The persistence of the Chekanov-Eliashberg algebra

We apply the barcodes of persistent homology theory to the Chekanov-Eliashberg algebra of a Legendrian submanifold to deduce displacement energy bounds for arbitrary Legendrians. We do not require the full Chekanov-Eliashberg algebra to admit an augmentation as we linearize the algebra only below a certain action level. As an application we show that it is not possible to $C^0$-approximate a stabilized Legendrian by a Legendrian that admits an augmentation.


Introduction
The (Lagrangian) Arnold conjecture states that the number of intersection points of a Lagrangian submanifold with its Hamiltonian image is bounded below by the sum of the Lagrangian's Betti numbers. Floer developed Lagrangian Floer theory to prove this bound in certain cases [16], but the bound does not always hold. Chekanov, using Hofer's norm [19] for the Hamiltonian isotopy, measured how "long" this bound persists by measuring how long Floer theory remains valid: how long d 2 = 0 holds, and how long the Floer theory remains invariant [5]. Lagrangian Floer theory also provides a similar temporary lower bound on the number of Reeb chords between a Legendrian and its contact Hamiltonian image [7].
In this article, we study the persistence of Reeb chords between a Legendrian submanifold Λ ⊂ (Y, ker α) of a contact manifold and its image under a contact isotopy. We replace Floer theory with the linearized Chekanov-Eliashberg algebra A(Λ) (also called the Legendrian contact DGA) induced by a choice of augmentation. We again measure things in terms of the Hofer norm of the contact Hamiltonian isotopy. In addition to measuring how long a certain linearized Chekanov-Eliashberg homology theory remains well-defined, we also measure how much of it persists.
We require that A(Λ) be rigorously well-defined, that the "handle-slide and birth/death bifurcation-analysis" (Section 2.2) proof of invariance holds, and (for Theorem 1.1) that there is a certain correspondence of J-holomorphic disks (Proposition 2.10). As of this writing, these requirements restrict our ambient contact manifold to be (Y, ξ) = (P × R z , ker α). Here (P, dλ) is an exact symplectic manifold tame at infinity (Gromov compactness holds), and the contact 1-form, which determines the Reeb flow, must be of the type α std := dz + λ [10]. Note that this includes one-jet spaces endowed with the canonical contact form.
1.1. Background. A Reeb chord on a Legendrian submanifold is a non-trivial integral curve of the Reeb vector field R α ∈ Γ(T Y ) defined by ι Rα α = 1 and ι Rα dα = 0. We are interested in estimating the number of Reeb chords from a given Legendrian Λ to its image under a contact isotopy. If there are no such Reeb chords, we say that the contact isotopy displaces Λ. Of course this notion depends on the choice of contact form.
The set of α-Reeb chords of Λ canonically generates A(Λ) as a free noncommutative algebra. The grading is a certain Maslov index. The differential ∂ counts J-holomorphic disks in the symplectization (R t × Y, d(e t α)) with Lagrangian boundary condition R t × Λ. The homotopytype of the DGA is invariant under Legendrian isotopy, which is a smooth isotopy of Legendrian submanifolds. We often notationally suppress the grading, the differential ∂, the dependence on J and α. See [10] and references therein for definitions.
Each Reeb chord c has a length (also called action) ℓ(c) := c α. For 0 < l ≤ ∞, let A l (Λ) be the unital sub-algebra generated by those generators with length bounded from above by l. The action preserving properties of the differential of the Chekanov-Elishberg algebra implies that A l (Λ) ⊂ A(Λ) is a unital sub-DGA. To that end, recall that the differential applied to a generator c consists of a sum of words of generators whose lengths all are strictly less than the length of c, as follows from an elementary application of Stokes' theorem.
An augmentation for the DGA A, ε : (A, ∂) → (k, ∂ k := 0), is a (graded) DGA-morphism to the ground field k viewed as a DGA. We will want to choose l such that A l (Λ) has an augmentation; since A l = k for l > 0 sufficiently small, this is always possible. If Λ is loose in the sense of Murphy [22] and c is the Reeb chord in a standard representative of the loose chart then there are a number of standard Legendrian contact homology arguments which show, up to unit, ∂(c) = 1. The contradiction ε • ∂(c) = ε(1) = 1 = 0 = ∂ k • ε(c), means that we cannot choose l ≥ ℓ(c).

The oscillation of a contact Hamiltonian
is the key ingredient in the Hofer norm of the corresponding contact Hamiltonian isotopy φ t α,Ht (which is defined by H t (φ t α,Ht (x)) = α d dt φ t α,Ht (x) ). Contact isotopies are generated by uniquely-defined contact Hamiltonians, that depend only on the choice of contact form; we thus sometimes say the oscillation of a contact isotopy when we mean the oscillation of the corresponding generating contact Hamiltonian. This article focuses on contact isotopies acting on Legendrian submanifolds. These are Legendrian isotopies. For a Legendrian isotopy ϕ t : Λ ֒→ Y we can consider the induced family of smooth functions that we identify with a family of functions h t : Define the oscillation of a Legendrian isotopy to be The perspectives between Legendrian isotopies ϕ t and ambient contact isotopies φ t α,Ht can be switched: Furthermore, there exists a smooth extension of h t to a globally defined contact Hamiltonian Finally, changing the Legendrian isotopy ϕ t by the precomposition ϕ t • ψ t of a smooth isotopy ψ t : Λ → Λ does not change h t . See e.g. the proof of [18,Theorem 2.6.2]. Henceforth, we abandon these distinctions and refer to the oscillation, using the compromise notation of H t osc .
For any compactly supported contact Hamiltonian H t : P × R → R, suppose that the inequality H t osc < min {l/2, σ ι i } is satisfied for some i ∈ {0, . . . , n}, and that φ 1 α std ,Ht (Λ) is transverse to the Reeb flow applied to Λ. Then there exists at least a number of Reeb chords with one endpoint on Λ, one endpoint on φ 1 α,Ht (Λ).
