such that $\pi^{-1}(\mbox{Star}(v_{i}))$ is equal to $L_{i}$ (regardless of the length of the edges) for each $i=1,\cdots,s$, and $\pi^{-1}(e_{j})$ is an edge of $L$ for each edge $e_{j}$ of $G$. Moreover, for any $e\in\mathcal{E}$, the midpoint of $\pi^{-1}(e)$ is separating in $L$.

Suppose that $v_{i}$ is in a circle of $G$. If $g_{i}\geq 1$, then the midpoint of the edge in $L_{i}$ is inseparating in $L$ unless this edge connects ends or intersects $\cup_{e\in\mathcal{E}}\pi^{-1}(e)$ (see Figure 12). If $g_{i}=0$, then we choose a circle $C_{i}$ of $G$ that contains $v_{i}$. Obviously, $\mbox{Star}(v_{i})\cap C_{i}$ contains two edges, then we make them correspond to the two bold edges in Figure 11 (c). Thus the conclusion (3) in this case is satisfied. Based on the above construction, the surjective projection $\pi$ satisfies the conclusions of Lemma 5.3.

$\Box$

Let $K_{i}$ be the set of the index-[math] or index-$2$ knots of $l_{i}$, and $K^{0}_{i}\subset K_{i}$ be the set of the singular fibers of $M_{i}$. Let $S_{i}$ be the set of the ends of $L_{i}$, and $S^{\prime}_{i}$ be the set of the vertices of $L_{i}$ adjacent to two ends. Since $L_{i}$ contains $z_{i}$ ends and $2y_{i}\leq m_{i}\leq z_{i}$, we can construct a bijection $\sigma_{i}:S_{i}\to K_{i}$ such that each end adjacent to a vertex in $S^{\prime}_{i}$ is contained in $\sigma^{-1}_{i}(K^{0}_{i})$.

We endow the orientation on the edges connecting an end in $L_{i}$ such that the ends mapping into index-[math] knots under $\sigma_{i}$ are sink ends, and the ends mapping into index-$2$ knots under $\sigma_{i}$ are source ends. Since $l$ contains both index-[math] knots and index-$2$ knots, $L$ must contain both source ends and sink ends.

Lemma 5.4**.**

$L\in\mathcal{S}$, where $\mathcal{S}$ is defined in Definition 4.5.

**Proof : **In fact, we only need to prove that for any separating point of $L$, each connected graph obtained by cutting $L$ along this point must contain ends. Now we prove it by contradiction. Suppose that there is a separating point $p$ of $L$ such that a connected component $L^{\prime}$ of $L\smallsetminus\{p\}$ does not contain any end. Obviously, $L^{\prime}$ must contain circles. Since $L$ contains at least two ends, the edge $e^{\prime}$ containing $p$ cannot connect ends.

If $\pi(L^{\prime}\cup e^{\prime})\cap\mathcal{E}\neq\varnothing$, then according to the construction of $\pi$, there is a point $p_{0}$ in $L^{\prime}\cup e^{\prime}$ such that $\pi(p_{0})$ is in an edge in $\mathcal{E}$. Then $p_{0}$ is separating in $L$ and there is a component of $L\smallsetminus\{p_{0}\}$ contained in $L^{\prime}\cup e^{\prime}$. By the condition (1) of the definition of the indexed link $l$, each component of $L\smallsetminus\{p_{0}\}$ must contain ends, which implies that $L^{\prime}$ must contain ends. This contradicts to the supposition, thus $\pi(L^{\prime}\cup e^{\prime})\cap\mathcal{E}=\varnothing$.

If $\mathcal{E}=\varnothing$, then either each vertex $v_{i}$ of $G$ is in a circle of $G$, or $G$ is a vertex. For the first case, by the conclusion (3) of Lemma 5.3, it is easy to observe that the midpoint of the edge in $L$ is inseparating in $L$ unless this edge connects ends. For the second case, we may assume that $G$ is the vertex $v_{1}$, then $L=L_{1}$. Since $\beta_{1}(L)\geq 1$, we have $g_{1}\geq 1$. According to the construction of $L$ in Lemma 5.3, the midpoint of the edge in $L$ is inseparating in $L$ unless this edge connects ends. However, $e^{\prime}$ cannot connect ends, which contradicts that $p$ is separating in $L$.

