NMS Flows on S 3 with no Heteroclinic Trajectories Connecting Saddle Orbits

In this paper we ﬁnd topological conditions for the non existence of heteroclinic trajectories connecting saddle orbits in non singular Morse-Smale ﬂows on S 3 . We obtain the non singular Morse-Smale ﬂows that can be decomposed as connected sum of ﬂows and we show that these ﬂows are those who have no heteroclinic trajectories connecting saddle orbits. Moreover, we characterize these ﬂows in terms of links of periodic orbits

Nevertheless, we can obtain some relevant information from the link such as the presence or not of heteroclinic trajectories connecting saddle orbits.
Next, we see how this paper is organized and the most significant results obtained in the different sections.
In Sect. 3 we carry out iterated connected sums of solid and thick tori and characterize the 3-manifold obtained.
In Sect. 4 we define and characterize the set of the links coming from iterated sum of tori. This set is formed by the NMS links of periodic orbits on S 3 associated to round 1-handles that are iterated sum of tori (Lemma 4.1).
In Sect. 5 we obtain a correspondence between flows characterized by links coming from iterated sum of tori and NMS flows on S 3 that are connected sum of flows (Theorem 5.3).
Finally, in Sect. 5.2, we show that such flows have no heteroclinic trajectories connecting saddle orbits (Proposition 5.4). Moreover, we characterize a kind of flows that necessarily have these type of trajectories (Proposition 5.4).
In fact, we divide the set of the flows with unknotted and unlinked saddle orbits in two complementary sets: the set of flows F I (S 3 ), having no heteroclinic orbits and its complement F I S 3 , the flows containing these type of trajectories.

Previous Results
In this section we collect some previous results that are necessary for the development of the subsequent sections.

Round Handle Decompositions
The notion of the RHD was introduced by Asimov [1] and modified by Morgan [6]; it establishes a correspondence between NMS flows and RHDs.
where each X i is obtained from X i−1 by attaching a round handle.
Proposition 1 [1,6] If a manifold X admits a RHD, then there is a NMS flow on X whose closed orbits coincide with the cores of the round handles and the flow is pointing outward on ∂ X i . Conversely, if X has a NMS flow, X admits a RHD whose core circles are the closed orbits of the flow.
Theorem 2.4 [6] Let X be a compact, orientable and irreducible manifold with ∂ X either empty or disjoint union of tori. If X admits a RHD then each ∂ X i is formed by tori.
For the case of dimension 3, the round handles are diffeomorphic to solid tori and correspond to 0-handles when there is a repulsive periodic orbit in the core, to 2-handles if there is an attractive periodic orbit in the core and to 1-handles if the orbit in the core is a saddle.
For the study of the RHD of a compact, orientable 3-manifold M is more convenient to use the fat RHD introduced by Morgan [6]: and C j is, either a 0 or 2-handle, or a fat round handle of the form: where A is a union of components of ∂ M i−1 , D s ⊕ D u is the Whitney sum of disk bundles D s and D u over S 1 and the image of ϕ : The fat round handle is obtained from the manifold A by attaching a round 1-handle by means of one or two attaching circles depending on whether the 1-handle is twisted or untwisted, let c 1 = ϕ {−1} × {0} × S 1 and c 2 = ϕ {+1} × {0} × S 1 denote these circles. So, while round 1-handles are always disk bundles over S 1 , the fat round 1-handles can be different types of manifolds depending on the way the 1-handles are attached.
The attachment of M N −1 with the last 2-handle gives the 3-manifold M.

