Geography of bilinearized Legendrian contact homology
GGEOGRAPHY OF BILINEARIZED LEGENDRIANCONTACT HOMOLOGY
FR´ED´ERIC BOURGEOIS AND DAMIEN GALANT
Abstract.
We study the geography of bilinearized Legendrian contacthomology for closed, connected Legendrian submanifolds with vanishingMaslov class in 1-jet spaces. We show that this invariant detects whetherthe two augmentations used to define it are DGA homotopic or not.We describe a collection of graded vector spaces containing all possiblevalues for bilinearized Legendrian contact homology and then show thatall these vector spaces can be realized. Introduction
Let Λ be a closed Legendrian submanifold of the 1-jet space J ( M ) of amanifold M . Given a generic complex structure for the canonical contactstructure on J ( M ), one can associate to Λ its Chekanov-Eliashberg differ-ential graded algebra ( A (Λ) , ∂ ), see [3, 7, 9]. The homology of ( A (Λ) , ∂ ),called Legendrian contact homology, is an invariant of the Legendrian iso-topy class of Λ, but it is often hard to compute. It is therefore useful toconsider augmentations of ( A (Λ) , ∂ ), because such an augmentation ε canbe used to define a linearized complex ( C (Λ) , ∂ ε ). The homology is denotedby LCH ε (Λ) and called linearized Legendrian contact homology of Λ withrespect to ε . The collection of these homologies for all augmentations of( A (Λ) , ∂ ) is also an invariant of the Legendrian isotopy class of Λ. The ge-ography of a similar homological invariant defined using generating familieswas described by the first author with Sabloff and Traynor [2]. Using thework of Dimitroglou Rizell [4] on the effect of embedded surgeries on Legen-drian contact homology, this geography can be shown to hold for linearizedLegendrian contact homology as well. On the other hand, the first authorand Chantraine showed [1] that it is possible to use two augmentations ε , ε of the Chekanov-Eliashberg DGA in order to define a bilinearized dif-ferential ∂ ε ,ε on C (Λ). The corresponding homology is called bilinearizedLegendrian contact homology and is denoted by LCH ε ,ε (Λ). The objectof this article is to describe the geography of bilinearized Legendrian contacthomology.When ε = ε , bilinearized Legendrian contact homology coincides withlinearized Legendrian contact homology. More generally, if the two augmen-tations are DGA homotopic, LCH ε ,ε (Λ) is isomorphic to LCH ε (Λ). Our a r X i v : . [ m a t h . S G ] J u l F. BOURGEOIS AND D. GALANT first result describes a crucial difference in the behavior of bilinearized Leg-endrian contact homology depending whether the two augmentations areDGA homotopic or not.
Theorem 1.1.
Let Λ be a closed, connected Legendrian submanifold of J ( M ) with dim M = n . Let ε , ε be two augmentations of the Chekanov-Eliashberg DGA of Λ with coefficients in Z . Then ε and ε are DGAhomotopic if and only if the map τ n : LCH ε ,ε n (Λ) → H n (Λ) is surjective. In other words, this means that the fundamental class of Λ induces a classin linearized Legendrian contact homology, while the class of the point in Λinduces a class in bilinearized Legendrian contact homology with respect tonon DGA homotopic augmentations.
Corollary 1.2.
Bilinearized Legendrian contact homology is a completeinvariant for DGA homotopy classes of augmentations of the Chekanov-Eliashberg DGA.
Our second result describes the geography of the Laurent polynomialsthat can be obtained as the Poincar´e polynomial for bilinearized Legendriancontact homology. We say that a Laurent polynomial with integral coeffi-cients is bLCH-admissible if it is the sum of a polynomial of degree at most n − − n is odd and vanishes when n is even. Theorem 1.3.
For any bLCH-admissible Laurent polynomial P , there existsa closed, connected Legendrian submanifold Λ of J ( M ) with dim M = n andthere exist two non DGA homotopic augmentations ε , ε of the Chekanov-Eliashberg DGA of Λ , with the property that the Poincar´e polynomial of LCH ε ,ε (Λ) with coefficients in Z is equal to P . This paper is organized as follows. In Section 2 we review the definition ofbilinearized Legendrian contact homology and state its main properties. InSection 3 we study fundamental classes in bilinearized Legendrian contacthomology, prove Theorem 1.1 and Corollary 1.2 and study the effect ofconnected sums on bilinearized Legendrian contact homology. In Section 4,we study the geography of bilinearized Legendrian contact homology andprove Theorem 1.3.
Acknowledgements.
We are indebted to Josh Sabloff for providing usa computer code that computes linearized Legendrian contact homology ofLegendrian knots in R . We thank Georgios Dimitroglou Rizell for a produc-tive discussion of DGA homotopies of augmentations. We also thank CyrilFalcon for his remarks on the original manuscript. An important refinementin the constructions from Section 4 emerged after an interesting conversationwith Sylvain Courte. FB was partially supported by the Institut Universi-taire de France and by the ANR projects Quantact (16-CE40-0017) andMicrolocal (15-CE40-0007). EOGRAPHY OF BILINEARIZED LCH 3 Bilinearized Legendrian contact homology
The 1-jet space J ( M ) = T ∗ M × R of a smooth, n -dimensional manifold M is equipped with a canonical contact structure ξ = ker( dz − λ ), where λ is the Liouville 1-form on T ∗ M and z is the coordinate along R . Let Λbe a closed Legendrian submanifold of this contact manifold, i.e. a closed,embedded submanifold of dimension n such that T p Λ ⊂ ξ p for any p ∈ Λ.The Reeb vector field associated to the contact form α = dz − λ for ξ issimply R α = ∂∂z . A Reeb chord of Λ is a finite, nontrivial piece of integralcurve for R α with endpoints on Λ. After performing a Legendrian isotopy, wecan assume that all Reeb chords of Λ are nondegenerate, i.e. the canonicalprojections to the tangent space of T ∗ M of the tangent spaces to Λ at theendpoints of each chord are intersecting transversally. Let us assume thatthe Maslov class µ (Λ) of Λ vanishes, see [7, section 2.2].We denote by A (Λ) the unital, noncommutative algebra freely generatedover Z by the Reeb chords of Λ. Each Reeb chord c is graded by itsConley-Zehnder ν ( c ) ∈ Z ; when Λ is connected, this does not depend on anyadditional choice since µ (Λ) = 0. The grading of c is defined as | c | = ν ( c ) − A (Λ) is naturally graded.Let J be a complex structure on ξ , which is compatible with its confor-mal symplectic structure. This complex structure naturally extends to analmost complex structure, still denoted by J , on the symplectization ( R × J ( M ) , d ( e t α )) by J ∂∂t = R α . We consider the moduli space (cid:102) M ( a ; b , . . . , b k )of J -holomorphic disks in R × J ( M ) with boundary on R × Λ and with k + 1punctures on the boundary that are asymptotic at the first puncture to theReeb chord a at t = + ∞ and at the other punctures to the Reeb chords b , . . . , b k at t = −∞ . For a generic choice of J , this moduli space is asmooth manifold of dimension | a | − (cid:80) ki =1 | b i | , see [7, Proposition 2.2]. Thismoduli space carries a natural R -action corresponding to the translation of J -holomorphic disks along the t coordinate. When { b , . . . , b k } (cid:54) = { a } , letus denote by M ( a ; b , . . . , b k ) the quotient of this moduli space by this freeaction.We define a differential ∂ on A (Λ) by ∂a = (cid:88) b ,...,bk dim M ( a ; b ,...,b k )=0 M ( a ; b , . . . , b k ) b . . . b k where M ( a ; b , . . . , b k ) is the number of elements in the correspondingmoduli space, modulo 2. This differential has degree − ∂ ◦ ∂ =0. The resulting differential graded algebra ( A (Λ) , ∂ ) is called the Chekanov-Eliashberg DGA and its homology is called Legendrian contact homologyand denoted by LCH (Λ). This graded algebra over Z depends only on theLegendrian isotopy class of Λ. F. BOURGEOIS AND D. GALANT