Remark 1.2. If H t is indefinite, by which we mean that H −1 t (0) = ∅ holds for all t, then the Reeb chords produced by the above theorem are all of length less than H t osc . Remark 1.3. If Λ is spin and orientable, we can define the Chekanov-Eliashberg algebra with Q or Z p -coefficients for p prime [12]. Our arguments work for these fields, so we can set k to equal Q or Z p in the above bound. Without this assumption on Λ, we set k equal to Z 2 .
Remark 1.4. Fix a generic choice of points pt i ∈ Λ disjoint from the Reeb chords, one for each connected component of Λ. If we wish to, for the definition ofσ n , we can impose the additional requirement that some disk in M n (c) must have its boundary pass through the union of rays R × {pt 1 , . . . , pt π 0 (Λ) }.
Remark 1.5. Our result can be improved to requiring only H t osc < min {l, σ ι i } when the contact Hamiltonian H t is lifted from a Hamiltonian on the symplectic manifold P. For contactizations, this is equivalent to having a contact isotopy which preserves the contact form.
Recall the standard Legendrian sphere in R 2n+1 whose front projection in R n+1 x 1 ,...,xn,z has an O(n)-symmetry in the x i -directions. It has one Reeb chord c along the z-axis of Legendrian Contact Homology index n and its Thurston-Bennequin invariant is tb = (−1) (n 2 +n+1)/2 . The front when n = 2 is depicted in Figure 1. See [9, Section 3] for a review of these concepts. When ℓ(c) = a we denote this sphere by Λ St (a).
(See e.g. [11].) Theorem 1.1 shows that a contact Hamiltonian that displaces Λ St (a) must be of oscillation at least a.   Consider an open subset U ⊂ Y 2n+1 having the homotopy type of a closed n-manifold. We say that a Legendrian Λ ⊂ Y can be squeezed into U if there exists a contact isotopy φ t of the ambient space such that Stabilized Legendrians [9, Section 3] of dimension at least two are loose and satisfy an hprinciple due to Murphy [22]; in particular, they are C 0 -dense in the space of Legendrian embeddings. By this h-principle, a loose Legendrian Λ ′ can be squeezed into U whenever this subset contains a formal Legendrian embedding in the same formal isotopy class as Λ ′ . Theorem 1.9. Let Λ stab ⊂ R 2n+1 be a stabilized Legendrian, and suppose for Λ ′ ⊂ R 2n+1 that A(Λ ′ ) admits an augmentation. Then Λ ′ cannot be squeezed into a standard contact neighborhood of Λ stab .
There has been recent interest in C 0 -limits of Hamiltonian diffeomorphisms (contact or symplectic) [3,24]. In particular, Rosen and Zhang prove that for the C 0 -limit of the images of a Legendrian submanifold under a sequence of contactomorphisms, if the limit is smooth and the sequence satisfies [24, Definition 4.1], then the limit is also Legendrian. The definition in particular requires the sequence of contactomorphisms to uniformly converging conformal factors. In [24, Remark 1.5] the authors suggest that the latter mechanism prevents such a sequence from being an "approximation by zig-zags" (recall that e.g. any, not even necessarily Legendrian, knot can be approximated by a Legendrian knot by adding more and more zig-zags). Theorem 1.9 could be interpreted as evidence that something stronger actually prevents this, as it shows that a stabilized Legendrian cannot be approximated by a Legendrian that admits an augmentation.
The homology condition in the definition is crucial. In R 2n+1 all Legendrians can be put in a neighborhood of any other Legendrian by first rescaling it to make it sufficiently small. Also, the 2-copy of Λ stab sitting in an arbitrary small neighborhood of Λ stab , has non-zero Z-homology but vanishing Z 2 -homology. The 2-copy has an augmentation which one can see either by an explicit construction of an exact Lagrangian filling by a cylinder I × Λ stab or, alternatively, by an explicit calculation of the Chekanov-Eliashberg algebra of the two-copy using the theory from [13].
Using this h-principle to approximate a non-Legendrian deformation of the initial Legendrian by the stabilized version we prove Theorem 1.10 below, in contrast to Theorem 1.1 above.  [7]. In this earlier article, our bound uses the conformal factor and is like Chekanov's bound [5]: the lower bound is either the full sum of Betti numbers, or 0 (we do not measure its persistence). First we compare the oscillation and the change in Reeb chord length. Then we study a length-filtered invariance property for the linearized Chekanov-Eliashberg algebra. We apply it to a two-component Legendrian, which includes the two-copy link of a single Legendrian. We interpret the results in terms of barcodes, the Barcode Proposition 2.8, which we then apply to prove Theorem 1.1. We end with the proof of Lemma 1.8.
Denote by Λ(t) a Legendrian isotopy parameterized by t. Let H t be the contact Hamiltonian H t : Y → R generating an ambient contact isotopy inducing Λ(t), and let X t denote the contact vector field.

2.1.
Reeb chord length and oscillatory energy. The filtration properties depend on the size of the oscillation of the contact Hamiltonian inducing the Legendrian isotopy. The main contact geometric property that we need is the following.
of Reeb chords with endpoint and starting point e(t), s(t) ∈ Λ(t) on a family of Legendrian submanifolds satisfies .
Proof. Cartan's formula gives us Using this we compute For the second equality we have use the fact that a Reeb chord (φ t α,Ht ) −1 • c(t) with endpoints on Legendrian Λ are critical points for the functional γ → γ (φ t α,Ht ) * α (with t fixed). For the last equality we combine Cartan's formula with the fact that the one-form ι X(t) dα pulls back to zero on any Reeb chord (by the definition of the Reeb vector field).