Links of Periodic Orbits in S 3
In order to classify the set of closed orbits of a NMS flow on S 3 , Wada [8] describes the different types of fat round 1-handles for the manifold S 3 . This result, was independently obtained in a different way by Yano [9]. Consider C together with ∂ − C = A×{0} and the core γ which is the 0-section of D s ⊕ D u , i.e., the core of D s ⊕ D u . The component C associated to a round 0 or 2-handle is just a solid torus. For 1-handles we have the following result: Lemma 1] The triple (C, ∂ − C, γ ) associated to a round 1-handle is one of the following types: and γ is an unknot in C. 1 and γ is an unknot in C. (c) C ∼ = τ 1 #τ 2 , where τ 1 and τ 2 are solid tori, ∂ − C = ∂τ 1 and γ is an unknot in C.
(d) C ∼ = F × S 1 where F is a disk with two holes, ∂ − C is a component or a union of two components of ∂C and γ = * × S 1 for some point * in I nt F.
The set of periodic orbits of a NMS flow in S 3 is characterized by Wada in terms of indexed links and six operations: (1) I (l 1 , l 2 ) = l 1 · l 2 · u, where u is an unknot with index 1. The link of periodic orbits of a NMS flow on S 3 is defined by the cores of the round-handle decomposition of the manifold. We call NMS links on S 3 those links that correspond to the set of periodic orbits of a NMS flow on S 3 and denote this set of links by L(S 3 ).
The cases (a), (b) and (c) in Lemma (2.5) define operations I, I I and I I I , respectively. We refer to them as type A operations and we denote by L A S 3 the set of the links obtained by applying only type A operations. For these links, all the components with index 1 are unknotted and unlinked. In these three cases, the round 1-handle C is a connected sum of thick and solid tori and their attachments imply at least one inessential circle.
On the other hand, operations I V, V and V I imply only essential attachments. We refer to them as type B operations.
We want to know which links are associated to round 1-handles that are iterated connected sum of tori. So, we begin by characterizing the 3-manifolds obtained from successive connected sum of tori. Let us apply this definition in order to make connected sums of thick and solid tori. Let τ 1 and τ 2 denote two solid tori D 2 × S 1 . Consider S 1 and S 2 2-spheres embedded in τ 1 and τ 2 , respectively. In order to make the connected sum, we have to reverse one of the 2-spheres. Let us assume that S 2 is reversed; then, a 3-ball with a toroidal hole is obtained. The boundary of this 3-ball is identified with S 1 and the torus τ 1 with a toroidal hole is obtained.
Then, τ 1 #τ 2 is a solid torus with a toroidal hole ( Fig. 1). Let (T × I ) denote the product of a 2-torus T by an interval I, called a thick torus. If we consider a 2-sphere embedded in a thick torus and we reverse it, we obtain a 3-ball with two linked toroidal holes inside; let us refer to them as a hopf hole (see Fig. 2). So, the connected sum of two thick tori, (T 1 × I )#(T 2 × I ), is a thick torus with a hopf hole.
Finally, the connected sum of a thick torus (T × I ) and a solid torus τ is a thick torus (T × I ) with a toroidal hole (see Fig. 3).
The realization of τ #(T × I ) yields a solid torus with a hopf hole if we reverse the 3-ball in the thick torus. When the tori are embedded in S 3 both results are equivalent; so, there is commutativity for the connected sum of tori. The identity element of the connected sum is S 3 . These operations verify commutativity and associativity on S 3 ; so, by iteration, it is easy to prove the following proposition:

Proposition 2 Let T denote a torus (thick or solid) without toroidal or hopf holes. Then
(T j × I ) = T with n toroidal holes and m hopf holes.
(2) 186 J Dyn Diff Equat (2012) 24:181-196 Proof Suppose that T is a solid torus. We prove by induction that By definition, when we make the connected sum of two solid tori τ and τ 1 , we reverse one of them into a 3-ball and identify its boundary with a 2-sphere embedded in the other torus.
Let us see that this operation does not depend on the torus chosen. If we reverse τ 1 we obtain the solid torus τ with one toroidal hole.
The same occurs if we reverse τ. Now, if the connected sum with another solid torus τ 2 is made, reversing τ 2 , a new toroidal hole appears; then, we have a solid torus τ with 2 toroidal holes. But if we make the connected sum reversing the torus with the toroidal hole, we have a 3-ball with two toroidal holes; then, when identifying its boundary with a 2-sphere embedded in τ 2 , we also obtain a solid torus with 2 toroidal holes. Now, suppose that n − 1 connected sum of solid tori have been made and we have that If the connected sum with another solid torus τ n is made, a new toroidal hole in τ appears; then: Similarly, it is easy to prove by induction that: So, these four equalities can be expressed as (2).
Let us remark that, as a consequence of this proposition, each time that the connected sum is made with a solid torus a new toroidal hole appears. If the connected sum is made with a thick torus a hopf hole appears.
In the next section we relate iterated connected sum of tori to the first three operations defined in [8, Theorem 1].