An augmentation of ( A (Λ) , ∂ ) is a unital DGA map ε : ( A (Λ) , ∂ ) → ( Z , ε ( c ) ∈ Z for all Reeb chords c of Λ,it satisfies ε (1) = 1, it extends to A (Λ) multiplicatively and additively, andit satisfies ε ◦ ∂ = 0.Such an augmentation can be used to define a linearization of ( A (Λ) , ∂ ).Let C (Λ) be the vector space over Z freely generated by all Reeb chords ofΛ. We also define the linearized differential ∂ ε on C (Λ) by ∂ ε a = (cid:88) b ,...,bk dim M ( a ; b ,...,b k )=0 M ( a ; b , . . . , b k ) k (cid:88) i =1 ε ( b ) . . . ε ( b i − ) b i ε ( b i +1 ) . . . ε ( b k ) . This differential has degree − ∂ ε ◦ ∂ ε = 0. The homology ofthe resulting linearized complex ( C (Λ) , ∂ ε ) is called linearized Legendriancontact homology (with respect to ε ) and denoted by LCH ε (Λ). The col-lection of these graded modules over Z for all augmentations of Λ dependsonly on the Legendrian isotopy class of Λ.Linearized Legendrian contact homology fits into a duality long exactsequence [6] together with its cohomological version LCH ε (Λ) and with thesingular homology H (Λ) of the underlying n -dimensional manifold Λ: . . . → LCH n − k − ε (Λ) → LCH εk (Λ) τ k −→ H k (Λ) → LCH n − kε (Λ) → . . . Moreover, the map τ n in the above exact sequence does not vanish. Theseproperties induce constraints on the graded modules over Z that can berealized as the linearized Legendrian contact homology of some Legendriansubmanifold, with respect to some augmentation. To be more precise, thePoincar´e polynomial of LCH ε (Λ) is the Laurent polynomial defined by P Λ ,ε ( t ) = (cid:88) k ∈ Z dim Z LCH εk (Λ) t k . When Λ is connected, the duality exact sequence and the non-vanishing of τ n imply that the above Poincar´e polynomial has the form(2.1) P Λ ,ε ( t ) = q ( t ) + p ( t ) + t n − p ( t − ) , where q is a monic polynomial of degree n with integral coefficients (corre-sponding to the image of the maps τ k ) and p is a Laurent polynomial withintegral coefficients (corresponding to the kernel of the maps τ k ). We shallsay that a Laurent polynomial of this form is LCH-admissible.The first author together with Sabloff and Traynor [2] studied generat-ing family homology GH ( f ), an invariant for isotopy classes of Legendriansubmanifolds Λ ⊂ ( J ( M ) , ξ ) admitting a generating family f . This invari-ant is also a graded module over Z and satisfies the same duality exactsequence as above. In this study, the effect of Legendrian ambient surgerieson this invariant was determined and these operations were used to producemany interesting examples of Legendrian submanifolds admitting generat-ing families. More precisely, for any LCH-admissible Laurent polynomial EOGRAPHY OF BILINEARIZED LCH 5 P , a connected Legendrian submanifold Λ P admitting a generating family f P realizing P as the Poincar´e polynomial of GH ( f P ) was constructed us-ing these operations. On the other hand, Dimitroglou Rizell [4] showed inparticular that Legendrian ambient surgeries have the same effect as aboveon linearized Legendrian contact homology (for more details in the case ofthe connected sum, see the proof of Proposition 3.5). This result can beused step by step in the constructions of [2] to show that, for any LCH-admissible Laurent polynomial P , there exists an augmentation ε P for Λ P such that LCH ε P (Λ P ) ∼ = GH ( f P ). Therefore, the geography question forlinearized Legendrian contact homology is completely determined by theabove LCH-admissible Laurent polynomials.Let us now turn to a generalization of linearized LCH introduced by thefirst author together with Chantraine [1]. Using two augmentations ε and ε of ( A (Λ) , ∂ ), we can define another differential ∂ ε ,ε on C (Λ), calledbilinearized differential: ∂ ε ,ε a = (cid:88) b ,...,bk dim M ( a ; b ,...,b k )=0 M ( a ; b , . . . , b k ) k (cid:88) i =1 ε ( b ) . . . ε ( b i − ) b i ε ( b i +1 ) . . . ε ( b k ) . As above, this differential has degree − ∂ ε ,ε ◦ ∂ ε ,ε = 0.The homology of the resulting bilinearized complex ( C (Λ) , ∂ ε ,ε ) is calledbilinearized Legendrian contact homology (with respect to ε and ε ) anddenoted by LCH ε ,ε (Λ). The collection of these graded modules over Z for all pairs of augmentations of Λ depends only on the Legendrian isotopyclass of Λ.Bilinearized Legendrian contact homology also satisfies a duality exactsequence [1], but one has to take care of the ordering of the augmentations:(2.2) . . . → LCH n − k − ε ,ε (Λ) → LCH ε ,ε k (Λ) τ k −→ H k (Λ) σ n − k −→ LCH n − kε ,ε (Λ) → . . . Moreover, unlike in the linearized case, there exist [1, section 5] connectedLegendrian submanifolds Λ with augmentations ε and ε such that themap τ n vanishes. Our goal in this article is to understand when the map τ n vanishes or not, and to study the geography of the Poincar´e polynomials P Λ ,ε ,ε ( t ) = (cid:88) k ∈ Z dim Z LCH ε ,ε k (Λ) t k . for bilinearized Legendrian contact homology.3. Fundamental classes in bilinearized Legendrian contacthomology
There are several notions of equivalence for augmentations of DGAs thatwere introduced in the literature and used in the context of the Chekanov-Eliashberg DGA. As the results of this section will show, it turns out that theequivalence relation among augmentations that controls best the behavior
F. BOURGEOIS AND D. GALANT of bilinearized LCH is the notion of DGA homotopic augmentations [13,Definition 5.13]. Let ε , ε be two augmentations of the DGA ( A , ∂ ) over Z . Recall that a linear map K : A → Z is said to be an ( ε , ε )-derivationif K ( ab ) = ε ( a ) K ( b ) + K ( a ) ε ( b ) for any a, b ∈ A . We say that ε is DGAhomotopic to ε , and we write ε ∼ ε , if there exists an ( ε , ε )-derivation K : A → Z of degree +1 such that ε − ε = K ◦ ∂ . It is a standard factthat DGA homotopy is an equivalence relation [10, Lemma 26.3].Note that the defining condition for a DGA homotopy admits a beautifuland convenient reformulation in terms of the bilinearized complex. Lemma 3.1.