If $\mathcal{E}\neq\varnothing$, then the graph obtained by cutting $G$ along the midpoint of each edge in $\mathcal{E}$ admits a connected component $G_{0}$ containing $\pi(L^{\prime}\cup e^{\prime})$. This is because that $\pi(L^{\prime}\cup e^{\prime})\cap\mathcal{E}=\varnothing$. Then either each vertex of $G_{0}$ is in a circle of $G_{0}$, or $G_{0}$ consists of a vertex and $G_{0}\cap\mathcal{E}$. Let $L_{0}=\pi^{-1}(G_{0})$, then $(L^{\prime}\cup e^{\prime})\subset L_{0}$. For the first case, by the conclusion (3) of Lemma 5.3, it is easy to observe that the midpoint of the edge in $L_{0}$ is inseparating in $L$ unless this edge connects ends or intersects $\cup_{e\in\mathcal{E}}\pi^{-1}(e)$. For the second case, we may assume that $G_{0}$ contains the vertex $v_{1}$, then $G_{0}$ is a neighborhood of $v_{1}$ in $G$, which implies that $L_{0}=L_{1}$ (regardless of the length of the edges). Since $\beta_{1}(L_{0})\geq 1$, we have $g_{1}\geq 1$. According to the construction of $L_{1}$ in Lemma 5.3, the midpoint of the edge in $L_{1}$ is inseparating in $L$ unless this edge connects ends or intersects $\cup_{e\in\mathcal{E}}\pi^{-1}(e)$. However, $e^{\prime}$ cannot connect ends and $e^{\prime}\cap(\cup_{e\in\mathcal{E}}\pi^{-1}(e))=\varnothing$, which contradicts that $p$ is separating in $L$. Therefore, $L\in\mathcal{S}$.

$\Box$

By Lemma 4.6, we can endow orientations on the edges of $L$ that do not connect ends, so that $L$ is an abstract Lyapunov graph with exactly one edge attached to each source and each sink vertex. Then the vertex of $L$ connecting three edges is a saddle vertex.

Lemma 5.5**.**

There is an NMS flow $\phi_{t}$ on $W$ related to incompressible torus decompositions such that $l$ is the indexed link of $\phi_{t}$.

**Proof : **Let $i\in\{1,\cdots,s\}$. It is note that $L_{i}$ has been oriented to an abstract Lyapunov graph containing $x_{i}$ saddle vertices. Moreover, each saddle vertex of $L_{i}$ connects three edges, and each of the sink and source vertices of $L_{i}$ connects only one edge.

Let $\mathcal{X}_{i}$ be the set of the saddle vertices of $L_{i}$. By Claim 5.2, $x_{i}\geq 1$, and when $b_{i}=0$, $x_{i}\geq 2$. Thus $\mathcal{X}_{i}\neq\varnothing$ and each saddle vertex is adjacent to at most two ends. From now on, we give the corresponding relationship between the subgraph $L_{v}$ of $L_{i}$ associated to a saddle vertex $v$ and a $\widetilde{C}(h_{v})$ in Lemma 2.8, where $C(h_{v})$ is a $1$-FRH of type (d) in Lemma 2.6. The subgraph $L_{v}$ of $L_{i}$ associated to $v$ is defined in Section 4.

If $v$ is adjacent to two ends $v^{\prime}$ and $v^{\prime\prime}$, then $\sigma_{i}(v^{\prime}),\sigma_{i}(v^{\prime\prime})\in K_{i}^{0}$. Namely, $\sigma_{i}(v^{\prime})$ and $\sigma_{i}(v^{\prime\prime})$ are two singular fibers of $M_{i}$. Suppose that $\frac{q^{\prime}_{1}}{p^{\prime}_{1}}$ and $\frac{q^{\prime}_{2}}{p^{\prime}_{2}}$ are the slopes of $\sigma_{i}(v^{\prime})$ and $\sigma_{i}(v^{\prime\prime})$ respectively. Then we make $L_{v}$ correspond to $\widetilde{C}(h_{v})\cong M(0,1;\frac{q^{\prime}_{1}}{p^{\prime}_{1}},\frac{q^{\prime}_{2}}{p^{\prime}_{2}})$. If $v$ is adjacent to one end $v^{\prime}$ and $\sigma_{i}(v^{\prime})\in K_{i}^{0}$, then we make $L_{v}$ correspond to $\widetilde{C}(h_{v})\cong M(0,2;\frac{q^{\prime}_{1}}{p^{\prime}_{1}})$, where $\frac{q^{\prime}_{1}}{p^{\prime}_{1}}$ is the slope of $\sigma_{i}(v^{\prime})$. If $v$ is adjacent to one end $v^{\prime}$ and $\sigma_{i}(v^{\prime})\notin K_{i}^{0}$, then we make $L_{v}$ correspond to $\widetilde{C}(h_{v})\cong M(0,2;)$. In addition, if $v$ is not adjacent to any end, we make $L_{v}$ correspond to $\widetilde{C}(h_{v})\cong M(0,3;)$.

