Contact Homology of Orbit Complements and Implied Existence

For Reeb vector fields on closed 3-manifolds, cylindrical contact homology is used to show that the existence of a set of closed Reeb orbit with certain knotting/linking properties implies the existence of other Reeb orbits with other knotting/linking properties relative to the original set. We work out a few examples on the 3-sphere to illustrate the theory, and describe an application to closed geodesics on $S^2$ (a version of a result due to Angenent).


Introduction
Let V be a closed 3-manifold with a contact form λ and associated Reeb vector field X λ . In this article we will be concerned with the following question about closed orbits: General Question: If one has a Reeb vector field with a known set of closed Reeb orbits L, can one deduce the existence of other closed Reeb orbits from knowledge about L? Let us call an affirmative answer to such a question an implied existence result.
We will see that in some instances one can obtain an affirmative answer to this question. These results agree with affirmative answers to analogous questions for certain similar conservative systems in low dimensions studied in e.g. [Ang05,GVV03,GVVW], but the present methods differ significantly. Specifically, to exhibit implied existence we study cylindrical contact homology on the complement of L for non-degenerate contact forms λ imposing as few conditions on the orbit set as we can manage. We will show in detail one approach to such a theory inspired by the intersection theory of [Sie08] for the necessary compactness arguments 1.1 . We use established techniques [CH08a,Bou02] to compute this homology in some cases, and use these computations as a tool to affirmatively answer the above question in certain cases.
1.1. A version of cylindrical contact homology on Reeb orbit complements. In this paper we denote by V a 3-dimensional manifold. A one-form λ on V is called a contact form if λ∧dλ is everywhere non-zero. Such a one-form uniquely determines its Reeb vector field X λ by the equations λ(X λ ) ≡ 1, dλ(X λ , ·) ≡ 0. It also determines a distribution ξ = ker λ, which is a contact structure, and two forms λ ± induce the same contact structure if and only if λ + = f · λ − for some nowhere vanishing function. The Conley-Zehnder index of a closed orbit for this vector field is a measure of the rotation of the flow around the closed orbit -see Proposition 2.4 for one characterization of the Conley-Zehnder index. The Conley-Zehnder index usually depends on a choice of trivializations, but in many cases we will consider, e.g. the tight 3-sphere, there is a global trivialization which is used to define Conley-Zehnder indices independently of choices.
1.1.1. The hypotheses (E) and (P LC). We introduce two types of technical hypotheses below, which are not mutually exclusive. They simplify the construction of contact homology on V \L which we use to deduce implied existence. We will give some examples of forms on the 3-sphere later which we hope will help to clarify these hypotheses.
We will consider pairs (λ, L) in which λ is a non-degenerate contact form on V and L is a link composed of closed orbits of the Reeb vector field of λ. Sometimes it may be convenient to refer to the subset of closed Reeb orbits with image contained in L; we will abuse language and denote this subset of closed Reeb orbits by L again 1.2 .
Following standard terminology 1.3 , an orbit is elliptic if the eigenvalues of its linearized Poincaré return map (a linear symplectic map on ξ) are non-real.
Definition 1.1. We will say (λ, L) satisfies the "ellipticity" condition (abbreviated by (E)) if • each orbit in L is non-degenerate elliptic (including multiple covers) • each contractible Reeb orbit y not in L "links" with L in the sense that for any disc with boundary y, L intersects the interior of the disc.
We shall see these hypotheses force the compactness of moduli of holomorphic cylinders in V \L necessary to define cylindrical contact homology. Before we continue, let us note a very simple example: Example 1.2. Consider R 4 = R 2 × R 2 with the usual polar coordinates (r i , θ i ) (i = 1, 2) on each R 2 factor of R 4 and let E be the ellipsoid determined by the equation where a, b are positive constants. The one form restricted to E defines a contact form which we denote λ. If the ratio a/b is irrational, then there are precisely two geometrically distinct closed Reeb orbits, P = S ∩ C × {0}, Q = S ∩ {0} × C. They are both non-degenerate and elliptic with Conley-Zehnder indices CZ(P k ) = 2 k 1 + a b + 1, CZ(Q k ) = 2 k 1 + b a + 1 Since the linking numbers (P k , Q ) = k, (P , Q k ) = k, the pairs (λ, P ), (λ, Q ), (λ, P Q ) all satisfy (E).
1.2 i.e. L refers to both a closed embedded submanifold which is tangent to the Reeb vector field X, as well as the set of solutions x : R/(T · Z) → V,ẋ = X(x) modulo reparametrization with image contained in L (this includes multiple covers of components of L). 1.3 e.g. [KH95]; see also Proposition 2.4 for another characterization of elliptic.  There is another way to control compactness of holomorphic cylinders. Again assume L is a link of closed orbits for λ, and let [a] be the homotopy class of a loop a in the complement. See Figure 1(1) for an example, and Figure 1(2) for a counter-example. The class [a] is meant to be analogous to a proper braid class studied in [GVV03,GVVW]. Both conditions (E) and (P LC) contain a "no contractible orbits" hypothesis. Such a hypothesis is necessary to preclude "bubbling", which is an obstruction to defining cylindrical contact homology in general [EGH00] 1.4 . We point to Examples 6.1, 6.2 below for concrete examples of orbits and (Morse-Bott) contact forms on the tight S 3 satisfying conditions (E) and (P LC) respectively.
In Section 4 we associate with a pair (λ, L) satisfying (E) and the homotopy class [a] of a loop a, or with a triple (λ, L, [a]) satisfying (P LC), an invariant 1. 4 The necessary hypotheses can actually be weakened with greater care. We remark that the "no contractible orbits" hypothesis in (P LC) is more restrictive.

CCH
[a] * ([λ] rel L). It is invariant in the sense that it depends only on the equivalence class ≡ in case (λ, L) satisfy (E) (that is, it depends on the Conley-Zehnder indices of the elliptic orbits in L), and in the case (P LC) it depends only on equivalence classes of ∼ (i.e. only on the contact structure ξ). CCH [a] * ([λ] rel L) will be an isomorphism class of graded vector spaces, so it makes sense to say whether or not it is zero.
It follows easily from the constructions of the invariants in Section 4 that: 1.2. A general implied existence result. The non-degeneracy assumption on the form λ in Theorem 1.4, and the implicit assumptions about contractible orbits are unfortunate, but can be weakened considerably by stretching-the-neck and compactness arguments given in Section 5.
Theorem 1.5. Suppose λ is a contact form with a closed orbit set L. Suppose either • L is non-degenerate and elliptic, or • (1) L is such that every disc F with boundary ∂F ⊂ L and [∂F ] = 0 ∈ H 1 (L) has an interior intersection with L, and (2) [a] is a proper link class relative to L. To clarify a possible point of confusion in the statement of Theorem 1.5, when we say CCH

If [a] is simple and CCH
[a] * ([λ] rel L) = 0 we mean that there is a λ ≡ λ resp. λ ∼ λ satisfying (E) resp. (P LC) for which CCH [a] * can be computed and is non-zero. When [a] is a proper link class, this theorem requires no non-degeneracy hypotheses whatsoever on λ, and in the ellipticity case requires only non-degeneracy of the orbits in L. For a concrete example where this is applicable see Example 6.2, and Section 6.4.
1.3. Some applications in S 3 . The results here are corollaries of Theorem 1.5 in Section 1.2 and the examples and computations of Section 6.
One application is the existence of analogs of "(p, q)-type orbits" if a certain pair of closed Reeb orbits exists and satisfy a "non-resonance" condition (namely the pair violates the "resonance" condition introduced in [BCE07] when the differential of cylindrical contact homology in the tight S 3 vanishes). In the following statement (·, ·) denotes the linking number of two knots in S 3 .
Corollary 1.6. Let λ be a tight contact form on the 3-sphere. Suppose that there is a pair of periodic orbits L 1 , L 2 such that L 1 L 2 is the Hopf link with self-linking 0 1.5 . Suppose L 1 and L 2 (and all multiple covers) are non-degenerate elliptic, and 1.5 By Theorem 4.1 in [EHM], all such links are transversely isotopic. The pair H 1 H 2 of Example 6.1 is one such example.
Remark 1.7. There are no hidden hypotheses on the contact form being either dynamically convex or non-degenerate (except in the explicit hypotheses on the orbits L 1 , L 2 ). There are other approaches to prove such a result using the surface of section constructed by [HWZ98], [HS10] and applying the result of [Fra92]; however, the construction of these surfaces of section require additional hypotheses, so it is not clear that they always exist. The result above applies even when there may be no such surface of section.
Proof. By Theorem 4.1 in [EHM], there is a transverse isotopy from L 1 L 2 to the link H 1 H 2 in Example 6.1, which extends to an ambient contact isotopy, and therefore a contactomorphism taking L 1 L 2 to H 1 H 2 . By applying the given contactomorphism we can assume (λ, L 1 L 2 ) ≡ (λ , H 1 H 2 ) with (λ , H 1 H 2 ) from Example 6.1. It follows immediately from Theorem 1.5 (each component is elliptic by hypothesis) and the computation in Example 6.1.
It is interesting to interpret this result in the case the contact form is obtained from a metric on S 2 . Angenent [Ang05] proved that if one has a simple, closed geodesic for a C 2,µ -Riemannian metric on S 2 , then whenever Poincaré's inverse rotation number ρ = 1, for every p/q ∈ (ρ, 1) ∪ (1, ρ) there is a geodesic γ p,q with the flat-knot type of a (p, q)-satellite geodesic. The following Corollary is a version of this result with stronger hypotheses on the rotation number and weaker conclusions about the resulting geodesics, but allowing the metric to be reversible Finsler. It is a direct consequence of Corollary 1.6 applied to the geodesic flow of a reversible Finsler metric on S 2 which will be proved in Section 6.2 Corollary 1.8. Let F be a reversible Finsler metric on S 2 . Suppose γ is a simple, closed geodesic with irrational inverse rotation number ρ = 1. Then for every pair of relatively prime integers p, q such that p + q 2q there is a geodesic γ p,q with the following topological property. If γ, γ denote the lifted double covers of the geodesic to S 3 (traversed "forwards" and "backwards"), then γ p,q has linking number p with γ and q with γ. In particular, the geodesics are geometrically distinguished (up to a double-count by traversing γ p,q oppositely).
Corollaries 1.6 and 1.8 will be improved to cover the degenerate and hyperbolic cases as well in collaboration with Umberto Hryniewicz and Pedro Salomão [HMSa].
We give other examples by considering fibered knots or links in the 3-sphere, in which conclusions can be drawn without non-degeneracy hypotheses 1.6 . One class of examples is obtained as follows. Let us say that a knot B in the tight 3-sphere is a tight fibered hyperbolic knot if the following conditions hold: (1) (fibered) The knot B is the binding of an open book decomposition of S 3 (2) (tight) The contact structure supported by the open book is the tight contact structure on S 3 (3) (hyperbolic) The monodromy map h of the associated open book decomposition is pseudo-Anosov (4) The map h * − I : H 1 (S; R) → H 1 (S; R) (where S is a page of the open book) is invertible We remark that the fourth condition actually follows automatically from the first; it will be a convenient fact (see section 6.5 where we also explain why this follows). We make some further brief remarks about this class of knots in section 6.5. For now we merely note that there are infinitely many examples, such as the Fintushel-Stern knot which is the Pretzel knot (−2, 3, 7). For tight fibered hyperbolic knots in S 3 (see Section 6.5): Corollary 1.9. Suppose λ is a tight contact form on S 3 . If its Reeb vector field has a closed orbit which is a tight fibered hyperbolic knot realizing the Thurston-Bennequin bound as a transverse knot, then there are infinitely many geometrically distinct closed Reeb orbits and the number of such orbits of period at most T is bounded below by an exponential function of T .
The required calculation is essentially due to Colin-Honda [CH08a], but turns out simpler because one does not need to consider holomorphic curves that intersect the binding.
Remark 1.10. Again, there are no hidden non-degeneracy hypotheses, and it applies to all tight contact forms (not only dynamically convex ones). Theorem 1.5 allows one to draw conclusions about free homotopy classes which must contain closed Reeb orbits.
Finally, cylindrical contact homology is usually thought to be only applicable to tight contact structures. However, after removing a certain orbit set, it may 1.6 In Corollary 6.3 we will also give another example, related to the example used in the previous Corollary, which does not assume non-degeneracy of the orbit set L. We postpone the statement to section 6 where it will be easier to describe the link L. be possible to apply cylindrical contact homology on the complement of the orbit set. For example, we can take the figure eight knot in the 3-sphere, which satisfies properties 1, 3, 4 in the definition of fibered hyperbolic supporting knots, but not property 2 (that is, it is fibered and hyperbolic, but does not support the tight contact structure). Hence it supports an over-twisted contact structure; Theorem 1.5 still applies from which one can deduce the following: Corollary 1.11. Let λ be a contact form for the (over-twisted) contact structure on S 3 supported by the open book decomposition with binding the figure eight knot, page diffeomorphic to a once-punctured torus, and monodromy map given by the matrix transformation 2 1 1 1 Suppose λ has a closed Reeb orbit transversely isotopic to the binding. Then the number of geometrically distinct periodic orbits of action at most N grows at least exponentially in N .
We note that in the above one can describe homotopy classes that must contain a closed orbit.
We will go over the details of this particular example in Section 6. An analogous result will hold for any (non-tight) fibered hyperbolic knot in S 3 , examples of which are plentiful [CL].
1.4.1. Comments. It is possible to study the cylindrical contact homology of stable Hamiltonian structures on complements of elliptic Reeb orbits (or in proper link classes) in general. For example, the mapping tori of Hamiltonian diffeomorphisms of surfaces fits in this category. In this case there are clearly no contractible Reeb orbits, and bubbling of spheres is not difficult to rule out in most cases. The portion of the loops space corresponding to the first return map are all simple homotopy classes and thus CCH * is well-defined and coincides with Floer homology. This particular case has been carried out explicitly (at least in the case the surface is a disc) and is called "braid Floer homology" in [GVVW].