2.2.
Handle-slides and birth/deaths. Lemma 2.1 uses only elementary calculus and applies to a Legendrian isotopy in any contact manifold Y. Henceforth, however, we need to assume as stated in the introduction, that (Y, ξ) = (P × R, ker{dz + λ}).
Given the Legendrian isotopy Λ(t) and constants 0 ≤ t − ≤ t + ≤ 1, denote the stable-tame DGA morphism constructed in [10] by A generic Legendrian isotopy has isolated singular moments during which exactly one of the following occurs: a unique rigid index −1 disk appears (handle-slide); a unique pair of Reeb chords appears/cancels (birth/death); or, the relative actions of two Reeb chords changes signs.
If the handle-slide disk u ∈ M A (a, b 1 · · · b k ) exists at time t then by [9,10] the induced DGA morphism for ǫ > 0 arbitrarily small is We extend the definition of length from Section 1. For any non-zero element in the algebra x ∈ A(Λ), let ℓ(x) ∈ [0, +∞) be the maximum of sums of lengths of Reeb chords in a nonzero word of Reeb chord generators that appears in x. We use here that we have a canonical basis of A given by the words of Reeb chords. Stokes' Theorem implies We review the induced algebraic continuation map at a birth-moment in the proof of [9, Lemma 2.13] below. Suppose a + , b + are the newly-born pair of points at time t which exists at time t + ǫ with |a + | = |b + | + 1. For all sufficiently small ǫ > 0, we can assume that the other (l + k) ≥ 0 chords satisfy i as ordered by their action/subscript. For the base case (the map Φ t−ǫ,t+ǫ restricted to the sub-DGA generated with no a − i generators), we define Define the algebra morphism f : A(Λ(t + ǫ)) → A(Λ(t + ǫ)) on words w + linear in b + by replacing the letter b + with a + : Here α + , β + is not divisible by b + . Observe that can be assumed to be arbitrarily small.
The map Φ t−ǫ,t+ǫ may be viewed as a sequence of artificial handle-slide maps: each The above considerations on Reeb chord lengths implies that Φ t−ǫ,t+ǫ does not increase action.

2.3.
Filtered invariance for the two-component link. Consider a Legendrian isotopy Λ(t) of a two-component link. (Each "component" may itself be disconnected and have interesting topology but we do not consider such sub-components individually.) There are two types of chords: pure chords Q pure = Q pure (t) which start and end on the same component; and, mixed chords which run between the two different components.
The sub-algebra A pure (Λ(t)) freely generated by pure chords Q pure is of course a DGA of its own, given as the free product of the DGAs for the two different components. Henceforth we will only consider an augmentation ε which vanishes on each mixed chord, i.e. which is induced by an augmentation of A pure (Λ(t)). Note that an augmentation of the second type always induces an augmentation of the first type by elementary topological reasons: the differential of a mixed chord must output words in which at least one chord is mixed. For an ordering Λ 0 (t) ⊔ Λ 1 (t) = Λ(t) of the two components, let Q mixed = Q mixed (t) denote the mixed chords starting at Λ 0 (t) and ending at Λ 1 (t). Let LCC ε * (Λ(t)) the induced linearized (chain) complex generated by Q mixed . The (linearized) differential ∂ ε counts holomorphic disks with a positive puncture at a mixed chord and the augmentation applies to all but one of the negative punctures (and thus the output is thus again a mixed chord of the first type). We refer to [1] for more details.
We let LCC ε * (Λ(t)) b a denote the linearised subcomplex generated by the subset of mixed chords in Q mixed having lengths contained in the interval [a, b). The arguments of [1] which imply that LCC ε * (Λ(t)) is well-defined combined with a standard filtered chain complex argument and Stokes' Theorem, imply LCC ε * (Λ(t)) b a is well-defined (but of course not necessarily invariant). We call [a, b) the action window.
We are also interested in the case when the DGA of Λ(t) might not have an augmentation, but at the same time, the sub-DGA A l pure (Λ(t)) ⊂ A pure (Λ(t)) generated by only the pure chords of length less than some fixed number l ≥ 0 admits an augmentation Consider the subspace A l 1 ⊂ A spanned by words of chords of which precisely one is a mixed Reeb chord contained in the subset Q mixed (which thus in particular starts on Λ 0 , ends on Λ 1 ,) and is of length contained in the interval [a, b), while the remaining pure chords all have length less than l. This subspace can naturally be identified with the free A l pure -bimodule generated by the chords Q mixed of lengths in the interval [a, b). This bimodule can be made into a chain complex, which we denote by LCC l,ε * (Λ(t)) b a . Since this complex is new to the literature, we describe its (linear) differential below.
There is an automorphism Φ ǫ : A l 1 → A l 1 given as the restriction of the algebra-map that is defined by c → c + ε(c) on each generator (which by assumptions on ε thus fixes the mixed generators). Let π ε : A → A l 1 ⊂ A be the canonical projection A → A l 1 induced by our canonical basis, post-composed with Φ ǫ . The linearized differential can then be expressed as the linear part of π ε • ∂ restricted to the vector subspace LCC l,ε * (Λ(t)) b a ⊂ A l 1 spanned by the mixed chords, which is a map Proof. The inequality b − a ≤ l implies that when counting the glued pairs of disks which contribute to (∂ ε ) 2 , the Reeb chord at which the gluing occurs cannot be a one for which the augmentation is not defined. This reduces the (∂ ε ) 2 = 0 -proof to the established case when the augmentation is globally defined.
We need to set-up a bifurcation analysis to prove a certain form of invariance for the complex LCC l,ε * (Λ(t)) b a as the parameter t varies. We start by considering the case of a singular moment t for the bifurcation of the DGA, at which no chord has length equal to l.