In fact, we can choose the above $\widetilde{C}(h_{v})$ good enough such that:

• •

$L_{v}$ is a Lyapunov graph of an NMS flow induced by the natural FRH decomposition of $\widetilde{C}(h_{v})$;

• •

$L_{v}$ corresponds to this natural FRH decomposition.

By Lemma 4.4, we get a gluing order of the subgraphs of $L_{i}$ associated to saddle vertices, then we can get the corresponding gluing order of $\{\widetilde{C}(h_{v})|v\in\mathcal{X}_{i}\}$. Since $\sigma_{i}:S_{i}\to K_{i}$ is a bijection and $K_{i}^{0}\subset K_{i}$, the slopes of the singular fibers in $\sqcup_{v\in\mathcal{X}_{i}}\widetilde{C}(h_{v})$ are the slopes of the singular fibers of $M_{i}$ exactly. Therefore, by gluing $\{\widetilde{C}(h_{v})|v\in\mathcal{X}_{i}\}$ with a series of homeomorphisms preserving fibers, we can get a Seifert manifold $N_{i}=M(g^{\prime}_{i},b_{i};\frac{q_{1}}{p_{1}},\frac{q_{2}}{p_{2}},\cdots,\frac{q_{m_{i}}}{p_{m_{i}}})$, where $\frac{q_{1}}{p_{1}},\frac{q_{2}}{p_{2}},\cdots,\frac{q_{m_{i}}}{p_{m_{i}}}$ are the slopes of the singular fibers of $M_{i}$ exactly.

Based on the above discussion, we get a FRH decomposition $N_{i}=(\partial_{-}N_{i}\times I)\cup_{v\in\mathcal{X}_{i}}\widetilde{C}(h_{v})$ of $N_{i}$. By Theorem 2.10, we can construct an NMS flow $\psi^{i}_{t}$ on $N_{i}$, such that the decomposition $N_{i}=(\partial_{-}N_{i}\times I)\cup_{v\in\mathcal{X}_{i}}\widetilde{C}(h_{v})$ is a FRH decomposition of $\psi_{t}^{i}$, and $L_{i}$ is the Lyapunov graph of $\psi^{i}_{t}$ corresponding to this decomposition. Note that $x_{i}\geq 1$, and when $b_{i}=0$, we have $x_{i}\geq 2$. Then $\psi^{i}_{t}$ admits saddle closed orbits, and when $\partial N_{i}=\varnothing$, $\psi^{i}_{t}$ admits at least two saddle closed orbits. By Lemma 2.11, $\psi^{i}_{t}$ is related to an incompressible torus decomposition $N_{i}=(\partial_{-}N_{i}\times I)\cup_{v\in\mathcal{X}_{i}}\widetilde{C}(h_{v})$. Similar to the proof of Proposition 5.1, we can obtain that $g^{\prime}_{i}=\beta_{1}(L_{i})=g_{i}$, which implies that $N_{i}\cong M_{i}$.

Due to the surjective projection $\pi:L\to G$, we can get a FRH decomposition $W=\cup_{k}C_{k}$ of an NMS flow $\phi_{t}$ on $W$ by suitablily gluing the above incompressible torus decompositions of $\psi^{1}_{t},\cdots,\psi^{s}_{t}$, so that $L$ is the Lyapunov graph of $\phi_{t}$ corresponding to the decomposition $W=\cup_{k}C_{k}$. By the definition of NMS flows related to incompressible torus decomposition, it is easy to prove that $\phi_{t}$ is related to an incompressible torus decomposition $W=\cup_{k}C_{k}$. Moreover, the indexed link $l^{\prime}$ of $\phi_{t}$ is isotopic to $l$, and the index of each knot in $l^{\prime}$ is the same as the index of the corresponding knot in $l$. Thus we can perturb $\phi_{t}$ such that $l$ is its indexed link. Lemma 5.5 is proved.

$\Box$

5.3. Proof of the sufficiency of Theorem 1.3

Let $l$ be an indexed link in $W$ satisfying the conditions of Theorem 1.3. Due to Lemma 5.5, we can construct an NMS flow on $W$ related to incompressible torus decompositions such that $l$ is the indexed link of this flow. Then the sufficiency of Theorem 1.3 is proved.