Two augmentations ε , ε are DGA homotopic if and only ifthere exists a linear map K : C (Λ) → Z of degree +1 such that ε − ε = K ◦ ∂ ε ,ε on C (Λ) .Proof. Suppose first that ε is DGA homotopic to ε . This implies in par-ticular that ε ( c ) − ε ( c ) = K ◦ ∂c for any c ∈ C (Λ). Since K is an ( ε , ε )-derivation, it directly follows from the definition of the bilinearized differen-tial that K ◦ ∂c = K ◦ ∂ ε ,ε c . It then suffices to take K to be the restrictionof K to C (Λ).Suppose now that there exists a linear map K : C (Λ) → Z of degree+1 such that ε − ε = K ◦ ∂ ε ,ε on C (Λ). The map K determines aunique ( ε , ε )-derivation K : A → Z via the relation K ( a . . . a n ) = (cid:80) ki =1 ε ( a . . . a i − ) K ( a i ) ε ( a i +1 . . . a n ) for all a , . . . , a n ∈ A . As above,these maps satisfy K ◦ ∂c = K ◦ ∂ ε ,ε c , so that ε − ε = K ◦ ∂ on C (Λ). Nowobserve that ε ( ab ) − ε ( ab ) = ε ( a ) ( ε ( b ) − ε ( b ))+( ε ( a ) − ε ( a )) ε ( b ) andon the other hand K ◦ ∂ ( ab ) = ε ( ∂a ) K ( b ) + ε ( a ) K ( ∂b ) + K ( ∂a ) ε ( b ) + K ( a ) ε ( ∂b ) = ε ( a ) K ( ∂b ) + K ( ∂a ) ε ( b ). Hence if a, b satisfy the DGA ho-motopy relation, then ab satisfies it as well. Since this relation holds on C (Λ), it follows that it is also satisfied on A . (cid:3) With this suitable notion of equivalence for augmentations, we can nowturn to the study of the fundamental class in bilinearized LCH, via the maps τ and τ n from the duality long exact sequence. The following propositiongeneralizes Theorem 5.5 from [6]. Proposition 3.2.
Let ε , ε be augmentations of the Chekanov-EliashbergDGA ( A , ∂ ) of a closed, connected n -dimensional Legendrian submanifold Λ in ( J ( M ) , ξ ) . The map τ : LCH ε ,ε (Λ) → H (Λ) from the duality longexact sequence vanishes if and only if ε and ε are DGA homotopic.Proof. Let f be a Morse function on Λ with a unique minimum at point m and g be a Riemannian metric on Λ. Since the stable manifold of m isopen and dense in Λ, for a generic choice of the Morse-Smale pair ( f, g ), theendpoints of all Reeb chords of Λ are in this stable manifold. The vectorspace H (Λ) is generated by m and we identify it with Z . By the resultsof [6], the map τ counts rigid J -holomorphic disks with boundary on Λ,with a positive puncture on the boundary and with a marked point on the EOGRAPHY OF BILINEARIZED LCH 7 boundary mapping to the stable manifold of m . This disk can have extranegative punctures on the boundary; these are augmented by ε if theysit between the positive puncture and the marked point, and by ε if theysit between the marked point and the positive puncture. Since mappingto m is an open condition on Λ, such rigid configurations can only occurwhen the image of the disk boundary is discrete in Λ. In other words, theholomorphic disk maps to the symplectization of a Reeb chord c of Λ. Sincethere is a unique positive puncture, this map is not a covering, and there isa unique negative puncture at c . There is a unique such J -holomorphic diskfor any chord c of Λ. The marked point maps to the starting point or to theending point of the chord c in Λ. If the marked point maps to the startingpoint of c , the negative puncture sits between the positive puncture andthe marked point on the boundary of the disk, which therefore contributes ε ( c ) to τ ( c ) at chain level. If the marked point maps to the ending pointof c , the negative puncture sits between the marked point and the positivepuncture on the boundary of the disk, which therefore contributes ε ( c ) to τ ( c ). We conclude that the map τ is given at chain level by ε − ε .If ε and ε are DGA homotopic, then by Lemma 3.1 the map τ isnull homotopic and therefore vanishes in homology. On the other hand, if ε and ε are not DGA homotopic, the Lemma 3.1 implies that the map ε − ε : C (Λ) → Z does not factor through the bilinearized differential ∂ ε ,ε . In other words, there exists a ∈ C (Λ) such that ∂ ε ,ε a = 0 but ε ( a ) − ε ( a ) (cid:54) = 0. But then the homology class [ a ] ∈ LCH ε ,ε (Λ) satisfies τ ([ a ]) (cid:54) = 0, so that τ does not vanish in homology. (cid:3) We are now in position to prove the first main result of this paper.
Proof of Theorem 1.1.
In the duality long exact sequence (2.2) for bilin-earized LCH, the maps τ k and σ k are adjoint in the sense of [6, Proposition3.9] as in the linearized case. The proof of this fact is essentially identicalin the bilinearized case: the holomorphic disks counted by τ k are still inbijective correspondence with those counted by σ k . In the bilinearized case,it is also necessary to use the fact that the extra negative punctures on cor-responding disks are augmented with the same augmentations, in order toreach the conclusion.In particular, τ n vanishes if and only if σ n vanishes. Since H (Λ) ∼ = Z ,the exactness of the duality sequence (2.2) implies that σ n vanishes if andonly if τ does not vanish. By Proposition 3.2, this means that τ n vanishesif and only if the augmentations ε and ε are not DGA homotopic. (cid:3) This difference in the behavior of bilinearized LCH can be used to de-termine DGA homotopy classes of augmentations. More precisely, the nextproposition shows that bilinearized LCH provides an explicit criterion todecide whether two augmentations are DGA homotopic or not.