Choose ǫ > 0 sufficiently small so that the chords which do not undergo a birth/death are preserved for all s ∈ [t − ǫ, t + ǫ]. In the event of a birth of pair of chords a + , b + we suppress the stabilization notation, using A(Λ(t − ǫ)) for S(A(Λ(t − ǫ))).
Lemma 2.4. Assume that no chord at time t = 0 has length equal to l, and that there exists an augmentation ε : be either a birth/death or a handle-slide as in (2.2). For sufficiently small ǫ > 0, there exists an augmentation ε ′ : A l pure (t + ǫ) → k for which the map is a composition of action-preserving (linear) chain maps c → c + f with ℓ(f ) < ℓ(c) (here we include the case f = 0) together with cancellations of death pairs.
Proof. Since ǫ is small, we can assume the same set of pure chord generators for A l pure (Λ(t−ǫ) and for A l pure (Λ(t+ǫ)), as well as the same set of mixed chord generators for LCC l,ε * (Λ(t−ǫ)) b a and for LCC l,ε ′ * (Λ(t + ǫ)) b a . The proof reduces to the the three cases when Φ t−ǫ,t+ǫ corresponds to either a single (real or artificial) handle-slide, a single stabilization, or a single destabilization.
Handle slide: There exists an inverse Φ t−ǫ,t+ǫ which also is action preserving. The augmentation can thus be taken to be ε ′ = ε • Φ −1 t−ǫ,t+ǫ | A l , and we thus get ε = ε ′ • Φ t−ǫ,t+ǫ | A l . In this case we define We need to check that φ ε is a chain map of the sought form.
When Φ t−ǫ,t+ǫ is defined by a disc for which all negative pure punctures action less than l, then statement follows from a standard argument; see e.g. [1, Section 2.4]. (We can interpret φ ε as a linearization of a DGA morphism in the standard sense.) Consider the handle-slide "disk" (real or artificial) with positive puncture e and negative punctures f 1 , . . . , f k . The inequality b − a < l and the description of the real and artificial handle-slides in Section 2.2 imply the following: if e is a pure chord of less length than l or a mixed chord between length a and b, then no f i can be a pure chord of length greater than l, unless possibly if at least one of the f i ∈ Q mixed is a mixed chord with action less than a. In this case, the induced linear map is the identity. That the identity map is a chain map can be seen from the point of view Gromov compactness: the handle-slide disk with a negative mixed chord of action less than a cannot be glued to a disk used to defined the boundary, since the action difference of the mixed chords of such a glued disk is greater than b − a.
Stabilization: We extend ε ′ to vanish on the new pair of generators, and φ ε is simply the canonical inclusion.
Destabilization: We let ε ′ take the same value as ε on the remaining generators. In the case when the death involves pure chords, the chain map φ ε is the identity. In the case when the death involves mixed chords, φ ε is simply the corresponding quotient. Note that these are chain map, even though it is possible that ε ′ • Φ t−ǫ,t+ǫ = ε. (However, since we are only considering the destabilization, as opposed to a death together with its artificial handleslides, this is irrelevant.) The main difference between the invariance of the usual linearized complex and the invariance of a sequence of complexes LCC l,ε * (Λ(t)) b(t) a(t) considered here is that generators of the latter can slide in and out of the action window [a(t), b(t)). We call such a phenomenon an entrance/exit moment. Such a moment can take place at either action level a(t) or b(t), and say that an entrance (resp. exit) is safe if it occurs at action level a(t) (resp. b(t)). When all entrances and exits are safe, there still is a composition of well-defined and canonical chain-maps for such a sequence of "moves," but chain maps need not be chain homotopy equivalences. We will call the chain maps corresponding to either a birth/death, a linear handle-slide, or safe entrance/exit an elementary chain map. When a chain map is a chain homotopy equivalence, we call it simple.
Note the different regions for the max and min in the below definitions.
If H t lifts from a Hamiltonian on the symplectic manifold P as in Remark 1.5, then we set l(t) ≡ 0.
Fix a, b, l such that 0 < b − a < l − l i . If A l pure (Λ(t i )) admits an augmentation ε i , then there is an augmentations ε i+1 of A l−l i pure (Λ(t i+1 )) and an elementary chain map Proof. We break up the isotopy into arbitrarily short intervals (t i+1 − t i ≪ 1 → l i ≪ 1) which isolate all birth/deaths and handle-slides, and longer intervals for which no such singular event occurs. For the short intervals, Lemma 2.5 follows from Lemma 2.4. So consider a longer interval.
There could potentially be a chord of action l + ǫ (for some 0 < ǫ ≪ 1) such that ε i has no extension to the DGA A l+2ǫ pure . By Lemma 2.1 the action of this chord might decrease by at most l i under the isotopy. This is why ε i+1 might be only defined for chords with actions less than l − l i . If we are as in Remark 1.5, then the Legendrian isotopy does not change the action of the pure chords, and so ε i+1 is still defined on A l pure .
When we apply Lemma 2.1 to the mixed chords, note that the start (resp. end) point of the chord always lies on Λ 0 (t)) (resp. Λ 1 (t)). In other words, the length ℓ(c(t)) of any of these chords at time t has differential that satisfies an even stronger bound To see that φ t i ,t i+1 is a chain map, we argue as follows. For t ∈ [t i , t i+1 ] consider the family of complexes LCC a+M (t) . The crucial statement is that all entrances and exits are safe, i.e. no generator can enter at the lower threshold a + M (t), nor leave at the upper threshold b + M (t). This is due to the upper bound of the speed of the increment of action of a mixed Reeb chord given by Inequality (2.6).