1.4.2. Outline. In section 2 we recall the geometric and analytic set-up. Section 3 describes an intersection theory of pseudoholomorphic curves in symplectizations [Sie09] and derives some compactness results from this intersection theory. In section 4 we describe the chain complexes, maps and homotopies. In section 5, we describe how to draw conclusions when the contact form might be degenerate or there may be contractible (in V \L) closed Reeb orbits. In section 6 some explicit examples on the 3-sphere are worked out in detail.
1.4.3. Acknowledgments. This work is a continuation of the author's Ph.D. thesis; I thank my adviser Helmut Hofer for his invaluable guidance and support. I thank Alberto Abbondandolo, Umberto Hryniewicz, Richard Siefring, Robert Vandervorst, and Chris Wendl for influential discussions, and Matt Hedden for answering some questions about fibered knots in S 3 . Finally, I thank Peter Albers and Pedro Salomão for many helpful suggestions, and the anonymous referee for a careful reading and for generously offering numerous detailed improvements to the readability.
2. Review of geometric and analytic set-up 2.1. Symplectizations. Let (V, ξ) be a co-oriented contact manifold. Then there is a symplectic manifold associated with it, W ξ= ξ ⊥ \0 (the annihilator of ξ in T * V minus the zero section) which is a sub-bundle and symplectic submanifold of T * V with respect to the exterior derivative of the Liouville form, dθ can . The symplectic manifold (W ξ , ω ξ = dθ can | W ξ ) is called the symplectization of (V, ξ). Because ξ is co-oriented there are two components, W ξ,± . We will consider only the component W ξ,+ and when we write W ξ we mean this component only. There is a natural R action (a, λ) → a * λ = e a · λ given by scalar multiplication in the fibers, and a natural projection to V by restricting the projection T * V → V to W ξ .
Any positively co-oriented contact form λ for ξ defines a global section of W ξ which gives an identification Given such a λ we denote by X the Reeb vector field, defined by We will later assume that λ is non-degenerate, a generic condition meaning that no closed periodic orbit has a Floquet multiplier equal to 1; for the moment this is not necessary. Given a contact form λ, we define a splitting of T W ξ : We might abuse notation and use X to denote X and ξ for ξ 2.1 .
2.1.1. Almost-complex structures. Given a contact form λ for ξ, consider the symplectic vector bundle (ξ, dλ) over V and let J (ξ) be the set of dλ-compatible almostcomplex structures. This defines a set of almost-complex structures on ξ via We denote the set of almost-complex structures on T W ξ that arise in this way J (λ); we can identify this with J (ξ). The almost-complex structures in J (λ) are called cylindrical almost-complex structures.

2.1
The subspaces R · X and ξ both depend on the choice of λ.
This is an exact symplectic cobordism between (V, λ − ), (V, λ + ). Let which fits together smoothly because J 0 is R-translation invariant. Let δ > 0 be an arbitrarily small but fixed number. Choose diffeomorphisms x) (using the coordinates ψ λ0 for W ξ ), and almost-complex structures J R = G (δ,R) * J R . We may concatenate the matching Hamiltonian structured ends of (W ξ , J 1 ), (W ξ , J 2 ) to get (W ξ , J 1 ) (W ξ , J 2 ) (see section 3.1). Then we have a diffeomorphism to the concatenation G : W ξ → (W, J 1 ) (W, J 2 ) defined by 2.2 By ∂ + we mean ∂a is outward pointing and by ∂ − we mean ∂a is inward pointing.
It follows from the construction of J R that G * J R converges to the concatenated almost-complex structure J 1 J 2 uniformly on compact subsets of W ξ \λ 0 (V ).
In (W ξ , J R ), denote the regions W ± = W ± (λ ± ), and denote by W 0 the region Let us denote the set of such almost-complex structures J R constructed in this way from J 1 , J 2 by J (J 1 , J 2 ) ∼ = [0, ∞).
2.1.4. Holomorphic maps and finite energy. A J-holomorphic map on a punctured Riemann surface (Σ, j, Γ) is a map U : Σ\Γ → W ξ that satisfies the Cauchy-Riemann equation DU + J · DU · j = 0 We only consider solutions of this equation having finite energy; for the definition of finite energy J-holomorphic curves we refer to [BEH + 03].
A crucial example for cylindrical almost-complex structure J are the trivial cylinders over a periodic Reeb orbit x of some period T : it is a solution of the form (for (s, t) ∈ R × S 1 ) ψ λ • U (s, t) = (T s + a 0 , x(T · t)). If a contact form is (Morse-Bott) non-degenerate 2.3 , then at any (non-removeable) puncture a finite energy J-holomorphic map is asymptotic to a half-trivial cylinder over a closed Reeb orbit in one of the ends W ± , see [HWZ95] or [BEH + 03]. A puncture is called positive (resp. negative) if it is asymptotic to the positive (resp. negative) half of a trivial cylinder in W + (resp. W − ). We will say that a finite energy holomorphic map is positively (resp. negatively) asymptotic to a Reeb orbit x if there is a positive (resp. negative) puncture z (perhaps already specified) at which U is asymptotic to a half-trivial cylinder over x in W + (resp. W − ).
When the contact form is (Morse-Bott) non-degenerate, then the asymptotic convergence to the trivial half-cylinder mentioned above is exponential in nature and one has an asymptotic formula [HWZ96,Sie08]. If one looks at the space of maps asymptotic to fixed closed Reeb orbits with a given rate of exponential convergence, then there is a Fredholm theory for the Cauchy-Riemann operator [Dra04,Wen10].
One more crucial property of finite-energy holomorphic curves is the compactness theory of [BEH + 03]. In section 8 of [BEH + 03], a notion of a k − |1|k + level holomorphic building and convergence of a sequence of holomorphic maps to that building is given. We refer to that paper for the definitions. Theorem 10.3 of [BEH + 03] extends to the case where W has cylindrical ends as well, so that the space of J R -holomorphic maps can be compactified for sequences with R ↑ ∞. The compactification consists of a holomorphic building in (W − (λ 0 ), J 1 ) (W − (λ 0 ), J 2 ) together with a J 0 (λ 0 ) holomorphic building, all of which glue together to W − (λ 0 ) W + (λ 0 ) = W ξ . We can think of these as k − |1|k 0 |2|k + level holomorphic buildings, which we describe next.
Let S be a decorated Riemann surface [BEH + 03] S, with each smooth component assigned a level labeled 1, 2 or (λ i , j i ) where i ∈ {1, 0, +} and 1 ≤ j i ≤ k i . Write S = S 1 ∪ S 2 , where S 1 consists of the levels labeled either (λ −,0 , ·) or 1, while S 2 consists of the parts of the domain labeled (λ 0,+ , ·) or 2. If k 0 = 0 then S 1 ∩ S 2 = ∅. Let (U ) be an assignment to each smooth component S a ⊂ S of a finite energy holomorphic map U a with domain S a such that 2.3 We will always assume that all contact forms have non-degenerate asymptotic orbits, so this condition is always satisfied in everything that follows.
• If one considers the subset of (U ) with domain in S 1 , it is a holomorphic building in W 1 := (W ξ , J 1 ) • If one considers the subset of (U ) with domain in S 2 , it is a holomorphic building in W 2 := (W ξ , J 2 ) • The maps glue together to give a continuous map (U ) : S → W 1 W 2 This data will be called a k − |1|k 0 |2|k + holomorphic building. It is called stable if both (U ) i = (U )| Si (i = 1, 2) are stable.
Let U k be a sequence of J R k holomorphic maps with the same asymptotic orbits and genus; denote the domains by Σ k (each a Riemann surface). Let (U ) be a building as above with domain S. If R k is bounded then there is a limit which is a J R∞ -holomorphic building, where R ∞ is an accumulation point of the sequence. Else, suppose without loss of generality that R k ↑ ∞. The sequence U k converges to the building (U ) if there exists a sequence of diffeomorphisms φ k : S → Σ k converging as decorated Riemann surfaces [BEH + 03] so that C1 For each level ( is the identification map described above). The asymptotic behavior of finite energy surfaces together with an application of Stokes' theorem shows -for finite energy solutions of the almost-complex structures that we consider -the energy E(U ) is bounded by the action of the positive asymptotic orbits. Then by [BEH + 03] Lemma 2.1. For J ∈ J ( J − , J + ), or J (λ), or J = J R ∈ J (J 1 , J 2 ), the space of J-holomorphic finite energy cylinders positively asymptotic to a closed orbit x of λ + is precompact in M J .
2.1.5. A restricted class of almost-complex structures. Suppose λ ± are two contact forms for ξ satisfying λ − ≺ λ + and that L ⊂ V is a link that is tangent to the Reeb vector fields for both. With the projection π : W ξ → V consider Z L = π −1 (L); which is a union of embedded cylinders. We have Lemma 2.2. Z L is an embedded symplectic submanifold of (W ξ , ω ξ ).
Proof. Z L is embedded, and each component of Z L is J(λ)-holomorphic for any λ with kernel ξ and Reeb vector field tangent to L, and any J ∈ J (ξ). By hypothesis on L there exist such λ. Since each J(λ) is ω ξ -compatible, we see Z L must be symplectic.
The proof of this fact is standard after the observations that (1) Z L is an embedded symplectic manifold, and (2) that regardless of J ± (λ ± ), Z L is J holomorphic in W ± . One considers the space of metrics for which, along Z L , T Z L and (T Z L ) ω (the symplectic complement) are orthogonal and then mimics the usual proof that the space of almost-complex structures is non-empty and contractible (e.g. [HZ94]). We therefore omit the details. Denote the space of smooth paths [0, 1] → J ( J − , J + : Z L ) by J τ ( J − , J + : Z L ). Notice that if J 1 ∈ J ( J − , J 0 : Z L ) and J 2 ∈ J ( J 0 , J + : Z L ) then for each J R ∈ J (J 1 , J 2 ) the set π −1 (L) is J R -holomorphic as well.
2.2. Review of some facts about Conley-Zehnder Indices. Let x be a closed Reeb orbit, x be the simple Reeb orbit with the same image as x, and Φ be a trivialization Φ : S 1 × C → x * ξ. We denote by CZ Φ (x) the Conley-Zehnder index of x with respect to the trivialization obtained from Φ by the obvious degree m x covering of S 1 × C (where m x is the covering number of x over x), unless it is specified that Φ is a trivialization of x * ξ itself (in which case we will compute with respect to Φ). If the trivialization is clear in a given context we drop the subscript Φ from CZ Φ .
One very useful fact about the Conley-Zehnder index is that it grows almostlinearly for Reeb orbits in dimension 3 (see e.g. [HWZ03] Theorem 8.3, or [Hut02]) in the following precise sense.
Proposition 2.4. Let γ be a periodic orbit of minimal period T , and Ψ a trivialization of the contact structure over that orbit. The Conley-Zehnder index CZ Ψ (γ k ) is monotone in k. Moreover, one has the following characterization of the index of its coverings γ k : • If γ is elliptic, there exists a unique θ so that the complex eigenvalues of Φ T are e ±2πiθ , and for all covering numbers k, CZ Ψ (γ k ) = 2 k · θ + 1. • If γ is hyperbolic then there exists an integer n so that CZ Ψ (γ k ) is k · n.
Another very useful fact is a relationship between the Conley-Zehnder index and the winding of eigensections of the asymptotic operator associated to the orbit x.
Definition 2.5. Let P be a simple, closed Reeb orbit, and k ≥ 1 be an integer. We define α − Φ (P, k) to be the winding number of the eigensection associated to the largest negative eigenvalue of the asymptotic operator of (P, k) (the k-fold covering of P ) with respect to Φ, and α + Φ (P, k) to be the winding associated to the least positive eigenvalue (see [Sie09]).
It is proved in [HWZ95] that the the eigenvalues can be ordered (with multiplicity) so that the winding of the corresponding eigensections is non-decreasing and increases by 1 every second eigenvalue, and that the Conley-Zehnder index is given by is the parity of the Conley-Zehnder index and is equal to either 0 or 1 depending on whether the winding numbers of the 'extremal' positive/negative eigenvalues agree or disagree: In particular, the Conley-Zehnder index is odd if and only if the winding of the eigensections associated with the eigenvalues nearest zero differ. This is important because it implies by the asymptotic formulas of e.g. [HWZ96,Sie08] that at odd index orbits that the holomorphic curves necessarily wind around the orbit differently (depending on whether they approach positively or negatively).
Using Proposition 2.4 we see easily that α − Φ (P, k) is given by kθ if P is elliptic, and k·n 2 if it is hyperbolic (even or odd), and α + Φ (P, k) is given by kθ if P is elliptic, and k·n 2 if it is hyperbolic.
2.3. Transversality for cylinders. In this section we assume that λ is a nondegenerate contact form on V . We will consider J-holomorphic cylinders in the symplectization and transversality of the Cauchy-Riemann operator for these cylinders. Here J = J will be a cylindrical almost-complex structure. The constructions of the chain complexes in Section 4 requires the Cauchy-Riemann operator to be transverse on enough moduli spaces. In particular we will prove the following transversality Theorem. In order to state it, we recall first that a Reeb orbit is called SFT-good if it is not an even cover of another orbit with odd Conley-Zehnder index.
Theorem 2.6. By [Dra04] there is a residual subset J gen (λ) of J (λ) such that for all somewhere injective J-holomorphic curves the Cauchy-Riemann operator is transverse for J ∈ J gen (λ). Let J ∈ J gen (λ): (1) At a J-holomorphic cylinder U of index Ind(U ) ≤ 1, the Cauchy-Riemann operator is transverse at U . In particular the only index ≤ 0 cylinders are trivial cylinders.