Links Coming From Iterated Sum of Tori
As commented before, Wada [8] describes the different types for the triple (C, ∂ − C, γ ) associated to a fat round 1-handle on S 3 . The results obtained in the previous section allows us to distinguish a special type of links coming from type A operations: the links on S 3 associated to fat round handles that are iterated connected sum of tori. We remark that in order to obtain a link on S 3 the components of the complement of C must be filled with 2-handles in the last step of the filtration in the RHD. This is equivalent to fill the components of the complement of T with attracting orbits. Moreover, we prove in Sect. 5 that the flow associated to these links is also a connected sum of flows.
Let us refer to this type of links on S 3 as links coming from iterated connected sum of tori. We include the hopf link in this definition, corresponding to 0 iterations. Let us denote by L I S 3 the set of these links and letL I S 3 denote the complement of the set L I S 3 with respect to L A S 3 .
In the following, h denotes the (0, 2)-hopf link, d denotes an unknotted and unlinked periodic orbit with index 0 or 2 and u denotes and unknotted periodic orbit with index 1. The power c p means the split sum of p components of type c.
where c can be an unknot d or a hopf link h and p ≥ 0 is the number of iterations.
Proof As we have proved in the Proposition 2, the fat round 1-handle associated to the iterated connected sum of tori is (T j × I ) = T with n toroidal holes and m hopf holes We obtain a NMS link on S 3 by attaching 0 or 2 handles. So there are attractive or repulsive orbits just in the core of the toroidal holes and saddle periodic orbits on the boundary of the 3-balls implied in each connected sum. In the case of hopf holes, one orbit must be attractive and the other must be repulsive.
Then, after p = n + m connected sums of tori, the corresponding link l on S 3 can be written as: where c is either a hopf link h or an unknot d. So, the link l is formed by the split sum of hopfs and components of type d and u with the following ratio: In the following result we characterize the links of the set L I S 3 in terms of Wada operations.

Lemma 4.2 Let l be a NMS link on S 3 , then l ∈ L I (S 3 ) if and only if l is obtained from
(0, 2)-hopf links by applying the following operations: Proof As type A operations commute (see [2]), in this proof we refer to combinations of operations and no matter the order of application of them.
Firstly, let us see that if l ∈ L I (S 3 ), it can be written in terms of the operations given above.
If l ∈ L I (S 3 ), then l is a (0, 2)-hopf link or l it is of the form (4): • If c is a hopf link, l has the form: Then, we have: with l 1 and l 2 links obtained from operations I and operations I I with the hopf link in the second argument.
• If c is a component d and m > 1, we can change the order of the components and then l is expressed as: This link can be written in terms of n + 1 operations I I and m − 1 operations I on hopf links. As above, we have: with l 1 and l 2 links obtained from operations I and operations I I with the hopf link in the second argument. • If c is a component d and m = 1, then l is of the form: In this case l can be written in terms of n + 1 operations I I on hopf links, so l can be expressed as where l 1 is a link obtained from operations I I on hopfs links in the second argument. where l 1 is a link obtained from operations I I on hopfs links in the second argument and one operation I I I on two hopfs links.
Conversely, let us see that every link obtained from (0, 2)-hopf links by applying the operations given above verifies condition (3) and then, the link comes from iterated sum of tori.

As condition (3) is verified, l ∈ L I (S 3 ).
In both cases, l can be written as: where l 1 has been obtained with operations I I with h in the second argument or a combination operations I I with h in the second argument and one operation I I I on two hopf links. • Suppose that l = I (l 1 , l 2 ), where l 1 and l 2 are of the form given in (5), (6), or (7). As the links l 1 and l 2 verify condition (3): Let us remark that operation I I I can not be iterated to obtain a link in L I (S 3 ); that is, we can combine operations I I I with operations I and I I, but operation I I I can not appear in the argument of another operation I I I. In fact, as we see in the following section, these type of links characterize flows with heteroclinic trajectories connecting saddle orbits.
In the following lemma, we characterize the links in the complement of L I (S 3 ):