We adopt to our set-up the barcodes from persistent homology theory (see e.g., [28, Section 1] and its references). Recall that the barcode of the complex LCC l,ε * (Λ) b a consists of a finite union of closed intervals contained inside [a, b), called bars. Each bar is characterized by a starting (lower) point s and endpoint e = s+L and length L. The rank of the inclusion-induced map H(LCC l,ε * (Λ) τ 0 a ) → H(LCC l,ε * (Λ) τ 1 a ) is the number of bars starting at s and ending e such that a ≤ s ≤ τ 0 ≤ τ 1 < e ≤ b. There exists a basis for the complex LCC l,ε * (Λ) b a whose elements are in bijection with all boundary points of the bars contained inside [a, b), where the action of the element is the same as the level of the boundary point. Furthermore, the starting points correspond to cycles, while endpoints contained inside [a, b) correspond to basis elements x for which ∂(x) is the starting point of the same bar.
Shifting the action-window for the complex in the positive direction is a chain map, and has the following effect on the barcode.  a+M (t) where ε t is the augmentation from Lemma 2.5. When the complex undergoes no bifurcation, entrance, or exit, the barcode is simply changed by a continuous change of action levels for its starting and endpoints induced by the change of action for the generators, as well as its extended length.
(1) The barcode is unaffected by a linear handle-slide. In the event of a birth/death of chords x, y a bar connecting ℓ(x) to ℓ(y) is added to/removed from the barcode.
In addition to the standard bifurcation in (1) the barcode undergoes the following: (2) the starting point of a bar can exit below, i.e. slide below action level a + M (t) at time t; in this case the bar gets replaced with a new bar starting at the same action level as where the old bar ended.
(3) the starting point or an endpoint of a bar can enter from above, i.e. slides below action level b + M (t) at time t; in this case either an endpoint is introduced, or a new bar is introduced.
Proof. This follows from Lemmas 2.4, 2.5 and 2.6. Note that all entrances/exits are safe.

2.4.
Completing the proof of Theorem 1.1. We use the notation introduced in Section 1 before the theorem statement. In particular, Λ is some Legendrian, not the link from the previous section.
With an arbitrarily small change in the Legendrian and the contact Hamiltonian, we can assume all moduli space below a pre-determined index are transversely cut out [10]. In particular Lemma 2.2 and [13, Theorem 3.6] (used below) hold. LetΛ = Λ ⊔ Λ ′ be the 2-copy of Λ where the second copy Λ ′ is translated in the positive Reeb direction by N ≫ max c ℓ(c). Here the max is taken over all pure Reeb chords. We perturb Λ ′ by a small Morse function with C 2 -norm bounded by ǫ ≪ min c ℓ(c). Here the min is taken over all Reeb chords. For each pure chord c of Λ there are two mixed chords p + c , p − c such that the projections P ×R → P of c, p + c , p − c are within ǫ, and such that |ℓ(p ± c )−(N ±ℓ(c))| < ǫ. The remaining mixed chords are called the Morse chords, which we denote by x. They correspond to the critical points of the Morse function and satisfy |ℓ(x) − N | < ǫ. (2) Let x be a Morse chord of index k. For every rigid disk with x as the unique positive (resp. unique mixed negative) puncture, the other mixed puncture is a negative puncture at some p − c (resp. positive puncture at some p + c ). Moreover, the existence of such a rigid disk implies that the moduli space M n−k (c) (resp. M k (c)) is non-empty.
(3) Assume that the Morse function has a unique max and min on each connected component. The disks having the max chord (resp. min chord) as a positive (resp. negative puncture) are small triangles that are in a one-to-one correspondence with gradient flow-lines that connect the max (resp. min) and an endpoint of a Reeb chord on Λ. Furthermore, the max is a cycle and the min is a cocycle for the linearized differential, if the same augmentation is used on both components Λ and Λ ′ .
(4) Assume that the Morse function has a unique max and min on each connected component, both of which moreover are located sufficiently close to a given generically chosen point pt i ∈ Λ, i ∈ π 0 (Λ). Any disk with the negative puncture at the max chord (resp. positive puncture at the min chord) corresponds to a moduli space M n (c) for which the boundary of the disc passes through the ray R × pt i .
(2): This is Part (3) of [13,Theorem 3.6], which identifies the disks on the two-copy with appropriate generalized pseudoholomorphic disks on one copy, together with the dimension formula [13, (3.11)] for the generalized pseudoholomorphic disks.
(4): This follows as a special case in the proof of (2). Note that the moduli spaces M n (c) defined for different choices of point constraints pt ′ i all can be canonically identified, if we assume that the moduli space defined with point constraint at pt i ∈ Λ is transversely cut out and that pt ′ i is sufficiently close to pt i .
We consider the linearized Legendrian contact homology complex as defined in Section 2.3, determined by the choice of ordering Λ 0 = Λ and Λ 1 = Λ ′ of the components ofΛ. Definē Λ(t) by fixing Λ and applying the N -vertical-shift of the isotopy to Λ ′ . By using [7, Lemma 2.3] we may replace H t by a contact Hamiltonian that is indefinite, in the sense that it does not assume just positive or negative values for any time t, without changing its oscillatory norm. This changes the isotopy by a translation of the z-coordinate, which is irrelevant for our displaceability considerations. Cutting it off with a bump-function to yield the contact Hamiltonian that generatesΛ(t) we produce a new contact Hamiltonian, still denoted by H t , which has the same oscillatory norm as the original H t .
We apply Proposition 2.8 with its parameters set as follows: Note that since H t has been made to vanish on Λ 0 (t), we have So it suffices to choose any l such that If we are in the case of Remark 1.5, l(1) = 0 and so it suffices that l ≥ H t osc + 2ǫ.