Proposition 3.3.
Let ε , ε be augmentations of the Chekanov-EliashbergDGA ( A , ∂ ) of a closed, connected n -dimensional Legendrian submanifold Λ F. BOURGEOIS AND D. GALANT
Figure 1.
Front projection of the Legendrian knot K . in ( J ( M ) , ξ ) . Then dim Z LCH ε ,ε n (Λ) − dim Z LCH ε ,ε − (Λ) = (cid:26) if ε (cid:54)∼ ε , if ε ∼ ε . Proof.
By the duality exact sequence (2.2), we have H (Λ) ∼ = Z σ n −→ LCH nε ,ε (Λ) → LCH ε ,ε − (Λ) → H − (Λ) = 0 . In other words,
LCH nε ,ε (Λ) / Im σ n ∼ = LCH ε ,ε − (Λ). Taking into accountthat dim Z LCH nε ,ε (Λ) = dim Z LCH ε ,ε n (Λ), we obtain the desired resultsince, as in the proof of Theorem 1.1, the rank of σ n is 1 when ε ∼ ε andvanishes when ε (cid:54)∼ ε . (cid:3) Corollary 1.2 follows immediately from the above proposition.
Example . Let us consider the Legendrian knot K studied by Melvinand Shrestha in [11, Section 3], which is topologically the mirror image ofthe knot 8 , and illustrated in Figure 1.It is shown in [11, Section 3] that the Chekanov-Eliashberg DGA ofthis Legendrian knot K has exactly 16 augmentations, which split intoa set A of 4 augmentations and a set B of 12 augmentations such that P K ,ε ( t ) = 2 t + 4 + t − if ε ∈ A and P K ,ε ( t ) = t + 2 if ε ∈ B . This impliesthat augmentations in A are not DGA homotopic to augmentations in B .However, the number of DGA homotopy classes of augmentations for K was not determined in [11], as linearized LCH does not suffice to obtain thisinformation.Using Proposition 3.3, these DGA homotopy classes can be determinedsystematically. It turns out that the augmentations in A are pairwise notDGA homotopic, because the Poincar´e polynomials of any such pair of aug-mentations are t + 2 and 2 + t − . On the other hand, the set B splits into6 DGA homotopy classes C , . . . , C of augmentations. The bLCH Poincar´epolynomials are given by t + 2 for two DGA homotopic augmentations in B ,by 1 for two non DGA homotopic augmentations both in C ∪ C ∪ C or in C ∪ C ∪ C , and by t + 2 and 2 + t − otherwise.These calculations are straightforward but tedious. A suitable Pythoncode run by a computer gives the above answer instantly. EOGRAPHY OF BILINEARIZED LCH 9
We conclude our study of the fundamental classes in bilinearized LCHwith a useful description of their behavior when performing a connected sum.To this end, it is convenient to introduce some additional notation aboutthe map τ n in the duality exact sequence (2.2). Its target space H n (Λ)is spanned by the fundamental classes [Λ i ] of the connected componentsΛ i of the Legendrian submanifold Λ. We can therefore decompose τ n as (cid:80) i τ n,i [Λ i ], where the maps τ n,i take their values in Z . Proposition 3.5.
Let Λ be a Legendrian link in J ( M ) equipped with twoaugmentations ε and ε . Let Λ be the Legendrian submanifold obtained byperforming a connected sum between two connected components Λ and Λ of Λ , and let ε and ε be the augmentations induced by ε and ε .If the map τ n, − τ n, constructed from the map τ n in the duality exactsequence (2.2) vanishes, then P Λ ,ε ,ε ( t ) = P Λ ,ε ,ε ( t ) + t n − . Otherwise, P Λ ,ε ,ε ( t ) = P Λ ,ε ,ε ( t ) − t n .Proof. As explained in [1, Section 3.2.5], the map τ n in the duality exact se-quence (2.2) for Λ counts holomorphic disks in the symplectization of J ( M )with boundary on the symplectization of Λ, having a positive puncture as-ymptotic to a chord c of Λ and a marked point on the boundary mappedto a fixed generic point p j of a connected component Λ j of Λ. This diskcan also carry negative punctures on the boundary; let us say that thoselocated between the positive puncture and the chord (with respect to thenatural orientation of the boundary) are asymptotic to chords c − , . . . , c − r ,while those between the marked point and the positive puncture are asymp-totic to c − r +1 , . . . , c − r + s . Let us denote by M ( c ; c − , . . . , c − r , p j , c − r +1 , . . . , c − r + s )the moduli space of such holomorphic disks, modulo translation in the R direction of the symplectization. The map τ n is then given by τ n ( c ) = (cid:88) j M ( c ; c − , . . . , c − r , p j , c − r +1 , . . . , c − r + s ) ε ( c − ) . . . ε ( c − r ) ε ( c − r +1 ) . . . ε ( c − r + s )[Λ j ] . On the other hand, the effect of a connected sum on bilinearized LCHcan be deduced from the results of Dimitroglou Rizell on the full Chekanov-Eliashberg DGA [4, Theorem 1.6]. There is an isomorphism of DGAsΨ : ( A (Λ) , ∂ Λ ) → ( A (Λ; S ) , ∂ S ) between the Chekanov-Eliashberg DGAof Λ and the DGA ( A (Λ; S ) , ∂ S ) generated by the Reeb chords of Λ as wellas a formal generator s of degree n −
1, equipped with a differential ∂ S satisfying in particular ∂ S s = 0. In this notation, S stands for the pair ofpoints { p ∈ Λ , p ∈ Λ } in a neighborhood of which the connected sumis performed. Any augmentation ε of the Chekanov-Eliashberg DGA of Λcan be extended to an augmentation of ( A (Λ; S ) , ∂ S ) by setting ε ( s ) = 0.Moreover, the pullback Ψ ∗ ε of this extension to the Chekanov-EliashbergDGA of Λ coincides with the augmentation induced on Λ from the origi-nal augmentation ε for Λ via the surgery Lagrangian cobordism between Λ and Λ. In particular, we have ε = Ψ ∗ ε and ε = Ψ ∗ ε . Applying thebilinearization procedure to the map Ψ, we obtain a chain complex isomor-phism Ψ ε ,ε between the bilinearized chain complex for Λ and the chaincomplex ( C (Λ , S ) , ∂ ε ,ε S ) generated by Reeb chords of Λ and the formalgenerator s . Since ∂ ε ,ε S s = 0, the line spanned by s forms a subcomplex of( C (Λ , S ) , ∂ ε ,ε S ). Moreover, the quotient complex is exactly the bilinearizedchain complex for Λ. We therefore obtain a long exact sequence in homology . . . → LCH ε ,ε k (Λ) → LCH ε ,ε k (Λ) ρ k → Z [ s ] k − → LCH ε ,ε k − (Λ) → . . . that corresponds to the long exact sequence obtained in [2, Theorem 2.1] forgenerating family homology. This exact sequence implies that bilinearizedLCH remains unchanged by a connected sum, except possibly in degrees n − n . The map ρ n is the part of the bilinearized differential ∂ ε ,ε S from the bilinearized complex for Λ to the line spanned by s . According tothe definition [4, Section 1.1.3] of ∂ S and the above description of τ n , thismap is given by ρ n = ( τ n, − τ n, ) s .If ρ n = 0, the generator s injects into LCH ε ,ε n − (Λ), resulting in an exactterm t n − in the Poincar´e polynomial. If ρ n (cid:54) = 0, the map LCH ε ,ε n (Λ) → LCH ε ,ε n (Λ) has a 1-dimensional cokernel, resulting in the loss of a term t n in the Poincar´e polynomial. (cid:3) Geography of bilinearized Legendrian contact homology
In this section, we study the possible values for the Poincar´e polynomial P Λ ,ε ,ε of the bilinearized LCH for a closed, connected Legendrian subman-ifold Λ in J ( M ) with dim M = n , equipped with two augmentations ε and ε of its Chekanov-Eliashberg DGA.When ε = ε , this geography question was completely answered in [2] forgenerating family homology. As explained in Section 2, this result extendsto linearized LCH via the work of Dimitroglou Rizell [4]. Moreover, bilin-earized LCH is invariant under changes of augmentations within their DGAhomotopy classes [13, Section 5.3]. Therefore, the geography of bilinearizedLCH is already known when ε ∼ ε .We now turn to the case ε (cid:54)∼ ε , and describe the possible Poincar´epolynomials for bilinearized LCH. Definition 4.1.