(2) At a J-holomorphic cylinder U of index Ind(U ) = 2, and such that both asymptotic orbits are SFT-good, the Cauchy-Riemann operator is transverse at U .
Therefore the corresponding moduli spaces (modulo the free smooth R-action) consist of isolated points (index 0 or 1 case) or of isolated intervals or circles (index 2 case).
2.3.1. Some Inequalities. Let π denote the projection of T V to the contact structure ker(λ). Given a finite energy surface U = (a, u) 2.4 , the section π • T u of Ω 0,1 (T Σ, u * ξ) is quite useful. It is proved in [HWZ95] that either it vanishes identically or it has isolated zeros. We let wind π (U) denote the oriented count of zeros of this section, which is proved in [HWZ95] to be non-negative. Also, we will use the following topological quantities: • Denote by Γ the set of punctures of the map U and by Γ ± the subsets of positive and negative punctures. Also, #Γ even denotes the number of punctures asymptotic to orbits of even Conley-Zehnder index, and #Γ odd denotes the number of punctures asymptotic to orbits of odd Conley-Zehnder index. • (this requires a choice of trivialization of U * ξ, but is independent of that choice). • Ind(U ) = CZ(U ) − χ(Σ) + #Γ, and 2c N (U ) = Ind(U ) − χ(Σ) + #Γ even We will use the following theorem, proved in [HWZ95], to show that certain curves are immersed in order to apply the automatic transversality Theorem 2.8. Then it vanishes only a finite number of times, and 0 ≤ wind π (U ) ≤ c N (U ).

2.3.2.
Review of some results about transversality. The Fredholm theory here was developed in [Dra04]. There the author defines a Banach manifold where p > 2, of W 1,p loc maps from a surface of genus g with k 'positive' punctures and l 'negative' punctures into R × V which converge exponentially with weight d in W 1,p to the periodic orbits x 1 , . . . , x k at positive punctures and to y 1 , . . . , y l at negative punctures. We will really only need the cases g = 0, k = 1, l = 1. Given this structure we have a section At a zero of this section, U , we define F U to be the projection of D∂ J (U ) to the vertical part T U E ∼ = T U B ⊕ E U . It is then proved in [Dra04] that F U is a Fredholm operator. We say the Cauchy-Riemann operator is transverse or regular at U (or U is a transverse solution, etc.) if F U is a surjective operator, and that a subset of solutions is transverse if each element is a transverse solution. Its index (computed in [Dra04]) gives the dimension of parameterized (j, J)-holomorphic maps near U when U is transverse. The dimension of the moduli space (allowing j to vary and dividing by automorphisms of the domain) is then given by the topological quantity Theorem 2.8. [Wen10] Suppose that U is an immersed finite energy surface. Letting g denote the genus of the domain of U , the linearized operator is surjective if Ind(U) ≥ 2g + #Γ even − 1.
We also need results for somewhere-injective curves. Following the notation of [Dra04], define the set M = M(x 1 , . . . , x n ; y 1 . . . , y n ) the set (C, J) of pairs consisting of non-parametrized curves C and compatible almost-complex structures J ∈ J (where J = J (λ) or J ( J − , J + )) for which C is J-holomorphic (for some parametrization) positively asymptotic to the Reeb orbits x i and negatively to the y j . In case J = J (λ) one quotients by the free R-action on solutions as well. We have the following from [Dra04] (Theorem 1.8 and its Corollary) which holds in any dimension: Theorem 2.9. The set M carries a structure of a separable Banach manifold. The projection map η : M → J , η(C, J) = J, is a Fredholm map with Fredholm index For regular values J of η, M J = η −1 (J) is a smooth, finite dimensional manifold whose dimension agrees with the above index Ind(C). By the Sard-Smale theorem, the set of regular values is a residual set. Consequently there is a residual set J gen ⊂ J of compatible almost-complex structures such that for every J ∈ J gen if U : S\Γ → R × V is a somewhere injective finite energy surface for J, then Ind(U) ≥ 1 provided π • T u does not vanish identically.
Theorem 2.9 can be proved if J = J ( J − (λ − ), J + (λ + ) : Z) as well because any somewhere injective curve for such a J will have a point of injectivity in W \(W + ∪ W − ∪ Z) with the exceptions of (1) components of Z, and (2) holomorphic curves for J(λ + ) contained entirely in a cylindrical end W ± . We will not be concerned about transversality of such curves anyway and therefore they can be safely ignored. This observation will be needed in section 4.
2.3.3. Proof of Theorem 2.6. Theorem 2.6 is a direct consequence of Proposition 2.10. There is a residual subset J gen of dλ-compatible almost-complex structures on ξ such that: (1) At any somewhere injective finite energy solution, the Cauchy-Riemann operator is transverse, by [Dra04].
(2) For any compatible J (not necessarily in J gen ), if U ∈ M J (x; y) is of index 1, then the linearization of the Cauchy-Riemann operator is surjective at U .
(3) Let J be any compatible almost-complex structure, and suppose U is a Jholomorphic finite-energy cylinder. Then its index is at least 0.
is an index 2 cylinder, and both asymptotic orbits are SFT-good, then the linearization of the Cauchy-Riemann operator is surjective at U . (5) For J ∈ J gen , all index zero holomorphic cylinders are trivial cylinders. The linearized Cauchy-Riemann operator is surjective at any trivial cylinder.
This is in fact what we will prove.
Proof. To begin the proof, take J gen to be as in Theorem 2.9.
(1) The first item is an assertion in Theorem 2.9.
(2) The map U is an immersion if wind π (U) = 0. For finite energy cylinders of index 1, Theorem 2.7 implies 0 ≤ wind π (U) ≤ 1 − 2(2) + 1 + 2(1) = 0 (since there must be one odd puncture and one even puncture). Thus, finite energy cylinders of index 1 are always immersed. We have g = 0 and #Γ even = 1 (since the index difference is 1, there is one even and one odd puncture), so by Theorem 2.8 the linearized Cauchy-Riemann operator is surjective.
(3) This was already observed in [HWZ95]; we include the short proof. π • T u vanishes identically on a cylinder if and only if the cylinder is a trivial cylinder. If a non-trivial finite energy cylinder has index less than 0, then by Theorem 2.7 0 ≤ 2wind π (U) < 0 − 4 + #Γ odd + 2#Γ even ≤ 0, which is impossible.
To prove (4) we will need the following lemma.
Lemma 2.11. Suppose U is a cylinder of index 2 with asymptotic limits x, y. If x and y have even Conley-Zehnder index and are SFT-good, then U is simply covered.
Proof. Let k be the covering number of U , so that U = U • τ k where U is a simple curve and τ k is the degree k holomorphic cover of R × S 1 . Choose a trivialization Φ for U * ξ, which induces trivializations Φ of ξ over U , x and y. We then have CZ Φ (x) − CZ Φ (y) = 2 by the index formula. Let CZ Φ (x) = 2l + 2, so CZ Φ (y) = 2l. Letting x and y denote the asymptotic limits of U , so that x, y are k-covers of x , y , by Proposition 2.4 we have k|2l +2, and k|2l, from which it follows that k = 2 or 1. Suppose k = 2. Then CZ Φ (x ) = l + 1, CZ Φ (y ) = l. Therefore, one of these has odd Conley-Zehnder index, so either x or y is a 'bad' orbit. Since both orbits are assumed SFT-good, k = 1.
(4) The parity of the Conley-Zehnder indices are either both even or both odd. If they are both even, then by Lemma 2.11 all curves in the index 2 component of the moduli space are simple. Transversality of the index 2 component of the moduli space for J ∈ J gen follows from Theorem 2.9.
In the case of an index-2 cylinder with odd asymptotic orbits, we observe Ind(U) = 2 ≥ 2g + #Γ even − 1 = 2(0) + 0 − 1 so automatic transversality applies by Theorem 2.7 and Theorem 2.8, in particular for any J chosen for the case of even, SFT-good orbits.
(5) Choose any J ∈ J gen . For simply covered index 0 cylinders, the inequality of Corollary 2.9 implies π • T u vanishes identically, which implies the cylinder is trivial. If a cylinder is multiply covered of index 0, the underlying simple cylinder has index 0 -by the monotone growth of the index of a cylinder under covering. Hence, it is a cover of a trivial cylinder so is trivial itself. Transversality is proved as in [Sch95].

Intersections and compactness
3.1. Intersections. We recall the intersection theory described in [Sie09].
For example, if V is a 3-manifold with a contact form λ, the associated stable Hamiltonian structure is H λ = (λ, dλ).
Let W be a 4-manifold. A cylindrical Hamiltonian structure on W is the data is a diffeomorphism and H ± is a stable Hamiltonian structure on V ± . We will usually refer only to W ± with the rest of the data implicit. A cobordism is a manifold W such that W \W ± is compact oriented with boundary.
Example 3.2. Suppose λ ± are contact forms with kernel ξ on V , and λ − ≺ λ + in W = W ξ . We can equip W ξ with the Hamiltonian structured ends , π(λ) Suppose W i are two such cobordisms. We say that we can concatenate W 1 and . Then we can define the manifold W 1 W 2 by compactifying the positive end of W 1 with {∞} × V + 1 , the negative end of W 2 with {−∞} × V − 2 (using the diffeomorphisms Φ ± to do so), and identifying them. This can be generalized to n-fold concatenations W 1 · · · W n for any n ≥ 2 as long as one can concatenate W i and W i+1 for each i.
Suppose W 1 is a cylindrical Hamiltonian structured manifold i.e. W 1 ∼ = R × V , and that we can form W 1 W 2 . Then obviously W 1 W 2 is homeomorphic to W 2 and is equipped with the same Hamiltonian ends. This identification in convenient and we will use it often in the following without comment.
For a closed T -periodic Reeb orbit δ, denote by Z δ the trivial cylinder over δ: , and where c * U denotes the action by translation of the R co-ordinate by c)). We say U is asymptotically cylindrical if at each puncture z ∈ Γ U is asymptotically cylindrical to some H-Reeb orbit γ m .
If U is a map into a 4-manifold W with Hamiltonian structured ends H ± , then we say (Σ, j, Γ, U, W ) is asymptotically cylindrical if for each puncture z ∈ Γ there is a neighborhood O z of z such that U | Oz is contained in one of the ends and is asymptotically cylindrical as defined above.
We denote by C ∞ g,p+,p− (V, H) the set of smooth asymptotically cylindrical maps of genus g with p + positive punctures and p − negative punctures, and set C ∞ (V, H) to be the union over all g, p − , p + ≥ 0. Similarly in [Sie09] the author defines asymptotically cylindrical maps in a 4-manifold W with Hamiltonian structured ends W ± and denotes these by C ∞ (W, H + , H − ). We will extend this notation to include continuous, piecewise smooth maps which are smooth on neighborhoods of the punctures and satisfy the same asymptotic conditions. Definition 3.4. Suppose U, V are asymptotically cylindrical maps, and z is a puncture for V asymptotic to γ m . Choose a trivialization Φ : There is a neighborhood O z of z that is mapped entirely into a cylindrical end and so that Let β be a cut-off function supported in O z which is identically 1 on a smaller neighborhood of z. Consider the perturbation which is well-defined for all > 0 sufficiently small. Let V Φ,β, be the map that results from making such a perturbation at each puncture of V . Then set ι Φ (U, V ) = int(U, V Φ,β, ). It depends only on the homotopy classes of the maps and the trivializations Φ, and is symmetric in the arguments U, V (see [Sie09]).
Definition 3.5. Let P be a closed, simple Reeb orbit, and let k 1 , k 2 be two positive integers.
Let z and z be punctures of U and V respectively. If U and V are asymptotic to covers of different Reeb orbits, then we set Ω Φ (z, z ) = 0. If U and V are positively asymptotic to covers P k1 , P k2 of the same Reeb orbit P at z and z then we set Ω Φ (z, z ) = Ω + Φ (P, k 1 , k 2 ); if they are negatively asymptotic at z and z to covers P k1 , P k2 of the same Reeb orbit P then we set It follows from Proposition 4.1 and Lemma 3.
Definition 3.6. For a pair of punctures z, z of U, V , if U, V are asymptotic to covers of different orbits at z, z then set ∆(z, z ) = 0; else, let P denote the underlying simple orbit, k 1 (z), k 2 (z ) the respective covering numbers of the orbit, and set , k 2 (z )) ≥ 0 It follows from equations (3.1), (3.2) that this difference is indeed trivialization independent. Sum this quantity over all pairs of punctures to get be homotopy classes of asymptotically cylindrical maps with representatives U, V (resp.). Set By the above comments this only depends on the homotopy classes and is therefore well-defined.
Asymptotically cylindrical maps U i in W i can be concatenated 3.1 if the asymptotics match in the obvious way so that the maps can be glued together (after choosing asymptotic markers) to form a (piecewise smooth) C 0 map U 1 U 2 in W 1 W 2 . One can also form n-fold concatenations U 1 · · · U n in the obvious way to make a map into W 1 · · · W n ; every holomorphic building defines such 3.1 The concatenation is well-defined only up to a Dehn twist of the domain Σ 1 • Σ 2 , since a gluing of these domains may require a choice of asymptotic markers. This data is determined when the concatenation occurs as a holomorphic building, which is the only case that interests us. Regardless, different choices of asymptotic markers yield the same intersection number (see the proof of Proposition 4.3 of [Sie09]). a concatenation. Proposition 4.3 of [Sie09] states (except for the last item which differs slightly; see the remark following the statement): Proposition 3.8. Let W, W 1 , . . . , W n be 4-manifolds with Hamiltonian structured cylindrical ends. Then: ( depends only on the homotopy classes (3) Using "+" to denote disjoint union of maps, we have Remark 3.9. The last assertion differs from the one made in [Sie09]; however, it is shown in the proof that the difference (where * denotes the intersection number minus the term 1 where we use ∆ ± to denote the sum of the terms in the definition of ∆ corresponding to the positive (resp. negative) punctures only (in particular ∆ = ∆ + + ∆ − ). It is easy to see that we get the above identity from our definition of " The statement in [Sie09] is extended to n-fold concatenations above.