Lemma 4.3 Let l be a NMS link on S 3 , then l ∈L I S 3 if and only if l can be written as:
Proof (⇒) If l ∈L I S 3 , l is a link in L A S 3 and does not verify condition (3); then, the number of c s is equal or less than the number of u s. Therefore where q ≥ 0 and p ≥ 2. Let us apply induction for showing that l can be written as l · u with l ∈ L A S 3 . For q = 0 : Suppose that it is true for q = n : If l n−1 ∈ L A S 3 then l −1 · u ∈ L A S 3 (see [2,Lemma 1]). Therefore, l n ∈ L A S 3 and for q = n + 1 : So, the links that do not come from iterated connected sum of tori can be written as l · u, with l ∈ L A S 3 .
(⇐) From [2, Lemma 1], we know that, if l ∈ L A S 3 then the link l = l .u ∈ L A S 3 . Then, if l contains n orbits of type u, it means that n type A operations have been applied. In each operation, the number of u components increases in 1 and the number of c components can increase in 1 or does not change. So after n operations, it contains a number n of u components and a number of c components equal or less than n + 1.
Then, l ·u contains a number of c components equal or less than n+1 and n+1 components of type u. So, l · u does not verify the condition (3).

Connected Sum of NMS Flows on S 3 and Heteroclinic Trajectories
In this section we study the NMS flows on S 3 associated with the links coming from type A operations and we obtain the following important results: 1. These flows are connected sum of NMS flows. Moreover, we see that the flows associated to links coming from iterated sum of tori can be successively decomposed as connected sums of basic flows, that is, flows corresponding to h s and d s. 2. The flows associated to links in L I (S 3 ) have no heteroclinic trajectories connecting saddle orbits and the flows associated to links inL I (S 3 ) have heteroclinic trajectories connecting saddle orbits.

Connected Sum of Flows
The definition of the connected sum of flows in 3-manifolds given in [3] is a generalization of the connected sum of flows defined for 2-dimensional manifolds (see [7]). Given two flows ϕ 1 and ϕ 2 on the 3-dimensional manifolds M 1 and M 2 , respectively, we can form a new flow ϕ 1 #ϕ 2 on the connected sum M 1 #M 2 in the following way: Herewith the flowsφ 1 andφ 2 give a flow ϕ on M 1 #M 2 with a saddle orbit at σ i . The flow ϕ is called a connected sum of the flows ϕ 1 and ϕ 2 . Also some care needs to be taken so that the new flow is stable. If a periodic orbit in either M i is tangent to boundary D i then a homoclinic orbit would be formed.
The boundaries of the 3-balls that appears when connected sum of tori are made, are transversal to the respective flows except in the points of the circles σ i on the equator (see [3]). We use this result to show that the flow is the connected sum of flows.
The attaching of a round 1-handle by means of two inessential circles is equivalent to taking the connected sum of two thick tori. Then, for the case of flows characterized by operation I we have the following result: Now, suppose that ϕ = ϕ 1 #ϕ 2 . Then, we have a 2-sphere splitting S 3 as connected sum of two 3-spheres with the respective flow transversal to the boundary except in the points of the circle σ i on the equator. The 2-sphere bounds 3-balls on both sides. The boundaries of the 3-balls are identified by means of an orientation reversing homeomorphism and a saddle orbit appears in the equator. The link of periodic orbits of ϕ consists of the set of periodic orbits of ϕ 1 , the set of the periodic orbits of ϕ 2 and a trivial orbit with index 1 on the circles σ i on the equator, then: Thus, a flow characterized by a link l = I (l 1 , l 2 ) is always the connected sum of two flows on S 3 .
If a flow is characterized by means of operations I I or I I I, it can also be connected sum of flows.
A link on a solid torus can be completed to a link on S 3 by adding a component k corresponding to an attractive or repulsive orbit in the core of the complement of this torus in S 3 , which is another solid torus. Conversely, if l is a link of periodic orbits of a NMS flow on S 3 and k corresponds to an attractive (repulsive) orbit, the link l − k characterizes the remaining flow on a solid torus.
Therefore, the following results are obtained immediately.