A Laurent polynomial with nonnegative integral coefficients P is said to be bLCH-admissible if it is the sum P = q + p of two Laurentpolynomials with nonnegative integral coefficients p and q such that (i) q is a polynomial of degree at most n − with q (0) = 1 , (ii) p ( − is even if n is odd and p ( −
1) = 0 if n is even. We first show that the Poincar´e polynomial of bilinearized LCH alwayshas this form.
Proposition 4.2.
Let ε , ε be augmentations of the Chekanov-EliashbergDGA ( A , ∂ ) of a closed, connected n -dimensional Legendrian submanifold Λ EOGRAPHY OF BILINEARIZED LCH 11 with vanishing Maslov class in ( J ( M ) , ξ ) . If ε and ε are not DGA homo-topic, then the Poincar´e polynomial P Λ ,ε ,ε corresponding to LCH ε ,ε (Λ) is bLCH-admissible.Proof. Considering the map τ k from the duality exact sequence (2.2), wehave the relation dim Z LCH ε ,ε k (Λ) = dim Z ker τ k + dim Z im τ k . Let p and q be the Poincar´e polynomials constructed using the terms in theright hand side of this relation: p ( t ) = (cid:80) k ∈ Z dim Z ker τ k t k and q ( t ) = (cid:80) k ∈ Z dim Z im τ k t k .Since im τ k ⊂ H k (Λ), q is a polynomial of degree at most n . By Propo-sition 3.2, since ε (cid:54)∼ ε , im τ (cid:54) = 0. But H (Λ) = Z as Λ is connected, sothat q (0) = 1. On the other hand, by Theorem 1.1, since ε (cid:54)∼ ε , τ n = 0 sothat the term of degree n in q vanishes and q is a polynomial of degree atmost n − H k (Λ) in theduality long exact sequence (2.2): we have dim Z H k (Λ) = dim Z im τ k +dim Z im σ n − k . Since the maps τ n − k and σ n − k are adjoint in the sense of [6,Proposition 3.9], we have dim Z im σ n − k = dim Z im τ n − k so that(4.1) dim Z H k (Λ) = dim Z im τ k + dim Z im τ n − k . Assume first that n is odd. By (4.1), q (1) = (cid:80) k ∈ Z dim Z H k (Λ), so that q ( −
1) has the same parity as (cid:80) k ∈ Z dim Z H k (Λ). Note that this holds forDGA homotopic augmentations as well, and that the Euler characteristic P Λ ,ε ,ε ( −
1) of the (bi)linearized complex does not depend on the augmen-tations. Equation (2.1) then implies that P Λ ,ε ,ε ( −
1) has the same parityas (cid:80) k ∈ Z dim Z H k (Λ), since ( − n − = 1 when n is odd. Substracting q ( −
1) from this, we deduce that p ( −
1) must be even.Assume now that n is even. By [8, Proposition 3.3], the Thurston-Bennequin invariant of Λ is given by tb (Λ) = ( − ( n − n − P Λ ,ε ,ε ( − tb (Λ) = ( − n +1 12 X (Λ) when n is even. Hence P Λ ,ε ,ε ( −
1) = X (Λ). Since n is even, the two termsin the right hand side of (4.1) contribute equally to X (Λ). Hence q ( − k ∈ Z , isequal to X (Λ). Substracting this from P Λ ,ε ,ε ( − p ( −
1) = 0as announced. (cid:3)
The duality exact sequence imposes less restrictions on
LCH ε ,ε (Λ) thanin the case of linearized LCH because it mainly relates this invariant to LCH ε ,ε (Λ) with exchanged augmentations. This fact, however, meansthat one of these invariants determines the other one. Proposition 4.3.
Let ε , ε be non DGA homotopic augmentations of theChekanov-Eliashberg DGA ( A , ∂ ) of a closed, connected n -dimensional Leg-endrian submanifold Λ with vanishing Maslov class in ( J ( M ) , ξ ) . If P Λ ,ε ,ε decomposes as q + p as in Definition 4.1, then P Λ ,ε ,ε ( t ) = q ( t ) + t n − p ( t − ) . b b b a a Figure 2.
Front projection of the maximal tb right handed trefoil. Proof.
Let us decompose P Λ ,ε ,ε ( t ) = ˜ q ( t ) + ˜ p ( t ) as in Definition 4.1. Thepolynomial p was defined as p ( t ) = (cid:80) k ∈ Z dim Z ker τ k t k in the proof ofProposition 4.2. But ker τ k is the image of the map LCH n − k − ε ,ε (Λ) → LCH ε ,ε k (Λ), which is accounted for by ˜ p ( t ). We therefore obtain ˜ p ( t ) = (cid:80) k ∈ Z dim Z ker τ k t n − k − = t n − p ( t − ).On the other hand, dualizing the exact sequence (2.2) and in analogywith the definition of q in the proof of Proposition 4.2, we have ˜ q ( t ) = (cid:80) k ∈ Z dim Z H k (Λ) / im τ k t n − k . By (4.1), this means that˜ q ( t ) = (cid:88) k ∈ Z dim Z im τ n − k t n − k = q ( t )as announced. (cid:3) We now describe a fundamental example in view of the constructionof Legendrian submanifolds and augmentations realizing bLCH-admissiblepolynomials.