Considering the oriented blow-up Σ [BEH + 03], because U is asymptotically cylindrical it extends over Σ to define a map U : Σ → W , with the property that if B is a boundary component of Σ, then U | B is a closed Reeb orbit in ∂W . Let us consider the subclass C 0 (W , H + , H − ) of those U ∈ C 0 (Σ, W ) such that for each boundary component B of Σ, U | B is a closed Reeb orbit in (∂ ± W ∼ = V ± , H ± ). By standard arguments the intersection number extends to the class C 0 (W , H + , H − ) and is homotopy invariant within that class.
In particular, the above discussion extends the intersection number to holomorphic buildings: a building (U ) defines a map (U ) Once the intersection number is extended this way, it is clear that uniformly. Therefore they represent the same homotopy classes in C 0 (W , H + , H − ) for k sufficiently large and therefore In a concatenation W 1 · · · W n , if both ends of each W i correspond to contact forms for which L consists of closed orbits, then the cylinder π −1 (L) = Z L is asymptotically cylindrical for each W i , and the concatenated map Z L . . . Z L identifies with Z L under the homeomorphism W 1 . . . W k ∼ = W ξ . So it makes sense to speak about the intersection of a holomorphic building (U ) and the cylinder Z L if all the ends of all pieces in the concatenation in which (U ) lives are such that L is a closed orbit for the form defining the Hamiltonian structure for that end. Moreover, if we index the levels of (U ) by by the level-wise additivity property (Property (4)  For the following, suppose U, V are asymptotically cylindrical holomorphic maps. The asymptotics formula and the similarity principle implies that if U, V have no components on which they are covers of the same holomorphic curve then the number of intersection points is finite. Let int(U, V) be the sum of all local intersections, which is non-negative by e.g. [McD91,MW95].
Theorem 3.11 ([Sie09] Theorem 4.4). Suppose U, V are asymptotically cylindrical pseudoholomorphic maps in an almost-complex cobordism with cylindrical ends, with no common components. Then The number δ ∞ (U, V ) is described in [Sie09] in terms of local quantities for holomorphic maps. We refer to [Sie09] for the definition. For later use we now recall what the quantity In the following formula, let L i ∼ U (w) mean that U is asymptotic to a cover of L i at w, let k w be the covering number of the L i to which U is asymptotic at w, let w λ,Φ (U, w) be the winding number of the loop (Φ kw ) −1 e U , where e U is the eigensection of the asymptotic operator appearing in the asymptotic formula of [Sie08] for U at the puncture w; let α ± λ,J,Φ (L i , k) be the extremal winding number of the asymptotic operator associated with (L k i , λ, J) and trivialization Φ k , and finally let p λ (x) ∈ {0, 1} be the parity of the orbit x with respect to the form λ. Then 3.1.3. Moduli spaces. Let (W, J) be a symplectic 4 manifold with Hamiltonian structured ends and compatible J adjusted to the ends. We use M J (x; y 1 , . . . , y k ) to denote the moduli space of finite energy J-holomorphic spheres with one positive asymptotic orbit x and perhaps several negative asymptotic orbits y 1 , . . . y k (modulo reparametrizations of the domain). Usually we are interested in moduli space of cylinders with one positive and one negative end, M J (x; y). If it is clear we may drop the subscript. If J is a cylindrical almost-complex structure J = J(λ) then we will actually mean the moduli space modulo the further R-action by translations. We assume that all Hamiltonian structures considered are non-degenerate, so we have the asymptotic formula [HWZ96, BEH + 03] which implies all solutions are asymptotically cylindrical and the intersection theory described above applies. Given a holomorphic curve Z, we denote For homotopies l → J l , we write In the following, Z will almost always be Lemma 3.12. Suppose Z is a finite energy surface with all asymptotic orbits elliptic (e.g. Z L = π −1 (L) under condition (E)), and [U ] ∈ M J (x; y 1 , . . . , y n ) is such that no component has image contained in a component of Z. Then [U ] ∈ M J (x; y, · · · : Z) if and only if U and Z never intersect and are never asymptotic to covers of the same asymptotic orbit with the same sign (in the case Z = Z L this is the same as saying x, y / ∈ D ± and U never intersects Z).
Let U be a possibly branched cover of C with 1 positive puncture and genus 0. Then we can identify the normal bundle to C with ξ, and with the exponential map and a unitary trivialization Φ : C * ξ → R×S 1 ×C we can identify a tubular neighborhood of C with (R×S 1 )×D , where D is a disc containing the origin in C. Let us denote this mapf = exp •Φ −1 : with the property that f (z, c) ∈ C if and only if c = 0 (possibly taking D smaller if needed). It is a local-diffeomorphism away from the branch points.
Sufficiently small sections of N can be identified with functions σ : S 2 \Γ(U ) → D by this trivialization, and it defines a map S σ : S 2 \Γ(U ) → W ξ by z → f (z, σ(z)). Choose any sectionσ of N such that • near the punctures it has the form 3.2 σ(s, t) = e λs e λ (t) where λ is the extremal eigenvalue of the asymptotic operator of the orbit to which U is asymptotic at that puncture, and e λ (t) represents an eigensection associated to λ, • it has only finitely many transverse zeros none of which occur at branch point, • it is small enough to be represented by a function σ : since S σ is homotopic to U via τ → S τ ·σ . In view of the representation ofσ near the punctures we have 3.2 In cylindrical coordinates at the puncture: The formula (3.3) for δ ∞ applies here, so the equations (3.4) imply δ ∞ (C, S σ ) = 0. Since all asymptotic limits are elliptic, it follows that each puncture w ∈ Γ(S σ ) contributes ∆(∞ w , w) ≥ 1 (∞ w denotes the puncture of C of the same sign as w; this inequality is demonstrated in the proof of Proposition 4.3 item (4) in [Sie09]). Therefore 1 2 ∆(C, S σ ) ≥ 1 2 #Γ(U ). It remains to compute int(C, S σ ). Since the mapf : R × S 1 × D → W is an orientation preserving local-diffeomorphism away from branch points, and all zeros occur away from branch points by construction, the computation reduces to a signed count of intersections of this map σ with 0 ∈ C (the mapping degree deg(σ; 0)). This degree is computed by the homology class of σ| ∂S ∈ H 1 (C\0) (where S is S 2 minus the small disc-like neighborhoods D z ± of the punctures Γ(U ) on which the asymptotic expression holds, i.e. σ(s, t) = e λs e λ (t)). The extremal winding numbers are given by α ∓ λ±,Φ (L i ; k w ) where k w is the covering number of L i at each puncture w of U , using α − with respect to λ + at positive punctures, and α + with respect to λ − at negative punctures. We recall from section 2.2 that there are irrational numbers where k + is the covering number of the positive puncture, and the k − z− are the covering numbers at the negative punctures z − , so so the mapping degree is at least −#Γ − . Summing everything together, Remark 3.14. If we drop the hypothesis (λ + , L) ≥ (λ − , L), then we lose this control on this intersection number since k + θ i Proof. If P is asymptotic to d ∈ L then the first assertion is immediate by the formula of Theorem 3.11 since the parity term will contribute 1/2 and all terms are non-negative. Suppose instead that P is asymptotic to an orbit x / ∈ L + . Then π V • P : C → V must intersect L by hypothesis, and therefore P must intersect Z L . Since both are J-holomorphic the intersection contributes 1 to the intersection, which bounds below [P ] * [Z] ≥ 1 by positivity of intersections.
We will call a puncture of a holomorphic building (U ) a free puncture if there is a path from the puncture to an end of the tree such that all nodes along this path represent trivial cylinders. Proof. Let U i be the vertices of (U ) which represent curves that are not branched covers of components of Z, and let V j be the vertices of (U ) which represent curves that are branched covers of components of Z. The level-wise additivity and disjoint union properties of the intersection number (Proposition 3.8) imply In the first case, the second sum is empty. Since [U i ] * [Z] ≥ 0, and is strictly positive if it is asymptotic to some x ∈ L or if π V • U i intersects L in its interior (Lemma 3.12), the first statement is clear. For the second case, using the estimate from Lemma 3.13 we have Consider a negative puncture z ∈ ∪Γ − (V j ) which is not a free puncture for (U ).
There is a first vertex on the path below this puncture which does not represent a trivial cylinder, which is either a U i or a V j (and which conversely determines (V j , z)). If this vertex is is not a V j , then it is a U i asymptotic to an orbit in L and therefore [U i ] * [Z] ≥ 1/2. Else, it is a V j . Therefore, given a term in the sum Vj Γ−(Vj ) − 1 2 which does not represent a free negative puncture, we find a uniquely determined term in the sum [U i ] * [Z] + Vj 1 2 that combines with it to make the total non-negative. By this grouping of terms we determine the extra 1 coming from the fact that there is at least one V j that is not used to cancel one of the terms in Vj Γ−(Vj ) − 1 2 (there must be one V j not on a lower level than some other V j ). Now the following proposition is a consequence of Lemma 3.16 and Lemma 3.12.
Proof. Let (U ) be a SFT-limit for the sequence U k ; by continuity of the intersection number (Lemma 3.10) we must have [(U )] * [Z L ] = 0. Let (U ) 2 be the building obtained by looking only at the top |1|k + -levels of (U ), and (U ) 1 be the building obtained by looking only at the bottom k − |1|k 0 levels of (U ). Then (U ) 1 represents the homotopy class [(U ) 1 ] of asymptotically cylindrical maps in W 1 = (W ξ , J 1 ), (U ) 2 represents the homotopy class [(U ) 2 ] of asymptotically cylindrical maps in W 2 = (W ξ , J 2 ), and [(U )] = [(U ) 2 (U ) 1 ]. The map (U ) 2 has a connected domain, while the map (U ) 1 may have a disconnected domain: let us denote by (U i 1 ) the connected components of (U ) 1 . Then The level-wise additivity and disjoint union properties of Proposition 3.8 imply Let k be the number of negative punctures of (U ) 2 . Then there are k-components (U i 1 ): label them so that for 1 ≤ i ≤ k−1, the component (U i 1 ) is planar (has no negative punctures), while (U k 1 ) is cylindrical (has one negative puncture asymptotic to y).
Suppose the second alternative of Lemma 3.16 holds for (U ) 2 . Then If for (U k 1 ) the positive asymptotic orbit is not in L, the stronger inequality holds. By Lemma 3.16, for the k − 1 planar components (U i 1 ): 1 ) has a positive asymptotic limit in L, then the last inequality is strengthened to ≥ 1/2. This is because if the second alternative to Lemma 3.16 holds for (U k 1 ), the inequality implies [(U k 1 ] * [Z] ≥ 1/2. If the first holds, then some level of (U k 1 ) is asymptotic to L: Lemma 3.16 asserts both that level will have strictly positive intersection, and that each level has non-negative intersection, so by level-wise additivity [(U k 1 )] * [Z] ≥ 1/2 again. It follows that whether or not (U k 1 ) has positive asymptotic orbit in L, the inequality [(U )] * [Z] ≥ 1/2 holds. Therefore the first alternative to Lemma 3.16 must apply to (U ) 2 .
Since the first alternative to Lemma 3.16 therefore asserts that (U ) 2 has no planar components and is never asymptotic to an orbit in L -because k = 1 it must be that every level of (U ) 2 is a single cylinder. Since [(U ) 1 ] * [Z] = 0 and #Γ − free ((U ) 1 , L) = 0 (this is defined in Lemma 3.16), the second alternative of Lemma 3.16 cannot hold, so the first alternative to Lemma 3.16 must apply. It implies that (U ) 1 also has no planar components and is never asymptotic to an orbit in L. Therefore In case (λ + , L) ≥ (λ 0 , L) ≥ (λ − , L) do not necessarily satisfy condition (E) (but are each such that L is elliptic), it will be useful in the proof of Theorem 1.5 to note the following. Let J n be a sequence of almost-complex structures splitting into J 1 J 2 .
Proposition 3.19. Suppose x, y / ∈ L and that the sequence U n ∈ M Jn (x; y : Z) has the SFT-limit (U ) = (U ) 2 (U ) 1 . Then [(U ) 1 ] * [Z L ] = 0, [(U ) 2 ] * [Z L ] = 0, and therefore by Lemma 3.16 each component U i of the limit satisfies the following properties: • No asymptotic limit of U i is an orbit in L • π V • U i has no interior intersections with L In particular, the compactified image π V • (U ) i ⊂ V \L (in particular it defines a homotopy from x to y since its domain is a compactified cylinder).
Applying Lemma 3.16 it follows that each term in the above sum is non-negative, and therefore each term is zero. Reading Lemma 3.16 once again, it now asserts that the components have the properties stated above. Proof. Condition (P LC) prevents bubbling off as follows. Let U k be a convergent subsequence with limit building (U ). Denote the connected components by U i (vertices of the bubble tree) indexed by i ∈ I. Let S be the underlying nodal domain of the building with components S i corresponding to the U i . By properties CHCE2, CHCE3 of SFT-convergence 3.3 there are maps φ i k : loc (up to a translation in cylindrical levels). Suppose U i is a plane for some i. Then by Lemma 3.15 and condition (P LC) loc and positivity of intersections implies that all maps C 0 -near U i intersect Z as well. Therefore int(U k , Z) = 0 and so [U k ] * [Z] > 0 contrary to 3.3 These properties are analogs of properties C2, C3 introduced earlier when considering the compactness theorem for cylindrical ends (without splitting along a contact-type hypersurface)we refer to Section 8 of [BEH + 03] for the precise statements. the hypothesis [U k ] ∈ M(x + ; x − : Z). This shows that the bubble tree is linear i.e. all components are cylinders.