Corollary 1
Let ϕ and ϕ 1 be NMS flows on S 3 characterized by the links l and l 1 respectively; let ϕ 2 be a NMS flow on the solid torus corresponding to a component d. Then: Corollary 2 Let ϕ be a NMS flow on S 3 characterized by the link l. Let ϕ 1 and ϕ 2 be NMS flows on solid tori corresponding to components d 1 and d 2 , respectively. Then: As commented before, another significant result is that we characterize those flows that can be successively decomposed as connected sums of basic flows; for basic flows we mean flows corresponding to h s and d s. In this case, we say that the flow is a decay of connected sums of flows. These flows are just the flows associated to links coming from iterated sum of tori. Proof (⇐) There exists an equivalence between connected sum of tori and connected sum of flows; the 3-balls implied in the connected sum of tori correspond to the 3-balls implied in the connected sum of flows because the boundaries of the 3-balls are transversal to the respective flows except in the points of the circles σ i on the equator and these circles correspond to the saddle orbits.
As a consequence of the Proposition 2, each time that a connected sum is made with a thick torus a hopf hole appears; if the connected sum is made with a solid torus a new toroidal hole appears. This is equivalent to the connected sum with a basic flow (see Fig. 5).
Let l be a link in L I (S 3 ) different from the polar flow. Then, from Lemma 4.2, l is one of the following links: where l 1 , l 2 ∈ L I (S 3 ) and, from Theorem (5.2) and its corollaries, the flows associated are the connected sum of the flows associated to l 1 and l 2 , l 1 and d, d 1 and d 2 , respectively. As l 1 , l 2 ∈ L I (S 3 ), they can be further decomposed as above until arrive to basic flows.
(⇒) If a flow ϕ is a connected sum of two flows ϕ 1 and ϕ 2 , the link of periodic orbits of ϕ is the split sum: where l 1 and l 2 are the links of periodic orbits of the flows ϕ 1 and ϕ 2 , respectively, and u represents the saddle orbit on the equator of the 2-sphere implied in the connected sum of flows.
If the flow ϕ is a decay of connected sum of flows, l 1 and l 2 can also be written as split sums of links and one orbit u: l = l 1 · l 2 · u = (l 11 · l 12 · u) · (l 21 · l 22 · u) · u and so on until arrive to basic components h s and d s. Each connected sum implies three components, one of them a u orbit. So, the number of u s is one less than the number of h s and d s. Then, l verifies the condition (3) and l ∈ L I (S 3 ).

NMS Flows with Heteroclinic Trajectories Connecting Saddle Orbits
Let F A (S 3 ) denote the set of NMS flows associated to the links L A (S 3 ) and let F I (S 3 ) denote the set of NMS flows associated to the links L I (S 3 ). In the following result we relate the existence of heteroclinic trajectories connecting saddle orbits to the type of link.

Theorem 5.4 Let ϕ ∈ F A (S 3 ). Then, ϕ has no heteroclinic trajectories connecting saddle orbits if and only if ϕ ∈ F I (S 3 ).
Proof (⇒) Let us show that if ϕ ∈F I (S 3 ) then ϕ has heteroclinic trajectories connecting saddles.
The link associated to a flow ϕ ∈F I (S 3 ) does not come from iterated connected sum of tori. So, the link can not be written as in Lemma 4.2. Therefore, it implies at least two type A operations and it is obtained by attaching the round 1-handle to a solid torus by means of one essential circle. This solid torus has toroidal or hopf holes inside generated by previous attachments.
There are two possibilities: 1. The round 1-handle is attached to the solid torus (with toroidal or hopf holes inside) by means one essential and one inessential circles (operation I I I is made). 2. The round 1-handle is attached to the solid torus (with hopf or toroidal holes inside) by means one essential circle and to another thick torus (with hopf or toroidal holes inside) by means of one inessential circle (operation I I is made).
In both cases there exists an invariant manifold of one saddle orbit that cuts the boundary of the torus by means of one transversal circle (see Fig. 6). The essential circle of the attachment of the new round 1-handle is a longitudinal circle and therefore one invariant manifold of the new saddle cuts the boundary of the solid torus by means of one longitudinal circle; consequently, both circles intersect and heteroclinic trajectories between the two saddles appear.
(⇐) As we have proved in Theorem 5.3, the flows characterized by links coming from iterated sum of tori can be decomposed as a decay of connected sum of basic flows. Then, each of them can be isolated by a 3-ball and there are no heteroclinic trajectories connecting saddle orbits (see Figs. 4, 5 and 6). In [3] we showed that NMS flows with one saddle orbit also "live" in general lens spaces. Now, from the results of this paper, this statement can be generalized to the case of flows with more than one saddle orbit.