Example . With n = 1, consider the right handed trefoil knot Λ withmaximal Thurston-Bennequin invariant, depicted in its front projection inFigure 2. The same Legendrian knot was already studied in Section 5.1of [1]. We consider it this time in the front projection, after applying Ng’sresolution procedure [12].The Chekanov-Eliashberg DGA has 5 generators: a and a correspondto right cusps and have grading 1, while b , b and b correspond to crossingsand have grading 0. The differential is given by ∂a = 1 + b + b + b b b ,∂a = 1 + b + b + b b b . This DGA admits 5 augmentations ε , . . . , ε given by b b b ε ε ε ε ε EOGRAPHY OF BILINEARIZED LCH 13 c c d d a Figure 3.
Replacement for the dotted rectangle in Figure 2.A straightforward calculation shows that P Λ ,ε i ,ε j ( t ) = 1 for all i (cid:54) = j .In view of Definition 4.1 and Proposition 4.2, this is the simplest Poincar´epolynomial that can be obtained using bilinearized LCH.In order to produce other terms in this Poincar´e polynomial, let us re-place the portion of Λ contained in the dotted rectangle in Figure 2 by thefragment represented in Figure 3. This produces a Legendrian link Λ (cid:48) .The additional generator a corresponds to a right cusp and has grading1. The 4 mixed chords between the unknot and the trefoil have a gradingthat depends on a shift k ∈ Z between the Maslov potentials of the trefoiland of the unknot. These gradings are given by | c | = k − , | c | = k, | d | = 1 − k, | d | = − k. The augmentations ε , . . . , ε can be extended to this enlarged DGA bysending all new generators to 0. The bilinearized differential of the originalgenerators is therefore unchanged. The differential of the new generators is,on the other hand, given by ∂c = 0 , ∂c = (1 + b b ) c , ∂d = d (1 + b b ) , ∂d = 0 , ∂a = 0 . If we choose ε L = ε or ε and ε R = ε , ε or ε , then the bilinearizeddifferential is ∂ ε L ,ε R c = 0 , ∂ ε L ,ε R c = 0 , ∂ ε L ,ε R d = d , ∂ ε L ,ε R d = 0 , ∂ ε L ,ε R a = 0 . The Poincar´e polynomial of the resulting homology is therefore P Λ (cid:48) ,ε L ,ε R ( t ) = t k + t k − + t + 1. We now perform a connected sum between the right cuspscorresponding to a and a in order to obtain the connected Legendriansubmanifold Λ (cid:48)(cid:48) represented by Figure 4. A Legendrian isotopy involving anumber of first Reidemeister moves is performed before the connected sum inorder to ensure that the Maslov potentials agree on the cusps to be merged.This connected sum induces a Lagrangian cobordism L from Λ (cid:48)(cid:48) to Λ (cid:48) , andwe can use this cobordism to pullback the augmentations ε L and ε R to theChekanov-Eliashberg DGA of Λ (cid:48)(cid:48) .By Proposition 3.5, since [ a ] ∈ LCH ε L ,ε R (Λ (cid:48) ) corresponds to the fun-damental class of the Legendrian unknot depicted in Figure 4, we obtainthe Poincar´e polynomial P Λ (cid:48)(cid:48) ,ε L ,ε R ( t ) = t k + t k − + 1. This corresponds to q ( t ) = 1 and p ( t ) = t k + t k − in Definition 4.1. − k Figure 4.
Front projection of the Legendrian knot Λ (cid:48)(cid:48) .In order to generalize Example 4.4 to higher dimensions, let us considerthe standard Legendrian Hopf link, or in other words the 2-copy of the stan-dard Legendrian unknot Λ (2) ⊂ J ( R n ). This will lead to a generalization ofthe trefoil knot from Figure 2, since the latter can be obtained from the stan-dard Legendrian Hopf link in R via a connected sum. Let us denote by (cid:96) the length of the unique Reeb chord of the standard Legendrian unknot andby ε the positive shift (much smaller than (cid:96) ) in the Reeb direction betweenthe two components Λ and Λ of Λ (2) . We assume that the top componentis perturbed by a Morse function of amplitude δ much smaller than ε withexactly one maximum M and one minimum m . In particular, among thecontinuum of Reeb chords of length ε between the two components, only twochords corresponding to these extrema subsist after perturbation. We alsoassume that thanks to this perturbation, all Reeb chords of Λ (2) lie abovedistinct points of R n . In order to define the grading of mixed Reeb chordsin this link, we choose the Maslov potential of the upper component Λ tobe given by the Maslov potential of the lower component Λ plus k . Proposition 4.5.
The Chekanov-Eliashberg DGA of Λ (2) ⊂ J ( R n ) has thefollowing 6 generators grading length c n (cid:96)c n (cid:96)c n + k (cid:96) + εc n − k (cid:96) − εm k − ε − δM n + k − ε + δ and its differential is given by ∂c = M + m c + c m ,∂c = c m ,∂c = m c , and ∂M = ∂m = ∂c = 0 .Proof. The front projection of each component in Λ (2) consists of two sheets,having parallel tangent hyperplanes above a single point of R n before the EOGRAPHY OF BILINEARIZED LCH 15 c c c M m c Λ Λ Figure 5.