Since x / ∈ L, it is easy to see as follows that no asymptotic orbit between levels is in L or not in [a]. If not, there would be a first level U i with positive asymptotic orbit x ∈ [a] and negative asymptotic orbit y either in L or not in [a]. But π V • U i provides a homotopy between x and y , so by condition (P LC) there must be an interior intersection with L in either case. But then the C ∞ loc -convergence property of SFT convergence cited in the previous paragraph implies by the same argument used above that for large k, int(U k , Z) > 0, which violates Theorem 3.11 since  A similar result holds for a sequence of J R ∈ J (J 1 , J 2 ). In the following, suppose λ + λ 0 λ − , that (λ +,0,− , L, [a]) each satisfy (P LC), and that Proof. The proof is similar to the previous proposition. Properties C2, C3 of SFTconvergence and condition (P LC) prohibit the bubbling of planes as in the above proof, so each level U i is connected and cylindrical. Similarly, the arguments in the second paragraph of the proof of Proposition 3.20 prevent any level from intersecting Z L , prevent any level from being asymptotic to any component of L, and force any asymptotic orbit x between levels to be in the homotopy class [a]. So the SFT-limit must have the form stated.
In the proof of Theorem 1.5, we will require a more general fact. Suppose that L satisfies the linking condition of Theorem 1.5, namely: L is such that every disc F with boundary ∂F ⊂ L and [∂F ] = 0 ∈ H 1 (L) has an interior intersection with L and that [a] is a proper link class for L (Definition 1.3). Suppose only that λ + , λ 0 , λ − are each non-degenerate and such that L is closed under the Reeb flows. Then • No asymptotic limit of U i is an orbit in L • π V • U i has no interior intersections with L In particular, the compactified images π V • (U ) ⊂ V \L.
Proof. The assertion that for each component U i that π V • U i has no interior intersections with L is argued using the C ∞ loc convergence properties of SFT-convergence in a familiar way. By properties CHCE2, CHCE3 3.4 of SFT-convergence there are maps φ i k : S i → R × S 1 from S i = the domain of U i to the domain of U k such that U k • φ i k | S i converges to U i in C ∞ loc (up to a translation in cylindrical levels). Since U i intersects Z L positively, it follows that U k must intersect Z L for all k large enough, but this contradicts [U k ] * [Z L ] = 0. So we conclude U i has no interior intersections with Z L .
The assertion that no asymptotic orbit of any component U i is in L (equivalently, no edge in the bubble tree represents an orbit in L) is argued as follows. Suppose some edge E in the bubble tree represents an orbit in L. Suppose first that the portion of the holomorphic building corresponding to the subtree below this edge is planar. Choose an edge E in this subtree such that no edge lower on the bubble tree lies in L (this is possible because the tree is planar so there are no free negative asymptotic orbits, thus every edge has a vertex below it). Let (P ) be the holomorphic building corresponding to the subtree below this edge E . Then π V • (P ) has an interior intersection with L, which must correspond to an interior intersection of one of its components (since no asymptotic orbit is in L), contradicting the first assertion which we have already proved.
Otherwise, the portion below E has one free negative puncture. E is therefore on the unique path from the free positive puncture to the free negative puncture. Without loss of generality, suppose E is the highest edge along this path (i.e. closest to the positive free puncture). Let (X) be the tree obtained by removing from the original bubble tree the portion below this edge E (by hypothesis E is not the positive free puncture so this is non-empty). By the previous step, any free negative puncture for (X) bounds a planar building (P ) such that each component satisfies the conclusions of this proposition. Hence, its compactified image satisfies π V • (P ) ⊂ V \L. It will now follow that the interior of π V • (X) ⊂ V \L, since no edge of (X) represents an orbit in L (except E) and we have already shown that no component of (U ) has an interior intersection with Z L . However, the negative asymptotic orbit of (X) is in L (since by hypothesis E represents an orbit in L), and π V • (X) realizes a homotopy between this orbit in L and the positive asymptotic orbit x ∈ [a] through V \L. But this contradicts the hypothesis that [a] is a proper link class, by definition. Therefore no such edge E can exist, which finally proves that no asymptotic orbit of any component U i is in L as claimed.

Cylindrical Contact Homology of Complements
The two crucial requirements for the definition of contact homology are compactness and regularity of the moduli spaces. Proposition 3.17, 3.20, 3.18, 3.22 supply the required compactness for the spaces M J (x; y : Z). In the cylindrical case, Theorem 2.6 provides the required transversality for the moduli spaces (to prove ∂ 2 = 0), but in the non-cylindrical case these arguments do not apply and we need to make some hypotheses. When considering non-cylindrical almost-complex structures, we will consider only simple homotopy classes of loops in V \L. Then following well-known arguments as in e.g.   Here m x denotes the covering number of the orbit x, and cov(U) denotes the covering number of the map U . The sign [U ] = ±1 is determined by comparing a choice of coherent orientation [BM04] with the canonical orientation given by the free R-action (it is well-defined since both asymptotic orbits are SFT-good). Since the moduli spaces we consider are open/closed subsets of the moduli space for (V, J), a coherent orientation chosen in the usual way will restrict to one for our moduli spaces and it does not require further comment (see [BM04] or [EGH00] for details). Though we suppressed it in the notation, it is an essential ingredient in the above definition. This sum will always be finite by compactness (Proposition 3.17, or 3.20) and transversality (Theorem 2.6).
Proposition 4.1. Under the above hypotheses ∂ 2 L = 0. Denote the chain complex (CC * (λ rel L), ∂ L (J)) by CC * (λ, J rel L) (or CC [a] * (λ, J rel L) for (P LC)). and negative puncture which is not a cover of a cylinder in Z has a point of injectivity outside W + ∪ W − ∪ Z where we are free to infinitesimally perturb the almost-complex structure.
Under hypothesis (E), the complex splits according to homotopy classes of loops in V \L: We remark that this is the usual definition of the cylindrical contact homology chain complex for the pair (λ| V \L , J| V \L ) on V \L. [a] * (λ − , J − rel L) defined in section 4.1. We will denote their boundary maps ∂ L (J ± ) when it is needed to clarify. Also, we use G (λ ± ) to denote the set of SFT-good Reeb orbits not in L.
Choose and fix a J ∈ J gen ( J − , J + : Z). We will define a map  [EGH00] sections 1.9.1, 1.9.2); the only parts that need to be verified are that the moduli spaces have the correct boundary structure, which is done as follows: • Each proof involves the count of two level holomorphic buildings of cylinders, each in M(1; 1 : Z L ). For example, in the proof ∂ 2 = 0, one counts pairs of index 1 cylinders in M J (1; 1 : Z L ) which can be concatenated, U 1 U 2 . By level-wise additivity (Proposition 3.8), this configuration also has intersection number 0 with Z L . • After choosing asymptotic markers, the pair can be glued, and by SFTcontinuity of the intersection number (Lemma 3.10), the glued cylinders must be in M(1; 1 : Z L ). • By index additivity and transversality, the glued solutions belong to a connected component of the moduli space diffeomorphic to an interval (after quotienting the free R-action in the R-invariant case). By homotopy invariance of the intersection number it is a subset of M(1; 1 : Z L ). • At the other end of the interval, there is an SFT-limit. The compactness Propositions 3.17 and 3.20 show that the limit is necessarily a broken trajectory with each cylinder in M(1; 1 : Z L ) again; transversality implies 4.3 Again, everything must be oriented appropriately, the details of which we omit. Also, by "index 0" we refer to the total Fredholm index including the parameter µ: this means that Ind(µ, U) = Ind(U) + 1, so we mean that Ind(U) = −1.
there are only two levels and determines the index of both cylinders, so it represents an object corresponding to another term in the algebraic identity.
• Comparison of the orientations shows that the terms representing opposite ends of the moduli space cancel, proving the required algebraic identity (Proposition 4.1, 4.2, or 4.3). In the cylindrical case the required transversality for non-simple curves is guaranteed by Theorem 2.6, so −+ (J 2 ) and a generic homotopy J τ ∈ J τ,gen (λ − , λ + : Z L ), there is a chain homotopy operator K(J τ ) i.e. −+ (J), and given two such maps and a generic homotopy between them J τ there is a chain homotopy K(J τ ) satisfying the above identity, so there are natural maps which obey the above composition law and are also isomorphisms.
Proof. The proof of Proposition 4.4 is complete once the composition law is verified and we prove that the chain maps give isomorphisms. The composition law is verified by a gluing/compactness argument, and we only need to check that we have compactness under stretching and enough transversality; the required compactness is guaranteed by Propositions 3.18, 3.22, and the required transversality is guaranteed by choosing J 1 , J 2 ∈ J gen and considering only simple homotopy classes of loops. Once the composition law is verified, one considers the compositions , both compositions can only be made if (λ + , L) ≡ (λ − , L), hence the hypothesis in Proposition 4.4). Using a cylindrical almost-complex structure we find that (Φ ±± ) * is the identity map. This proves the maps (Φ ±∓ ) * are both injective and surjective and therefore isomorphisms.
Remark 4.5. We would like to point out that the condition "[a] is simple" can be relaxed to the condition "[a] is such that, for one of λ ± , all closed Reeb orbits in [a] are simple". First, the chain complexes are defined even for [a] not simple. Then the condition that one of λ ± has only simple closed orbits in [a] guarantees that holomorphic cylinders in a cobordism used to define the chain maps above are somewhere injective [HWZ96,Sie08] (hence also have an open set of somewhere injective points), and therefore chain maps/homotopies can be constructed for almost-complex structures in J gen just as above. The conclusion of Proposition 4.4 holds even if the contact form at the opposite end of the cobordism has multiply covered Reeb orbits in [a]. We will use this observation in Section 6. 4.3. Action filtration. As in Morse theory, the chain complexes can be filtered by action. We consider the subspace CC (λ, J rel L) with N < ∞ one only requires that λ satisfies (E) up to action N . This means that the assertion (E) (resp. (P LC)) about contractible in V \L Reeb orbits holds for orbits with action less than or equal to N . Then even though CC as above, and they are filtered by the action in the same way. This follows exactly as the construction of the above chain complexes. The proofs of compactness are carried out as straightforward extensions of Propositions 3.17, 3.18, 3.20, 3.22, using the fact that any contractible (in V \L) Reeb orbit has action greater than N to rule out bubbling for moduli spaces of holomorphic cylinders. The Fredholm theory remains the same. We will use these filtered chain complexes later.

Forms with degeneracies
It is possible to obtain existence results which include cases where the form λ may be degenerate or have closed orbits which are contractible in V \L, namely Theorem 1.5. The strategy is to take a sequence of small perturbations of a degenerate form to sufficiently non-degenerate approximating forms and deduce the existence of closed Reeb orbit for the approximating forms by a stretching argument. Then, using the action filtration to bound the action of the orbits found this the way (which provides C 1 -bounds), the Arzela-Ascoli theorem is used to find the desired Reeb orbit for the original degenerate form.
We will also briefly justify the use of the Morse-Bott complex (when λ satisfies (P LC) or (E) but is Morse-Bott non-degenerate) to compute CCH Lemma 5.1. Suppose (λ, L) is such that L consists of non-degenerate closed Reeb orbits for λ (including multiple covers). Given any N > 0, there is a sequence λ n = f n · λ → λ (i.e. f n → 1 in C ∞ (V ; R)) such that (λ n , L) ≡ (λ, L) and such that all periodic orbits of the Reeb vector field for λ n of action at most N are non-degenerate.
Proof. Around each Reeb orbit x of action at most N choose neighborhoods (1) For all such x, V x ∼ = S 1 × D 2 (2), and in these coordinates U x ∼ = S 1 × D 2 (1) (2) The Reeb vector field X λ is transverse to the pages {t} × D 2 (2) of V x (3) For all t ∈ S 1 , any orbit starting at a point {t} × D 2 (1) (i.e. starting at a point in U x ) stays inside V x for at least time N + 1, and thus at least until the first return to {t} × D 2 (2) which occurs before N + If x / ∈ L is an orbit of action at most N , then we choose V x so that, in addition, V x ∩ L = ∅. Moreover, select the neighborhoods V Li around the components L i of L such that the V Li are mutually disjoint and contain no other orbits of period at most N in V Li (other than iterates of L i ). This is possible because the L i and covers are assumed non-degenerate.
By the Arzelà-Ascoli theorem, the set Γ N of periodic orbits of action at most N is compact. By the compactness of the orbit set, it is not difficult to see that the image of these orbits in V (also denoted Γ N ) may be covered by finitely many of the above sets U i , i.e. Γ N ⊂ U 1 ∪ · · · ∪ U k . Order them so that U 1 , . . . , U m are the neighborhoods of the form U Li ⊂ V Li for components L i of L among {U 1 , . . . , U k }.
We modify λ on V 1 , . . . , V k so that there are no degenerate orbits of period at most N in these V i by the method of [CH08a] Lemma 7.1. Choose any of the neighborhood pairs (U i , V i ). Let U be any neighborhood of λ. The proof in [CH08a] shows how one can find a form λ ∈ U such that (1) λ differs from λ only on a compact subset of V i , (2) any λ orbit that originates in U i of action at most N remains in V i for time N + 1 − , (3) all closed orbits of period at most N contained in U i are non-degenerate, (4) there are no closed orbits of action at most N originating in the complement V \ k j=1 U j (for λ n sufficiently near λ); therefore all closed orbits are still contained in k j=1 U j . Proceeding one neighborhood pair at a time, one finds a form λ i ∈ U such that k j=1 U j still covers all the closed Reeb orbits, but all closed Reeb orbits in i j=1 U j are non-degenerate (the above procedure shows how to make all Reeb orbits in U i non-degenerate without introducing new ones in V \ k j=1 U j , and by choosing the perturbation small enough all Reeb orbits contained in U j for j ≤ i remain non-degenerate). The form λ k will be the desired form.