In the following we endow A l (L) with the same augmentation on both components, and suppress it from notation. Proposition 2.10 (1) implies that LCC l * (Λ) N +ǫ N −ǫ is quasi-isomorphic to the Morse homology of Λ. These homology classes need not survive in LCC l * (Λ) when increasing the action range. (For example, if Λ has an augmentation and can be pushed off its Reeb-flow image, then LCC ∞ * (Λ) +∞ −∞ is acyclic.) So if x (a linear combination of Morse chords) represents a degree k homology generator of this Morse subcomplex, its action value is a starting or ending point of a bar in the barcode for LCC l * (Λ(t)) b a where l, a, b are as above.
Suppose x, viewed as Morse chain, lies in the Morse subcomplex of cycles which generates k j=0 H ι j (Λ; k) where ι j is defined right before Theorem 1.1. Assume σ k ≥ H t osc . We claim ℓ(x) cannot be a bar endpoint. This is because the action of any chord p − c which could be the start point of a bar ending at ℓ(x) satisfies Since ǫ is arbitrarily small, ℓ(p − c ) lies outside the action window [a, b). So ℓ(x) must be a starting point. Note Proposition 2.8 prevents ℓ(p − c ) from ever entering the action window for 0 ≤ t ≤ 1. So ℓ(x) is always a starting point, and by Proposition 2.8, remains in the action window.
The endpoint of the bar is, if the cycle x is killed, the action of a chord p + c . Proposition 2.10 (2) implies the bar has (extended) length at least For k = 0, n we do not know if we must consider p − c or p + c . Therefore, we do not know if we need to boundσ k orσ n−k from below by H t osc which is why we assume σ k := min(σ k ,σ n−k ) ≥ H t osc . For k = n (resp. k = 0) Proposition 2.10 (3) implies we only need to bound from belowσ k (resp.σ n−k ). Proposition 2.10 (4) implies the modification specific to the definition of σ n . Corollary 2.9 and the hypothesis σ ι 0 ≥ σ ι 1 ≥ · · · ≥ σ ι k ≥ H t osc from Theorem 1.1 imply that at t = 1 there are number of bars which persist. So there are (at least) the same number of mixed chords corresponding to the starting points of the bars.
2.5. Proof of Lemma 1.8. Consider the contact isotopy Its contact Hamiltonian is equal to H t (x, y, z) = − z 1−t , with oscillation inside the subset {|z| ≤ b} equal to 2b. Using this contact isotopy we can thus rescale the subset {|z| ≤ a/2} to its image of the time-s map for any 0 ≤ s < 1. We compute the total oscillation for the isotopy t ∈ [0, s], 0 ≤ s < 1 restricted to the image of the subset {|z| ≤ a/2} to be equal to In other words, since Λ St (a) ⊂ {|z| ≤ a/2}, taking 1 − s > 0 to be sufficiently small we may move Λ St (a) into an arbitrarily small neighborhood of {x ∈ D 2 , y = 0, z = 0} by a contact isotopy of total oscillation equal to a − δ. Since, for any δ ′ > 0, we can find a neighborhood of {x ∈ D 2 , y = 0, z = 0} which is displaceable by a Hamiltonian of oscillation at most δ ′ (for instance, we can take a lift of a Hamiltonian isotopy of the symplectic manifold ({(x, y)}, ω 0 )) we now deduce that Λ St (a) is displaceable with a total oscillation a + ǫ. Appropriate cut-offs using smooth bump functions can then be used to make the contact Hamiltonians compactly supported.
3. Proof of Theorem 1.9 Recall that two Legendrians are said to be formally isotopic if they are connected by a smooth isotopy which is covered by a Lagrangian bundle monomorphism [22, Definition 1.1].
Lemma 3.1. Assume that two Legendrian spheres Λ, Λ ′ ⊂ R 2n+1 , n ≥ 2, agree in a neighborhood of a point, and that they are formally Legendrian isotopic. Then there exists a formal isotopy which is fixed in a possibly smaller neighborhood of the same point, in the sense that both the underlying smooth isotopy as well as the Lagrangian frames are constant there.
Proof. By [2, Proposition 2.1] the underlying smooth isotopy may be deformed, through a continuous path of smooth isotopies, to one which fixes a neighborhood of the point. Since π 1 (U (n)) = Z is non-trivial the path of Legendrian frames at the point fixed during this new formal isotopy is not automatically contractible. However, applying a suitable S 1 -family of contactomorphisms of R 2n+1 obtained as lifts of symplectomorphisms from U (n) ⊂ Symp(R 2n , ω 0 ) to the initial formal isotopy, the latter frame over the point may be assumed to give rise to a contractible loop in U (n).
Lemma 3.2. Let Λ be an arbitrary Legendrian embedding of dimension n ≥ 2, and let Λ ′ be a Legendrian formally isotopic to Λ St . The cusp connected sum between Λ and Λ ′ is formally Legendrian isotopic to Λ.
Proof. Cusp connected sum is an operation supported in a neighborhood of an isotropic arc γ with endpoints on the two different Legendrians, where the two Legendrians are separated by e.g. the hyperplane {x 1 = 0}; see e.g. [6]. In the dimensions under consideration this operation is well-defined by [6,Proposition 4.9], and does not depend on the choices made.
First note that the cusp connected sum of Λ with the standard Legendrian sphere Λ Std is Legendrian isotopic to Λ. The Legendrian isotopy is easy to construct explicitly if the representative of the standard Legendrian is chosen to be the flying saucer (in the front projection) as shown in Figure 1.
We may assume that Λ ′ coincides with the representative of the standard sphere in a neighborhood of the isotropic arc γ along which the surgery is performed. The formal isotopy from Λ ′ to Λ St may further be assumed to have support that is disjoint from the neighborhood of the union Λ ∪ γ by Lemma 3.1, together with a general position argument (for the interior of the arc).