Quiver corresponding to the standard Hopf link.perturbation by the Morse function. The number of Reeb chords above thatpoint is the number of pairs of sheets, which is − = 6. The chordsbetween the two highest or the two lowest sheets belong to a continuumof chords of length ε between the two components, which is replaced bytwo chords M for the maximum M and m for the minimum m afterthe perturbation by the Morse function. Their lengths are therefore ε ± δ .Their gradings are given by the Morse index of the corresponding criticalpoint plus the difference of Maslov potentials minus one, so that we obtain n + k − k − c ij , where i numbers the com-ponent of origin for the chord and j numbers the component of the endpointof the chord. Each of these chords corresponds to a maximum of the localdifference function between the heights of the sheets it joins. We thereforeobtain the announced gradings and lengths.The link Λ (2) and its Reeb chords determine a quiver represented in Fig-ure 5, in which each component of the link corresponds to a vertex and eachReeb chord corresponds to an oriented edge. When computing the differen-tial of a generator, the terms to be considered correspond to paths formedby a sequence of edges in this quiver with the same origin and endpoint asthe generator, with total grading one less than the grading of the generatorand with total length strictly smaller than the length of the generator.For ∂c , the only possible terms are M , m c and c m . Indeed, c cannot appear in such terms because two other chords from Λ to Λ would be needed as well. The resulting total length would be smaller thanthe length of c only in the case of m c m , but this term is of grading 2lower than c . The generators c and c can appear at most once due totheir length, and due to total length constraint, only m can appear (onlyonce) as a factor, leading to the possibilities m c and c m . Finally, if M appears, then no other chord can appear as a factor by the previousdiscussion, leading to the possibility M .Let us show that each possible term in ∂c is realized by exactly oneMorse flow tree [5], which in turn corresponds to a unique holomorphic curve.To obtain M , we start at the chord c and follow the negative gradientof the local height difference function, in the unique direction leading tothe chord M . At this chord, we have a 2-valent puncture of the Morseflow tree and we continue by following the negative gradient of the localheight difference function corresponding to one of the components Λ or Λ
26 F. BOURGEOIS AND D. GALANT (depending on which hemisphere the maximum M is located). This gradienttrajectory will generically not hit any other Reeb chord and will finally hitthe cusp equator of that component, which is the end of the Morse flow tree.To obtain m c , we start at the chord c and follow the negative gradientof the local height difference function, in the unique direction leading to thechord c . At this chord, we have a 2-valent puncture of the Morse flowtree and we continue by following the negative gradient of the local heightdifference function corresponding to the highest two sheets, which is theMorse function used to perturb the Hopf link. Generically, this gradienttrajectory will reach the minimum m so that we obtain a 1-valent punctureof the Morse flow tree at m . The term c m is obtained similarly.For ∂c , the only possible term is c m . Indeed, when n >
1, thechord c is the only one available to start an admissible path from Λ to itself, because the empty path is not admissible. When n = 1, theempty path is admissible but there are two holomorphic disks having c as a positive puncture and no negative puncture, which cancel each other.Due to its length, the only chord we can still use is m and after this, noother chord can be added. Let us show that this possible term for ∂c isrealized by exactly one Morse flow tree. We start at the chord c and followthe negative gradient of the local height difference function, in the uniquedirection leading to the chord c . At this chord, we have a 2-valent punctureof the Morse flow tree and we continue by following the negative gradient ofthe local height difference function corresponding to the lowest two sheets,which is the Morse function used to perturb the Hopf link. Generically, thisgradient trajectory will reach the minimum m so that we obtain a 1-valentpuncture of the Morse flow tree at m . The calculation of ∂c is analogous.For ∂c , there are no possible terms because no other chord can leadfrom Λ to Λ . For ∂M , the only chord which is short enough to appear is m but its grading k − n > n + k −
2. When n = 1, there are two gradient trajectories from themaximum to the minimum of a Morse function on the circle, which canceleach other. Finally, ∂m = 0 because it is the shortest chord and it joinsdifferent components. (cid:3) Corollary 4.6. If k = 1 , the Chekanov-Eliashberg DGA of Λ (2) ⊂ J ( R n ) has two augmentations ε L and ε R , such that ε L ( m ) = 0 , ε R ( m ) = 1 and vanishing on the other Reeb chords. When n > , there are no otheraugmentations.Proof. When n > m is the only generator of degree 0, so that themaps ε L and ε R are the only two degree preserving algebra morphisms A → Z . In order to show that these are augmentations, we need to check that1 , m / ∈ im ∂ . This follows from the fact that there is no term 1 and that m always appears a a factor of another generator in the expression of ∂ inProposition 4.5. (cid:3) EOGRAPHY OF BILINEARIZED LCH 17
After this preliminary calculation, let us consider a combination of sev-eral such links in view of obtaining more general Poincar´e polynomials thanthose in Example 4.4. To this end, consider the 2 N -copy of the standardLegendrian unknot Λ (2 N ) ⊂ J ( R n ) for N ≥
1. This link contains the com-ponents Λ , . . . , Λ N numbered from bottom to top. If (cid:96) denotes the lengthof the unique Reeb chord of Λ i and ε denotes the positive shift between anytwo consecutive components, we require that 2 N ε is much smaller then (cid:96) .We perturb the component Λ i for i = 2 , . . . , N by a Morse function f i withtwo critical points and of amplitude δ much smaller than ε , such that alldifferences f i − f j with i (cid:54) = j are Morse functions with two critical points. Inorder to define the gradings of mixed Reeb chords in this link, we choose theMaslov potential of the component Λ i to be given by the Maslov potentialof the lowest component Λ plus i − i andΛ j shows that the chords of Λ (2 N ) are given bygrading length c i,i n (cid:96)c i,j n + j − i (cid:96) + ε ( j − i ) c j,i n − j + i (cid:96) − ε ( j − i ) m i,j j − i − ε ( j − i ) − δM i,j n + j − i − ε ( j − i ) + δ where the indices i and j take all possible values between 1 and 2 N , suchthat i < j . Proposition 4.7.
The algebra morphisms ε L and ε R defined by ε L ( m i,i +1 ) =1 when i is even, ε R ( m i,i +1 ) = 1 when i is odd and vanishing on all otherchords are augmentations of the Chekanov-Eliashberg DGA of Λ (2 N ) .Proof. Let us to show that m i,i +1 / ∈ im ∂ for all i = 1 , . . . , N −
1. If m i,i +1 was a term in ∂a for some a in the Chekanov-Eliashberg of Λ (2 N ) , then a would have to be a linear combination of chords from Λ i to Λ i +1 . Indeed, ∂c does not contain the term 1 for any chord c of Λ (2 N ) , say from Λ i to Λ j ,because it would give rise to a term 1 in Proposition 4.5 for the LegendrianHopf link composed of Λ i and Λ j . Therefore, ∂ does not decrease the numberof factors in terms in acts on. Since a must be a single chord from Λ i toΛ i +1 , if there were a term m i,i +1 in ∂a , then there would already be sucha term in Proposition 4.5 for the Legendrian Hopf link composed of Λ i andΛ i +1 . Hence m i,i +1 / ∈ im ∂ as announced.This implies that ε L and ε R are augmentations, because any element ofim ∂ is composed of monomials having at least one factor which is not ofthe form m i,i +1 , and in particular not augmented, so that ε L and ε R vanishon im ∂ . (cid:3) Λ Λ Λ Λ N − Λ N − Λ N Figure 6.
Quiver corresponding to the 2 N -copy of the stan-dard Legendrian unknot. Proposition 4.8.