However, we have to take care about the perturbations so that they satisfy the conclusions stated above. By hypothesis all periodic orbits originating in U 1 , . . . , U m of period at most N are non-degenerate, by the choice of the V Li and the choice of the ordering. In this case, in the above process we do not need to perturb the form for the first m steps, i.e. we can take λ m = λ. In subsequent steps, the new perturbations λ i , i > m differ from λ m only on i j=m+1 V j , which is disjoint from L by choice; in particular there is a neighborhood N containing L on which λ i | N = λ| N . Therefore in the end (λ k , L) ≡ (λ, L).
To summarize, for any neighborhood U of λ, there is a λ such that (1) λ has only non-degenerate periodic orbits of period at most N contained in U 1 ∪ · · · ∪ U k . (2) λ has no periodic orbits of period at most N which The desired sequence is obtained by choosing a countable neighborhood basis U n at λ and choosing λ n ∈ U n as above.
Let [(λ, L)] denote the subset of the set of (positive) contact forms µ for ξ (which we will denote Λ(ξ)) such that (µ, L) ≡ (λ, L), equipped with the C ∞ topology as a subspace of Λ(ξ). It is a closed subset of Λ(ξ) so its topology can be obtained from a complete metric space structure on it. Let us denote by Λ gen (ξ) the set of nondegenerate contact forms -as is well-known this is the intersection of a countable set of open dense sets (a G δ ), and therefore dense in Λ gen (ξ). Then in fact Finally, we note the following Lemma 5.3. If λ is a contact form and L is a closed link for the Reeb vector field, then in any C ∞ -neighborhood U of λ there is a λ such that (λ , L) ∼ (λ, L) and each component of L (including multiple covers) is non-degenerate for λ . This can be achieved by an arbitrarily small perturbation to the 2-jet of λ along L (leaving the 1-jet unperturbed so L is still tangent to the Reeb vector field).

5.2.
The proof of the implied existence Theorem 1.5. We prove Theorem 1.5: Proof. Let λ be any form such that L is closed for the Reeb vector field and elliptic non-degenerate. Suppose (λ, L) ≡ (λ , L) is non-degenerate and satisfies (E), and by rescaling if necessary that λ ≺ λ. Choose a constant c such that also cλ ≺ λ . Choose J ∈ J gen (λ), J ∈ J gen (λ ), and J 1 ∈ J gen (J, J : Z L ), J 0 ∈ J gen (J , J : Z L ) (to be specific J 0 is equal to J on W − (cλ)). Then we have the path J R , R ≥ 0 of almost-complex structures in the symplectization splitting along λ to (W ξ , J 0 ) (W ξ , J 1 ).
Claim: for each R sufficiently large there is a J R -holomorphic cylinder U such that [U ] * [Z L ] = 0 with positive asymptotic orbit having λ-action at most N and asymptotic limits in [a]. To see why this is so, we may find a sequence J n → J R with each J n ∈ J gen (since J gen is Baire), and each such defines a morphism Such a chain map is chain homotopic to the morphism obtained from a cylindrical almost-complex structure J, which can be identified with inclusion [a] * (λ, J rel L) generated by orbits of action ≤ N such that [q] = 0 when considered as an element of CCH [a] * (λ, J rel L). Since Φ(J n ) is chain homotopic to the identity and q is closed but not exact, it is easy to deduce that Φ(J n )(q) = 0, so there must be a J n finite energy holomorphic cylinder U n with [U n ] * [Z L ] = 0 and positive asymptotic orbit of action at most N . Taking a SFT-limit as n → ∞, by Proposition 3.17 (and using the fact that (λ, L), (cλ, L) satisfy (E)) we obtain a J R -holomorphic cylindrical building, which has a component U R which is a J Rholomorphic cylinder with [U R ] * [Z L ] = 0, positive asymptotic orbit of action at most N , and asymptotic limits in [a] as claimed.
Since R was arbitrary, we make take a sequence of J Rn finite energy cylinders, U n , with R n ↑ ∞, each with positive asymptotic orbit of action at most N , [U n ] * [Z L ] = 0, and both asymptotic limits in [a]. To continue, first suppose that λ is non-degenerate up to action N . Then by SFT-compactness we obtain a limit holomorphic building (U ) = (U ) 0 (U ) 1 in (W ξ , J 0 ) (W ξ , J 1 ). Proposition 3.19 asserts that the compactified image π V • (U ) ⊂ V \L. The image is a piecewise-smooth cylinder with one boundary component equal to x and the other boundary component equal to y. Suppose (U ) 1 has k free negative punctures. Let To conclude in the case λ is degenerate, select λ n → λ in C ∞ with (λ n , L) ≡ (λ , L) such that λ n is non-degenerate up to action N by Lemma 5.2. For each n sufficiently large we obtain a closed λ n -Reeb orbit x n as above (since eventually λ n ≺ λ). Applying the Arzela-Ascoli theorem, we obtain a closed λ -Reeb orbit x as a C ∞ limit of the x n . x cannot be a l-fold cover of a component of L for any l ≥ 1, because if it were then one could show that 1 is an eigenvalue for the corresponding linearized lth-return map of the Reeb flow, contradicting the hypothesis that each cover of L is elliptic non-degenerate. It follows easily that [x] = [a], since x n → x in C ∞ , any loop C 0 -close enough to x is in the same homotopy class as x, and every x n ∈ [a]. The orbit x also has action at most N .
If instead L satisfies the linking condition of the Theorem and [a] is a proper link class for L, select any (λ , L, [a]) satisfying (P LC). Assuming first λ is nondegenerate, repeat the above argument using Proposition 3.23 instead. To pass to the degenerate case, use Lemmas 5.3, 5.2 to similarly find approximating nondegenerate Reeb vector fields and a sequence of orbits x n ∈ [a] with x n → x with x a closed λ Reeb orbit. It is clear that x cannot be a cover of a component of L since no sequence of loops in [a] can converge to a cover of a component of L: this is because [a] is a proper link class relative to L.

A brief justification of Morse-Bott computations.
Suppose that λ satisfies (E) and is Morse-Bott non-degenerate (a similar argument will work if λ satisfies (P LC) instead). Choose a Morse function on the Reeb orbit sets of λ. The Morse-Bott chain complex CC [a] * (λ, J rel L) is generated by critical points of the Morse function chosen on the orbit sets, removing the generators corresponding to Reeb orbits with image in L. The differential counts generalized holomorphic cylinders (holomorphic cylinders between orbit sets with cascades along the gradient flow lines of the Morse function, described in detail in [Bou02]) of index 1 which neither intersect nor are asymptotic to L.
Using the perturbation of Lemma 2.3 of [Bou02] one finds, given any N > 0 and C ∞ -neighborhood of λ, a form λ N (which in fact only differs from λ on an arbitrarily small tubular neighborhood the orbits sets of action at most N + 1) such that • λ N is in the C ∞ -neighborhood chosen • All closed orbits that have period at most N + 1 correspond to critical points of a Morse function on the orbit sets of λ and are non-degenerate. • There are no contractible (in V \L) Reeb orbits of period at most N (because a sequence of such orbits for a sequence of forms converging to λ would produce a contractible in V \L Reeb orbit for λ of action at most N by Arzela-Ascoli, contradicting hypotheses) Such forms can be given explicitly in terms of the Morse functions on the orbit set in [Bou02]. Given such a λ N , one may construct the cylindrical chain complex (for generic J N and Morse functions) For any fixed N , choosing a sequence of such λ n → λ (with corresponding J n → J), the work of [Bou02] guarantees that for large n the cylindrical contact chain complex generated by orbits of action at most N is canonically identified with the Morse-Bott complex for (λ, J) generated by the orbit sets of action at most N , so this portion of the homology can be canonically identified. It is not difficult to see that a sequence U n of J n -holomorphic cylinders with U n * Z L = 0 will converge to a generalized holomorphic cylinder of the type used to construct the differential above. Similarly, if any sequence converges to such a generalized holomorphic cylinder, it is easy to see that lim U n * Z L = 0. The results of [Bou02] imply that for large n CC  λ n ) by a closed, non-exact element (if N was chosen large enough and λ n is chosen near enough λ). Since the composition above is chain homotopic to ι, we get a closed, non-exact element in CC can be used to find a surjective map on homology

Examples in S 3
We will use two approaches to computing contact homology on some orbit complements in S 3 . The first is to use an integrable model for which the dynamics are known precisely, usually because of some symmetry, and compute using the symmetry to make deductions about holomorphic curves as well. The Morse-Bott technique introduced in [Bou02] is a particularly useful tool when computing using such an approach. The second technique is to use open book decompositions (an approach used in [CH08a] to compute contact homology for a large class of examples). In this section we will give sample computations using both approaches.
Since H 2 (S 3 ; Z) = 0 we can use Q as the coefficient ring for all chain complexes, and a global trivialization of the contact structure is used to compute all Conley-Zehnder indices (and thus to grade the chain complexes) for the tight contact structure.
First we will show that S is a smooth, star-shaped hypersurface with respect to the origin in R 4 . Let H(x, y) be any smooth function such that H • γ ≡ 1 and ∇H = 0 along γ. Then S is realized as the set H(r 2 1 , r 2 2 ) = 1 (at least in some neighborhood of S). We compute r 1 ∂ r1 + r 2 ∂ r2 ¬d(H(r 2 1 , r 2 2 )) = r 1 ∂ r1 + r 2 ∂ r2 ¬2r 1 (∂ x H)(r 2 1 , r 2 2 )dr 1 + 2r 2 (∂ y H)(r 2 1 , r 2 2 )dr 2 = 2 r 2 1 (∂ x H) + r 2 2 (∂ y H) However, since dH •γ ≡ 0 we mush have that ∇H(γ(t)) is proportional to j ·γ (t) = (y (t), −x (t)) (since this vector is perpendicular to (x , y )) and therefore, using x(t) = r 2 1 , y(t) = r 2 2 , the previous line is proportional to 2(x(y ) + y(−x )) = 2(x · y − y · x ) > 0 The proportionality constant is nowhere zero, so we see that the original quantity is nowhere vanishing and we can conclude that dH = 0 and hence S is a smooth hypersurface in R 4 . The radial vector field r 1 ∂ r1 + r 2 ∂ r2 is transverse to S and therefore S is star-shaped with respect to the origin. Since the radial vector field is Liouville, the hypersurface is of contact type. Slope (γ ) ⊥ Slope= θ 1 The Hamiltonian vector field of H is So y (t)∂ θ1 − x (t)∂ θ2 positively generates the characteristic foliation. It is clear that r 1 , r 2 are integrals for the flow, so S foliates (on the complement of the Hopf link H 1 H 2 = (r 1 = 0 r 2 = 0)) by tori, parameterized by points on the curve (x(t), y(t)) = γ(t) on which the flow has slope (y (t), −x (t)). In particular, we find for each t such that (y (t), −x (t)) = C · (p, q), C > 0 is rational a torus foliated by closed Reeb orbits. The signs of p, q will be determined by the direction of the vector γ (t). The condition that x (t)/y (t) is strictly monotone implies that this torus is Morse-Bott, so we have a Morse-Bott flow on S. We shall see that closed orbits in different tori represent different homotopy classes of loops in S 3 \(H 1 H 2 ) (which is homotopy equivalent to T 2 ). In the following, let us denote the homotopy class of loops [a] such that for l ∈ [a] (l, H 2 ) = p and (l, H 1 ) = q by [a] = (p, q) ∈ Z 2 : this classifies the set of homotopy classes of loops in S\ (H 1 H 2 ).
Let us more explicitly describe the set of closed characteristics in S. First, the knots H 1 , H 2 corresponding respectively to the sets r 2 = 0, r 1 = 0 are closed Reeb orbits on S. Using that the characteristic foliation is positively generated by y (t)∂ θ1 − x (t)∂ θ2 , one computes the Conley-Zehnder indices: The characteristic foliation on the torus at (r 2 1 , r 2 2 ) = γ(t) is (positively) generated by The curve γ divides into three distinct arcs: on the first arc t ∈ [0, t 1 ) (for some 0 ≤ t 1 < 1), x (t) > 0 and y (t) > 0; on the second arc t ∈ (t 1 , t 2 ) (where t 1 < t 2 ≤ 1), x (t) < 0 and y (t) > 0; on the third arc, t ∈ (t 2 , t 3 ] (where t 2 ≤ t 3 ≤ 1), x (t) < 0 and y (t) < 0. Note that the first and third arcs may be empty. We analyze the Reeb orbits belonging to points on each arc separately.
Then for those t for which −x /y is rational and equal to (−q)/p in least terms (with p, q > 0), the leaves of the foliation form closed Reeb orbits which link p times with H 2 and −q times with H 1 , i.e. it is in the homotopy class (p, −q).
This arc is non-empty if and only if θ 1 < 0. In this case, since −x (t)/y (t) is initially θ 1 , and strictly increases. Thus, we find, for each relatively prime positive pair of integers (p, q) such that θ 1 < (−q)/p < 0 a unique torus (corresponding to that t (p,−q) for which −x (t (p,−q) )/y (t (p,−q) ) = (−q)/p) foliated by Reeb orbits in the homotopy class (p, −q).
Then, for those t for which −x /y is rational and equal to q/p in least terms (with p, q > 0, the leaves of the foliation form closed Reeb orbits which link p times with H 2 and q times with H 1 , i.e. it is in the homotopy class (p, q).
Thus, we find, for each relatively prime positive pair of integers (p, q) such that max{0, a unique torus (corresponding to the point γ(t (p,q) ) where −x (t (p,q) )/y (t (p,q) ) = q/p) foliated by Reeb orbits in the homotopy class (p, q).
There is also the possibility that −x (t)/y (t) is decreasing, which happens if and only if 0 < θ 2 < θ 1 . In this case, we find for each the torus (r 2 1 , r 2 2 ) = γ(t (p,q) ) foliated by orbits in the homotopy class (p, q). On the third arc: both x (t) and y (t) are negative, and y (t)/x (t) increases strictly until t = 1.