Lemma 3.3. Any loose Legendrian Λ ⊂ R 2n+1 , n ≥ 2, is Legendrian isotopic to a representative which satisfies the following property for an arbitrary choice of numbers A > δ > 0: • There exists a Legendrian fibre F = {x = x 0 , z = z 0 } for which there are precisely two Reeb chords with one endpoint on Λ and one endpoint on the fibre, both which moreover are transverse; • the two Reeb chords between Λ and F both start on Λ, and their length difference is greater than equal to A > 0; and • the Legendrian Λ can be displaced from the fibre by a contact isotopy of oscillation less than δ > 0.
(See Figure 4 for an example.) Proof. In view of Lemma 3.2 and Murphy's h-principle for loose Legendrians [22] it suffices to construct a Legendrian sphere in the formal Legendrian isotopy class of the standard sphere that satisfies the properties in the statement. Indeed, it is then a simple matter of taking a cusp connected sum with Λ and that sphere.
We begin by constructing the sphere Λ of dimension n = 2 that satisfies the assumptions, and which is formally isotopic to Λ St . In this dimension there is a unique formal isotopy class of Legendrian spheres [22]. Considering loose spheres as depicted in Figure 4, with sufficiently many zig-zags, one can readily produce sought examples.
Increasing the number of zig-zags allows us to increase the z-coordinate while keeping y 1 , y 2 small. By Lemma 1.8 the displacing Hamiltonian can then be made small. Thus, a high number of zig-zags makes more optimal the constants A > δ > 0.
The construction of the spheres in higher dimensions R 2n+1 , n > 2, can be done by induction. Assume that we have produced the sought embedded sphere Λ n−1 in dimension n − 1. Figure 4. Introducing many zig-zags we may assume that the difference of length between the chords a and b from Λ to the fibre F becomes arbitrarily large, while the Lagrangian projection of Λ is still contained inside a fixed subset of R 4 .
Consider the standard Legendrian n-sphere Λ n St , which we assume to have its cusp-edge contained above the unit sphere in R n = {(x 1 , . . . , x n )}. We then perform a stabilization over a closed domain U ⊂ B n ⊂ R n with smooth boundary and Euler characteristic χ(U ) = 0. The resulting Legendrian is loose and formally isotopic to the standard sphere; see [8,Lemma 2.2]. We can find such a domain U which moreover is of the form [−ǫ, ǫ] × U n−1 near the hyperplane {x 1 = 0}, where U n−1 again has vanishing Euler characteristic.
We can assume the stabilization of Λ n St intersects the hypersurface in an (n − 1)-dimensional Legendrian sphere, which again is loose and formally isotopic to the standard sphere; this intersection is itself the stabilization of Λ n−1 St by U n−1 . After a suitable Legendrian isotopy in R 2n−1 lifted to R 2n+1 we have thus managed to construct a loose Legendrian n-sphere in the formal isotopy class of Λ n St which coincides with the cylinder in some neighborhood of {x 1 = 0}. The sought fibre F can be found in the same region.
We now prove Theorem 1.9 when n ≥ 2; note that a stabilized Legendrian is loose in these dimension. Given that the statement is shown in higher dimension, we obtain the statement for knots as follows. The front-spinning construction [11] applied to a stabilized Legendrian knot in R 3 produces a loose Legendrian torus inside R 5 . A potential example of a knot that is squeezed into a neighborhood of a stabilized can thus be spun to an example of an augmentable torus that can be squeezed into a neighborhood of a loose torus. To that end, note that the front spinning preserves Legendrian isotopy classes as well as augmentability [14].
Remark 3.4. Lenny Ng pointed out to us that the dimension 1 case can also be proved with rulings. Consider the local picture of an odd-covering of the stabilization. Any attempt at pairing the sheets, necessary for the (local) construction of the ruling, leaves one copy and its two stabilization cusps unpaired. A 1-dimensional knot with an augmentation must have a ruling [17,25].
In the following we consider the case of a loose Legendrian Λ loose ⊂ R 2n+1 of dimension n ≥ 2. Place the loose Legendrian in the position satisfying the conclusion Lemma 3.3. By assumption, we can isotope Λ into a standard neighborhood of Λ loose contactomorphic to a neighborhood of the zero section inside J 1 Λ loose [18, Theorem 6.2.2].
By a fibre-wise rescaling of Λ inside this jet-bundle, together with a general position argument, we can then assume that all mixed Reeb chords between the fiber F (see Lemma 3.3) and Λ all end on F and start on Λ, and come of two types: an odd number of Reeb chords of action roughly equal to length l > 0 and an odd number of Reeb chords of length roughly equal to A + l, where A ≫ 0 is arbitrarily large. Here we use the assumption that the degree of the bundle projection Λ ⊂ J 1 Λ loose → Λ loose is of odd degree to infer that both clusters of Reeb chords are odd.
We claim that there must be a bar in the complex LCC ∞ * (Λ ∪ F ) +∞ −∞ of length at least equal to A. Indeed, the subcomplex consisting of chords of length strictly less than A is odddimensional and hence not acyclic. (We use that Λ and F have augmentations to be able to set l = ∞.) Since Λ is displaceable from the fibre by a contact isotopy of some fixed small oscillation δ > 0 by Lemma 3.3, we now arrive at a contradiction with Part (2) of the Barcode Proposition 2.8.

Proof of Theorem 1.10
Consider a smooth isotopy Σ t ⊂ (X 4 , ω) of a compact symplectic embedded surface inside a symplectic four-manifold, where ∂Σ t = ∅. Under the additional assumptions that Σ t is fixed near the boundary, it is possible to find a global generating Hamiltonian H t : X → R which moreover vanishes near ∂Σ t ; see [26, Proposition 0.3].
Lemma 4.1. The Hamiltonian H t : X → R for which φ t Ht (Σ 0 ) = Σ t can be taken to vanish along all of Σ t . Hence, after a deformation by a suitable cut-off function, we may assume that the uniform norm of H t is arbitrarily small on all of X.