For those t for which −x /y is rational and equal to q/(−p) in least terms (with p, q > 0, the leaves of the foliation form closed Reeb orbits which link −p times with H 2 and q times with H 1 , i.e. it is in the homotopy class (−p, q).
This arc is non-empty if and only if θ 2 < 0. In this case, −x (t)/y (t) tends to −∞ as t → t + 2 , and strictly increases to the limiting value θ 2 as t → 1. Thus, we find, for each relatively prime positive pair of integers (p, q) such that q/(−p) < θ 2 a unique torus (corresponding to that t (p,−q) for which −x (t (−p,q) )/y (t (−p,q) ) = q/(−p)) foliated by Reeb orbits in the homotopy class (−p, q).
Finally, there are the points t 1 , t 2 . If t 1 = 0, then x (t 1 ) = 0 and y (t 1 ) > 0 so there will be a torus of closed orbits at γ(t 1 ) representing the homotopy class (1, 0) (i.e. linking once with H 2 and zero times with H 1 ). If t 2 = 1, then x (t 2 ) < 0 and y (t 2 ) = 0 so there will be a torus of closed orbits at γ(t 2 ) representing the homotopy class (0, 1) (i.e. linking 0 times with H 2 and once with H 1 ).
Example 6.2. We consider the same form λ as above, and (for simplicity) assume that 0 < θ 1 < θ 2 . Consider instead the two-component link T = T 1 T 2 whose components are • T 1 is the one leaf of the foliation of the torus T (p,q) by the Reeb vector field.
• T 2 is the one leaf of the foliation of the torus T (p ,q ) by the Reeb vector field. where both pairs (p, q) and (p , q ) are positive, relatively prime and Choose any relatively prime pair (p , q ) such that Let This shows that the linking numbers q ) ] cannot be homotopic to either T 1 or T 2 in the complement of T 1 T 2 since these linking numbers are homotopy invariant in S 3 \T 1 T 2 . The linking numbers of the orbits in T (p ,q ) with T 1 , T 2 show that orbits in different orbit sets do in fact lie in different homotopy classes of loops, so (as in the previous example) the Morse-Bott complex is simply that of a Morse function on the orbit set. Therefore for [T (p ,q ) ]
Corollary 6.3. Let λ be a tight contact form on the 3-sphere. Suppose there is a pair of knots L = L 1 L 2 such that there is a contactomorphism taking this pair to the pair of torus knots described in Example 6.2: Then for any p , q relatively prime and such that Proof. Applying the contactomorphism we can assume without loss of generality that (λ, L ) ∼ (λ , T ) from Example 6.2. Then this follows from Theorem 1.5 and the computation in Example 6.2, since the homotopy class [a] of K (which has the homotopy type of the leafs of the foliation of T (p ,q ) by the Reeb vector field in that example) is simple and a proper link class relative to T 1 T 2 . The hypotheses q = q , p = p imply that that the linking numbers (T 1 , [a]) = (T 1 , T 2 ) and (T 2 , [a]) = (T 2 , T 1 ) which ensures that [a] is a proper link class, though [a] might be a proper knot class even if these inequalities are not satisfied.
Remark 6.4. Similarly, one can consider the cases where at least one of θ 1 , θ 2 is negative and obtain an analogous result for certain cases where p, p , q, q may be negative.
It is also true that (λ , T, [H i ]) satisfies (P LC) for the knots H 1 , H 2 in Example 6.1 (the components of the Hopf link) if p = 1 and q = 1. In this case, it is easiest to compute CCH where the generators q 1 , q 2 have degrees |q 1 | = 2 1 + q p , |q 2 | = 2 1 + p q 6.2. Application to reversible Finsler metrics on S 2 . The geometric set-up described in the next three paragraphs is well-known; a clear treatment can be found in section 4 of [HP08]. Let us recall the double-covering map S 3 ∼ = SU (2) → SO (3): By taking the second and third columns 6.2 , we obtain a double-covering map G :   which gives an explicit formula for the double-covering map S 3 → S 1 T S 2 . Let • 0 : T S 2 \0 → T * S 2 \0 denote the Legendre transform with respect to the round Riemannian metric of unit curvature. In general, F denotes the one obtained from a Finsler metric F . • F is a Finsler metric on S 2 , while F * is the co-metric obtained via F • −1 F . • Set M * F = {F * (p) = 1} be the unit co-sphere bundle with respect to F * , and let M * 0 be the unit co-sphere bundle for F = | · | the norm for the round Riemannian metric. , p)).
• G is the map SU (2) → S 1 T S 2 = M 0 considered above. • λ is the Liouville form on T * S 2 , while λ 0 is the standard contact form on S 3 . Note that M * F is contact type.
The Reeb flow of 2(f •G)λ 0 is smoothly conjugate to the double cover of the geodesic flow on M F . 6.1 This can be checked by a linking argument which we omit. 6.2 One could take different pairs of columns instead.
There is a global frame X 1 , X 2 , X 3 on M * with the property that X 1 is the Reeb vector field, and X 2 , X 3 trivialize ξ = ker λ| M * F . This can all be pulled back via the double-cover r F • 0 • G : S 3 → M * F , so we may use this frame to compute Conley-Zehnder indices. The linearized flow along a closed Reeb orbit z(t) can be computed with the following result (see e.g. [HP08]). Let φ t denote the Reeb flow. Let x 0 X 2 (z(0)) + y 0 X 3 (z(0)) be in ξ z(0) . Let x(t), y(t) solve dφ t (z(0)) = x(t)X 2 (z(t)) + y(t)X 3 (z(t)), x(0) = x 0 , y(0) = y 0 Then x(t), y(t) solves the linear system where K : S 2 → R is determined by the Finsler structure and agrees with the curvature in the Riemannian case. Poincaré's rotation number of a geodesic γ can be described as [Ang05] 1/ρ = lim n→∞ y 2n nL where y k is the k th positive zero of y(t) which solves y + K(γ(t)) · y = 0 and L is the length of the geodesic. This limit can also be described by Let z be the closed Reeb orbit which double covers γ, such that z has period T . We will now compare the Conley-Zehnder indices CZ(z k ) to the inverse rotation number ρ. The Conley-Zehnder index of z k satisfies (see, for example, the Appendix of [HWZ03]) |CZ(z k ) − 2∆| < 2 In the above, 2π∆ is the change in argument of the vector (x(t), y(t)) over the paths domain [0, kT ], where (x, y) satisfies the linearized equation associated to z above. However, if (x, y) solves this equation, then y solves the equation (recall G denotes the double cover) y + K(G • z(t))y = 0 Moreover, if there are k zeros of y, then k − 1 < 2∆ < k + 1. This claim is a consequence of the following observations. First, we notice that all crossings of the x-axis are transverse: this is because if y = 0 and y = 0 then y ≡ 0 by uniqueness of solutions. Second, at any crossing, by the differential equation we see that if y > 0 then x > 0, and if y < 0 then x < 0; thus each zero contributes a transverse counterclockwise crossing of the path (x(t), y(t)) against the x-axis. So |2∆−k| < 1; hence (for any k): Since z has period T , z k has period kT . Then 0 ≤ lim k→∞ |CZ(z k ) − #{zeros of y ∈ [0, kT ]}| k ≤ lim 3/k = 0 p, q. We remark that the symmetry between p, q above corresponds to a doublecount of geodesics: a closed geodesic has two distinct lifts corresponding to the direction the geodesic is traversed, and reversing the direction of the geodesic changes the homotopy class of the lifted Reeb orbit from (p, q) to (q, p).
To compare with the geodesics found in [Ang05] in the case F is Riemannian (which are found under weaker regularity hypotheses and without any hypothesis about irrationality of ρ), we note that there are representatives in this homotopy class which are precisely double-covers (via G) of the curve (γ p+q,2p , γ p+q,2p ) in the unit tangent bundle, where γ p+q,2p is a (p + q, 2p)-satellite of the geodesic γ. However, we cannot assert using Corollary 1.6 that the geodesics we have found are the (p + q, 2q)-satellites found in [Ang05], because there are many different flat knot types lifting to transverse curves in this homotopy class in S 3 \(w z). . This means that B is a transverse link in V (the binding), π : V \B → S 1 is a fibration, and every component B ⊂ B has a neighborhood U such that π| U \B ∼ = S 1 ×Ḋ 2 has the form (t, (r, θ)) → θ (where (r, θ) are polar coordinates on the punctured disc). The open book supports a contact structure ξ if there is a contact form λ (called a supporting contact form) with ker(λ) = ξ such that • B is a closed for the Reeb vector field • The form dλ is symplectic on the pages π −1 (θ), i.e. dλ| π −1 (θ) > 0, ∀θ ∈ S 1 . In particular, the Reeb vector field is transverse to the pages, so there are no contractible orbits in V \B. It turns out that every contact structure is supported by some open book (in fact, infinitely many [Gir02]).
We first check an elementary topological fact: Lemma 6.6. The binding B satisfies the second condition of Theorem 1.5, namely Every disc F with ∂F ⊂ B and [∂F ] = 0 ∈ H 1 (B) intersects B in the interior. except when the page is a disc.
Proof. Let F be any smoothly map D 2 → V with ∂F ⊂ B. Suppose that F −1 (B) ⊂ ∂D 2 . Consider the covering map coming from the open book decomposition. For any r < 1 we may lift F | D 2 (r) to a map G : D 2 (r) → R × int(S), uniquely determined if we choose once and fix G(0) ∈ π −1 {F (0)}. Thus we can in fact define G : int(D 2 ) → R × int(S). We may homotopy G to G : int(D 2 ) → {0} × int(S) ∼ = int(S) by scaling the R-coordinate. Let N (B) be the tubular neighborhood of B such that N (B)\B ∼ = S 1 × int(N (∂S)) where N ((∂S) is a collar neighborhood of ∂S chosen small enough so that the monodromy h| N (∂S) is the identity, so we have an inclusion map ι : S 1 × int(N (∂S)) → N (B) and F | ∂D 2 (r) = ι • π • G| ∂D 2 (r) (where r is taken large enough so that F | ∂D 2 (r) ⊂ B). Applying the H 1 -functor and using the fact that G is homotopic to G and [F | ∂D 2 (r) ] = [F | ∂D 2 ] = 0 ∈ H 1 (N (B)) ∼ = H 1 (B) we have From this it is easy to conclude that [G | ∂D 2 (r) ] = k · [∂S] ∈ H 1 (N (∂S)) for some k = 0, and therefore is homotopic to a k-fold cover of ∂S with k = 0; hence G | ∂D 2 (r) is homotopic to k · ∂S in S. However, G | ∂D 2 (r) is null-homotopic in S, while the k-fold cover of ∂S is not if k = 0 (unless the page S is a disc). This contradiction shows that we cannot have F −1 (B) ⊂ ∂D 2 , so F intersects B in the interior. any disc bounding a contractible Reeb orbit either links non-trivially with the binding (since the Reeb vector field is transverse to the pages), or is asymptotic to a component of the binding (in which case by Lemma 6.6 it has an interior intersection with the binding), and by hypothesis [a] is a proper link class. If the page is not the disc or the annulus, there should be many such [a].
If the binding is elliptic non-degenerate then it satisfies (E) as well.
It follows, for simple proper link classes [a] (relative to B), that CCH [a] * (ξ rel B) is well-defined (where ξ is the contact structure supported by the open book). In particular, Theorem 1.5 can be applied to any proper link class [a] for the binding B.
In [CH08a], the authors achieve the much more difficult task of computing the cylindrical contact homology of the manifold (V, ξ) when it is defined. For V \B the situation is considerably simpler. First, cylindrical contact homology of the complement will always defined (at least for simple homotopy classes), and independent of the equivalence classes [λ] of ≡ in proper link classes relative to B. Second, we only need to count a small subset of the holomorphic cylinders and can ignore holomorphic planes (since they will all intersect the binding), so the computation is almost always far easier. Since one can read off computations from [CH08a], we will content ourselves to go through a particular example in some detail to illustrate the approach concretely.
Recall that the Nielsen fixed point classes of h are the equivalence classes of fixed points under the equivalence relation ∼, where x ∼ y if there is a path γ from x to y such that γ is homotopic to h(γ) rel endpoints. A fixed point for h (or more generally a periodic point) determines a loop in the mapping torus. It is a well-known fact that different Nielsen equivalence classes of fixed points determine different free homotopy classes of loops in the mapping torus. More generally, two loops in the mapping torus arising from two k-periodic orbits of h are freely homotopic if and only if there are points on the corresponding orbits which are h k -equivalent (in this case we will call the orbits Nielsen equivalent as well). Hence, to Nielsen equivalence classes we may assign a unique homotopy class [a] in V \B ∼ =h.e. Σ(S, h). ∈ SL 2 (Z) ⊂ GL 2 (R) which fixes the integer lattice. It therefore descends to a map h : T 2 → T 2 of T 2 = R 2 /Z 2 . Since, moreover, it fixes the integer lattice it also defines a map on T 2 \{0} = (R 2 \Z 2 )/Z 2 which preserves the area form inherited from R 2 . One may also wish to think of such a map on a torus with one boundary component, which we model by removing a -disc around the each point in the integer lattice and denote T 2 \D 2 ( ). The map h can be represented by the time-1 flow x y → 2 1 1 1 t x y generated by the symplectic vector field X(x, y) = log 2 1 1 1 · x y , log 2 1 1 1 = 1 √ 5 log 3 + √ 5 2 and it is straightforward to compute that it is generated by the Hamiltonian flow of a quadratic form Q(x, y) = Ax 2 + Bxy + Cy 2 = 1 √ 5 log 3 + √ 5 2 x 2 − xy − y 2 of signature (1, 1). Take φ : R → [0, ∞) to be any smooth function with the following properties: and denote φ = φ( 1 ·). Consider the Hamiltonian function φ 2 (x 2 + y 2 ) · Q(x, y) and denote byh its time-1 map. Since it agrees with h outside of a disc of radius we may defineh on T 2 \D 2 ( /2) by replacing the map h withh inside the disc of radius . In this way we get an area-preserving maph on T 2 \D 2 ( /2) which is the identity on a neighborhood of the boundary.