Proof. The initial Hamiltonian isotopy produced by the aforementioned proposition says nothing about the behavior of the Hamiltonian along Σ t . However, we claim that a suitable Hamiltonian isotopy that fixes Σ t setwise can be used to correct the value of H t | Σt .
Consider a standard neighborhood of Σ 0 which is a trivial symplectic disk-bundle with a product symplectic form. There is a one-to-one correspondence between Hamiltonian isotopies that take fibers to fibers and Hamiltonian isotopies of the base Σ 0 : on the level of Hamiltonians, it is simply given as the precomposition on the Hamiltonian on the base with the bundle projection.
Using the Hamiltonian flow together with the above identification, we get a family of standard neighborhoods of Σ t . In these identifications we then add the precomposition of the bundleprojection and the Hamiltonian −H t | Σt : Σ t → R on the base to the original Hamiltonian H t .
(Here a suitable smoothing must be performed outside of the standard neighborhood.) The newly produced Hamiltonian H t then satisfies φ t In particular, it vanishes along Σ t and still takes Σ 0 to Σ t .
Assume that the cusp-edge of the front projection of Λ St (1) lives above the unit circle in the (x 1 , x 2 )-plane; see Figure 1. Let Π : R 5 → R 4 = {(x i , y i )} denote the canonical projection.
We begin the proof of Theorem 1.10 in Steps 1-3 below by constructing a (non-Legendrian) smooth and arbitrarily C 0 -small push-off of Λ St (1) that can be displaced by a contact Hamiltonian of any fixed small oscillation. (Compare to the result in [21] for non-Lagrangian submanifolds.) In Step 4, we show how with a small oscillation, the stabilized sphere in the Theorem's statement is Legendrian isotopic to a Legendrian C 0 -close to the initial push-off. We then apply the ambient isotopy of the non-Legendrian's displacement from Steps 1-3 to complete the Legendrian isotopy and the proof.
Step 1: Arguing similarly as in the proof of Lemma 1.8 in Section 2.5, but while taking some additional care, one can readily find a Legendrian isotopy Λ t from Λ 0 = Λ to Λ 1 = Λ ′ for which the isotopy Π(Λ t ) has support in the interior of U := {(x 1 , x 2 ) ∈ D 2 1−2ǫ \ D 2 2ǫ } ⊂ R 4 , and such that Λ ′ is displaceable by the lift of a Hamiltonian on R 4 of very small oscillation. This is because Π(Λ ′ ) may be assumed to live in a small neighborhood of the Lagrangian disc {(x 1 , x 2 , y 1 , y 2 ) ∈ D 2 1 × {0}} ⊂ R 4 . To live in such a small neighborhood, the Reeb chord of Λ t must certainly shrink; however, the Lagrangian projection Π(Λ t ) still can be assumed to be fixed near the corresponding double-point. We may assume that all Λ t have a unique Reeb chord and that the projection to the (x 1 , x 2 )-plane is a diffeomorphism inside Π −1 (U ) ⊂ R 5 .
The deformation can be performed by, for example, considering a Lagrangian standard neighborhood of Π(Λ t ) ⊂ R 4 and pushing it off as a section consisting of a suitable family of one-forms whose exterior derivative is a symplectic form on Λ t inside U. Note that this deformation Λ t necessarily must be anti-symplectic somewhere inside V \ U by Stokes' theorem: a closed chain inside R 4 cannot have nonzero symplectic area. However, the family of one-forms can still be taken to be fixed inside V \ U, which ensures the third bullet point above.
Step 3: We can now apply Lemma 4.1 inside U ⊂ R 4 to Π( Λ t ) in order to deduce the existence of a Hamiltonian on R 4 of arbitrarily small oscillation that generates the isotopy Π( Λ t ) ⊂ R 4 . Since Λ 1 is assumed to be arbitrarily C 0 -close to Λ ′ , and Λ ′ is displaceable by the lift of a Hamiltonian on R 4 having small oscillation, this finishes the construction of the push-off with a displacement of small oscillatory norm.
Step 4: Consider a standard contact neighborhood (T * ≤ǫ S 2 × [−ǫ, ǫ], dz − pdq) of Λ St (1) in which Λ St (1) corresponds to the zero section, and which contains the non-Legendrian sphere Λ 0 which can be displaced with small oscillation. Here we may assume that ǫ > 0 is arbitrarily small.
We add a stabilization to Λ St (1) inside the above standard neighborhood to create a loose Legendrian Λ loose in the same formal isotopy class as Λ St (1). By Murphy's h-principle [22] we can find a Legendrian isotopy confined to the above standard neighborhood that takes Λ loose to a Legendrian that approximates Λ 0 arbitrarily well in the C 0 -norm; here we need to use the fact from [27] that Λ 0 admits a C 0 approximation by a Legendrian sphere in the first place. What remains is to argue that the oscillation of this Legendrian isotopy can be assumed to be of order ǫ. This can be achieved by applying a fibre-wise rescaling by a small positive number to the whole isotopy, thereby making it confined to an arbitrarily small neighborhood of the zero-section. We then just need to estimate how much oscillation is needed to do the initial shrinking of Λ loose , together with the expansion back to the approximation of Λ 0 . The crucial estimates of the oscillation of the fibre-wise rescaling, i.e. the contact isotopy (q, p, z) → (q, (1 − t)p, (1 − t)z), t ∈ [0, 1), were considered in the proof of Lemma 1.8 in Section 2.5. Its generating contact Hamiltonian with respect to the standard tautological contact form dz − pdq is given by H t = − z 1−t . The ǫ-neighborhood of the zero-section can thus be shrunk to a λ · ǫ-neighborhood, where 0 < λ < 1, with a contact Hamiltonian of oscillation 2(1 − λ)ǫ.