This abstract open book gives rise to S 3 with binding a figure-eight knot. The contact structure supported by this open book, however, is over-twisted. 6.4.2. Realizing the monodromy map by a Reeb flow. Let h : S → S be a map that preserves the area form ω. Let β be a primitive ω = dβ. Consider the mapping torus of h, which is [0, 2π] × S quotiented by (1, x) ∼ (0, h(x)). We wish to find a contact form with the Reeb vector field ∂ t (we will use t to denote the S 1 "coordinate"), using β to construct it. Lemma 6.9. [CHL08] If h * − I is an isomorphism on H 1 (S; R), we may find a primitive β such that h * β − β is zero in H 1 (S; R). Hence by Lemma 6.8 h can be realized by a Reeb flow on its mapping torus.
Consider specifically the mapsh on T 2 \D 2 ( /2) with the area form inherited from the area form for R 2 (thus T 2 \D 2 ( /2) has area 1 − π 2 /4). We can assume that the form β has the form ( r 2 2 − 1 2π )dθ in D 2 ( ). Let z be the S 1 coordinate of the mapping torus, so that the contact form dt + β in these coordinates is To construct the open book, this is identified with a neighborhood of the boundary of S 1 × D 2 ( /2) with coordinates (s, φ, t) ∈ D 2 ( /2) × S 1 by s = r, φ = z, t = −θ in which the form is and the pages are φ = const.
We will now extend this form to the interior of D 2 ( /2) × S 1 using a well-known argument [TW75]. The contact form will be where the functions f, g have the following properties (and T > 0 is a parameter): • f (s 2 ) = 1 and g(s 2 ) = 1 2π − s 2 /2 for s ≥ /2 • f (s 2 ) = s 2 and g(s 2 ) = 1−T s 2 for s ≤ δ for some δ > 0 as small as required • f g − f g > 0 • g < 0: This ensures that the Reeb vector field is everywhere positively transverse to the pages φ = const and is the reason we need to assume T > 0. Moreover, then there is some constant c = c ,T ≥ 0 such that −g (s) ≥ c > 0. This bounds the first return time of the Reeb flow to a page φ = const near the binding B by the constant 2π/c.
The explicit construction of such functions is straightforward but slightly tedious, so we will omit it. We remark as in [TW75] that it is equivalent to drawing a smooth curve in R 2 to the point (1, 1 2π ) from the point (0, 1) with respective velocities at these points (1, −T ), and (0, −1), such that the position vector is never collinear with the velocity vector (and, by the fourth condition, the velocity vector must point downward at all times as well).
We see that the binding has Conley-Zehnder index equal to CZ(B k ) = 2 kT + 1 with respect to the framing given by the angular coordinate φ (i.e. with respect to the vectors ∂ r ((r, 0), t), ∂ r ((r, π/2), t) at the point (0, 0, t) ∈ B). Therefore we will select T irrational so that B is elliptic non-degenerate. As constructed, it has an open neighborhood such that no periodic orbit enters the neighborhood. The Reeb flow thus constructed is degenerate, but after a small perturbation can be made non-degenerate without altering the dynamics too much. The pages form a surface of section with bounded return time. Moreover, there is a page S ∼ = T 2 \0 such that the restriction to a subsurface S ∼ = T 2 \D 2 ( /2) of the first return map is the maph , and to a further subsurface S ∼ = T 2 \D 2 ( ) we haveh | S = h| S .
Given > 0 sufficiently small to perform the above construction, let λ be such a contact form.
Lemma 6.10. For each k ≥ 1, there is a k such that if < k then for each non-trivial Nielsen fixed point class there is a unique non-degenerate closed Reeb orbit for λ in the corresponding homotopy class [a] in S 3 \B ∼ = Σ(T 2 \0, h).
Proof. The fixed point set of h k for each k is a finite set of points. Therefore, there is a disc D containing the puncture 0 which contains no fixed points of h l for l ≤ k. Choose an even smaller disc 0 ∈ D such that k l=1 h l (T 2 \D) ∩ D = ∅ Then if D 2 ( ) ⊂ D then the periodic points of period at most k ofh and h on T 2 \D are the same.
We may also choose 0 ∈ D ⊂ D such that h l (D ) ⊂ D for 1 ≤ l ≤ k. Select k so that D 2 ( k ) ⊂ D . Then, if < k , all l-periodic orbits ofh contained in D must enter D , and are therefore in fact contained in D . So we only have to examine the casesx ∈ D andỹ ∈ T 2 \D. Let h : R 2 → R 2 lifth in the sense that π • h = π •h (for π : R 2 → T 2 the universal cover), which is uniquely determined by demanding h (0) = 0. Letx be a l-periodic point in D andỹ a l-periodic point in T 2 \D, andγ a path between these two points, with lifts x, y, γ respectively such that γ(0) = x, γ(1) = y and x is contained in the preimage of D containing the origin in R 2 . It is easy to see that the orbit h i (x) remains in this component for all i and therefore h k (x) = x. On the other hand, h l (y) = h l (y) = y. After these observations it is straightforward to deduce thatx,ỹ are not Nielsen equivalent forh l : a homotopy rel endpoints betweenγ andh l •γ would lift to one between γ and h l • γ rel endpoints, but the endpoints of these two paths do not even coincide so there can be no such homotopy in the first place. Thus,ỹ,x lie in different equivalence classes.
One can show that the loops corresponding to l-periodic points x and y are homotopic in the mapping torus if and only if there are iteratesh i (x),h j (y) which areh l -equivalent. Since x is the onlyh l fixed point in itsh l -equivalence class (which follows by checking directly that the fixed points of the matrix map h l lie in different equivalence classes) this shows that the corresponding orbit is the unique closed Reeb orbit in this homotopy class. It will be hyperbolic and therefore non-degenerate. Therefore, if [a] is the homotopy class of a fixed point of h, the triple (λ, B, [a]) satisfies (P LC) (and (λ, B) satisfies (E) as well), though λ may not be nondegenerate. Therefore we will need to perturb, and the following Lemma asserts that small non-degenerate perturbations retain important properties: Lemma 6.11. (Approximation lemma) In any C ∞ neighborhood of λ , there is a non-degenerate contact form λ with the following properties: • Let φ t denote the Reeb flow. There are constants c, C and a page S of the open book decomposition such that for any x ∈ S, there is a least 0 < t such that φ t (x) ∈ S and c < t < C. In particular, every closed orbit has non-trivial linking number with B. Proof. By Lemma 6.10 the (degenerate) contact form λ has the following properties: • Let φ t denote the Reeb flow. There are constants c, C and a page S of the open book decomposition such that for any x ∈ S, there is a first t > 0 such that φ t (x) ∈ S and c < t < C . • For each Nielsen fixed point class [a] such that ([a], B) ≤ k, if < k then there is a unique non-degenerate closed Reeb orbit for λ in [a]. By Lemma 5.2, in any C ∞ -neighborhood of λ , we can find a non-degenerate contact form λ such that (λ , B) ≡ (λ , B). In particular, if λ is chosen close enough, the page S will still be a surface of section for λ because of the Conley-Zehnder index in a neighborhood of the binding, and because λ has bounded return time on the (compact) complement of this neighborhood in S if λ is chosen close enough. Therefore there are c, C such that if x ∈ S then for some c < t < C (in fact, the first such t > 0) we have φ t (x) ∈ S.
The second assertion can be deduced from the λ non-degeneracy of the orbits which correspond to fixed points of the unperturbed map h.
The following proposition follows immediately: Proposition 6.12. Let [a] denote a homotopy class in S 3 \B corresponding to a simple Nielsen fixed point class. Then there is an such that for the contact form λ in Lemma 6.11 we have (λ , B, [a]) is non-degenerate, satisfies (P LC) and for J ∈ J gen CCH [a] * (λ , J rel B) ∼ = Q It follows that (see the following Remark 6.13) CCH [a] * is defined and CCH [a] * (ξ rel B) ∼ = Q Remark 6.13. While the closed orbit above is assumed simple, it is conceivable that the homotopy class [a] it is in is not (i.e. contain multiply covered loops). However, Remark 4.5 explains why CCH Let us denote by ξ o the (over-twisted) contact structure supported by this open book. One may apply Theorem 1.5 to deduce, given any contact form for ξ o with a closed Reeb orbit transversely isotopic to B, information about the homotopy classes of Reeb orbits in S 3 \B. However, arguing a little more carefully we can also deduce the following growth rate of periodic orbits in the action (which proves Corollary 1.11): Proposition 6.14. Suppose µ is a contact form for ξ o , and its Reeb vector field has a closed periodic orbit transversely isotopic to the binding B. Then • the number of geometrically distinct closed Reeb orbits of period at most N grows at least exponentially in N , and • the number of closed orbits x such that (x, B) ≤ k grows exponentially in k.
Proof. Choose any λ = λ as in Lemma 6.11, so that (λ , B, [a]) satisfies (P LC) for each homotopy class [a] corresponding to a non-trivial Nielsen fixed point class for h. Lemma 6.11 asserts that there is a surface of section S such that each point x ∈ S has first return time t in bounded by t < C for a constant C independent of x. In particular, for each closed Reeb orbit we have a bound T (x) < C · (x, B). The number of simple non-trivial Nielsen fixed point classes of h k is bounded below by Ae bk for positive constants A, b. These fixed point classes satisfy 1 ≤ ([a], B) ≤ k and therefore T < C · k. Therefore, the number of closed Reeb orbits of action at most Ck is at least Ae bk . By scaling b = b/C we have the number of closed Reeb orbits of action at most k is at least Ae bk . Let µ be a contact form on S 3 with kernel ξ o with an orbit transversely isotopic to B, and if necessary pull-back by a contact isotopy to arrange that this orbit in fact coincides with B. First suppose that µ ≺ λ . Arguing as in the proof of Theorem 1.5, for each non-trivial simple Nielsen fixed point class [a] such that ([a], B) ≤ k we obtain a µ orbit in the same homotopy class with period bounded by its λ period which is itself bounded by C · k. Therefore we have an exponential lower bound on the number of geometrically distinct periodic orbits of period at least N for µ as well. Any µ can be rescaled so that dµ ≺ λ , and one establishes lower bounds similarly (replacing b with d · b).
6.5. Fibered hyperbolic knots. The arguments in the above example can be replaced by any open book with one boundary component and pseudo-Anosov monodromy map which is Reeb-realizable.
In most examples one must smooth the prong singularities of the pseudo-Anosov map, a procedure we will not discuss (one way to smooth the singularity is discussed in [CC09], section 3.2). Because φ * − I is invertible (as we observed in the introduction), we can apply Lemma 6.9 cited above to realize the (smoothed) pseudo-Anosov map by a Reeb vector field, continuing the contact form over the binding as in the last section. The reader should note that the argument that the number of non-trivial Nielsen fixed point classes grows exponentially is generally more complicated. Moreover, there may be several non-degenerate fixed points in certain Nielsen equivalence classes, but an appeal to Euler characteristics can be used if needed to show that the homology is non-trivial in non-trivial classes. For details we refer to [CH08a], Section 11.1. Consider now specifically a tight fibered hyperbolic knot realizing the Thurston Bennequin bound in the tight S 3 . By [Hed08], tight fibered knots realizing the Thurston-Bennequin bound are determined by the topological knot type up to transverse isotopy, so we may assume without loss of generality that such a knot is in fact the binding of an open book decomposition supporting the tight S 3 with (pseudo-Anosov) monodromy map φ. By definition, the open book supports the tight contact structure, so the kernel of the constructed contact form (which supports this open book) is the tight contact structure on S 3 (up to isotopy). Thus Corollary 1.9 is proved in a similar manner as Corollary 1.11/Proposition 6.14 by considering the pseudo-Anosov monodromy map φ in place of the matrix map h.
We include some general remarks about tight fibered hyperbolic knots. First, there are approaches to decide whether or not a given knot is fibered ( [Gab86], [Ni07]). Second, a knot forms the binding of an open book decomposition for the tight S 3 if and only if it has a transverse representative which realizes the Thurston-Bennequin bound (2g −1 where g is the three-genus) in the tight S 3 [Hed08]. Third, a sufficient (and necessary) condition for the monodromy map to be pseudo-Anosov is that the knot is a hyperbolic knot [Thu98]. Finally, to see that the first property in the definition implies the fourth property as we claimed in the introduction, evaluating the Alexander polynomial ∆(t) = det(t · I − h * ) at t = 1 shows that h * − I is invertible since ∆(1) = ±1 for any knot. Matt Hedden pointed out to the author that there is an abundance of knots satisfying all these properties including, for example, the Fintushel-Stern knot (see Figure 6.2, 12n 0242 ), also known as the Pretzel knot (−2, 3, 7). An easy way to see that there are infinitely many examples is to notice that one can positively stabilize many times to make the genus g of the page arbitrarily large, thus increasing the self-linking number of the binding (it will be 2g − 1) which distinguishes the knots. Positive stabilization does not change the manifold or contact structure, so 1, 2 continue to hold. It is shown in e.g. [CH08b] that if one positively stabilizes an open book appropriately enough times that the monodromy can be made pseudo-Anosov (thus will satisfy 3), so this infinite family will possess infinitely many tight fibered hyperbolic representatives. Using the search form in [CL] and the observations above, the 35 tight fibered hyperbolic knots with crossing number at most 12 were found and are listed in Figure 6.2 (by knot diagram and Alexander-Briggs notation) (1) 10 139 (2) 10 145 (3) 10 152 (4) 10 154 (5) 10 161 (6) 11n 077 (7) 11n 183 (8) 12n 0091 (9) 12n 0105 (10) 12n 0136