The bilinearized differential ∂ ε L ,ε R of Λ (2 N ) is given by ∂ ε L ,ε R c i,i = i c i,i − + i c i +1 ,i ,∂ ε L ,ε R c i,j = M i,j + j c i,j − + i c i +1 ,j ,∂ ε L ,ε R c j,i = i c j,i − + j c j +1 ,i ,∂ ε L ,ε R m i,j = j m i,j − + i m i +1 ,j ,∂ ε L ,ε R M i,j = j M i,j − + i M i +1 ,j , with i < j and where i and j are the modulo reductions of i and j . In theabove formulas, any generator with one of its indices equal to or N + 1 or of the form m i,i or M i,i should be replaced by zero.Proof. The link Λ (2 N ) and its Reeb chords determine a quiver represented inFigure 6, and as in the proof of Proposition 4.5, the terms in the differentialof a chord from Λ i to Λ j must form a path from vertex i to vertex j .Let us compute ∂ ε L ,ε R c i,i . The only possible terms in ∂c i,i that couldlead to a nonzero contribution to ∂ ε L ,ε R c i,i are c i +1 ,i m i,i +1 and m i − ,i c i,i − .Indeed, there are no other chords of Λ i so a change of component is needed.Since only chords of the form m i,i +1 are augmented by ε L and ε R , there mustbe exactly one chord from Λ j to Λ k with j > k . Moreover, since neither ε L nor ε R augment consecutive chords in the quiver determined by Λ (2 N ) , wemust have | j − k | = 1 and j = i or k = i . Considering the Legendrian Hopflink composed of Λ i and Λ i +1 , Proposition 4.5 gives the term c i +1 ,i m i,i +1 ,while considering the Legendrian Hopf link composed of Λ i − and Λ i , it givesthe term m i − ,i c i,i − . With the first term, since m i,i +1 has to be augmentedby ε R , we obtain the contribution c i +1 ,i when i is odd. With the second term,since m i − ,i has to be augmented by ε L , we obtain the contribution c i,i − when i − ∂ ε L ,ε R c i,i = i c i,i − + i c i +1 ,i as announced.Let us compute ∂ ε L ,ε R c i,j with i < j . All terms in ∂c i,j involving a singlechord from Λ i to Λ j correspond to terms with a single factor in the expressionfor ∂c in Proposition 4.5. We therefore obtain the term M i,j . The otherterms must involve augmented chords; since ε L and ε R do not have consec-utive augmented chords, these other terms could come from m j − ,j c i,j − , c i +1 ,j m i,i +1 , m j − ,j c i +1 ,j − m i,i +1 or analogous terms with c k,l replaced with m k,l or M k,l . The latter two possibilities lead to elements with a too smallgrading, so that the unaugmented chord is of the type c k,l . The possibilities m j − ,j c i,j − and c i +1 ,j m i,i +1 are each realized by a single holomorphic disk, EOGRAPHY OF BILINEARIZED LCH 19 corresponding to the contribution m c + c m in the expression for ∂c in Proposition 4.5. The remaining possibility m j − ,j c i +1 ,j − m i,i +1 has a toosmall grading. Summing up, the possibility m j − ,j c i,j − leads to the term c i,j − when j is odd and the possibility c i +1 ,j m i,i +1 leads to the term c i +1 ,j when i is odd, so that we obtain ∂ ε L ,ε R c i,j = M i,j + j c i,j − + i c i +1 ,j asannounced.The computation of ∂ ε L ,ε R c j,i with i < j is similar. Since there are noother chords from Λ i to Λ j , the only contributions involve augmented chordsand come from m i − ,i c j,i − , c j +1 ,i m j,j +1 or m i − ,i c j +1 ,i − m j,j +1 . The lastpossibility has a too small grading, while the first two possibilities are eachrealized by a single holomorphic disk, corresponding to the contributions c m and m c in the expressions for ∂c and ∂c in Proposition 4.5.The possibility m i − ,i c j,i − leads to the term c j,i − when i is odd and thepossibility c j +1 ,i m j,j +1 leads to the term c j +1 ,i when j is odd, so that weobtain ∂ ε L ,ε R c j,i = i c j,i − + j c j +1 ,i as announced.The computation of ∂ ε L ,ε R m i,j and ∂ ε L ,ε R M i,j with i < j − m k,l and M k,l since all other chords are much longer. Letus start with ∂ ε L ,ε R m i,j . Arguing as above, since m i,j is the shortest chordfrom Λ i to Λ j , the only contributions involve augmented chords and comefrom m i − ,i m j,i − , m j +1 ,i m j,j +1 or m i − ,i m j +1 ,i − m j,j +1 . The last possibil-ity has a too small grading, and the first two possibilities are each realizedby a unique Morse flow tree [5], which in turn corresponds to a uniqueholomorphic curve. Both Morse flow trees start with a constant gradienttrajectory at m i,j , which is the minimum of the difference function f j − f i .The only possibility to leave m i,j is to have a 3-valent vertex, correspond-ing to the splitting of the gradient trajectory into two gradient trajectories,for f j − f k and for f k − f i , for some k strictly between i and j . Thesetrajectories converge to the corresponding minima m k,j and to m i,k , so weobtain the desired trees for k = i + 1 and k = j −
1. Summing up, weobtain as above ∂ ε L ,ε R m j,i = i m j,i − + j m j +1 ,i as announced. The com-putation of ∂ ε L ,ε R M i,j is completely analogous, except for the description ofthe Morse flow trees. Both Morse flow trees start with a gradient trajec-tory from M i,j to a priori any point of the sphere. In order to reach M i +1 ,j or M i,j − it is necessary for the gradient trajectory to end exactly at themaximum of the corresponding height difference function. There, we havea 2-valent puncture of the Morse flow tree and we continue with a gradienttrajectory converging to the minimum m i,i +1 or m j − ,j . Again, we obtain ∂ ε L ,ε R M j,i = i M j,i − + j M j +1 ,i as announced. (cid:3) Proposition 4.9.
The Poincar´e polynomial P Λ (2 N ) ,ε L ,ε R is given by N (1 + t n ) .Proof. We need to compute the homology of the complex described in Propo-sition 4.8.Let us first consider the subcomplex spanned by the chords m i,j with i 1, the generators m k − , l − , m k, l − , m k − , l − and m k, l − form an acyclic subcomplex. When k = l − 1, wejust have a subcomplex with the 3 generators m l − , l − , m l − , l − and m l − , l − , which has homology spanned by [ m l − , l − ] = [ m l − , l − ] indegree 0. We therefore obtain N − k = 1 , . . . , N − 1, the generators m k − , N and m k, N form an acyclic subcomplex. Finally,the generator m N − , N survives in homology and has degree 0. The totalcontribution of the chords m i,j to the polynomial P Λ (2 N ) ,ε L ,ε R is thereforethe term N .Consider now the subcomplex spanned by the chords M i,j with i < j and c i,j for all i, j = 1 , . . . , N . For any k, l = 1 , . . . , N with k < l − c k − , l − , c k, l − , c k − , l − , c k, l − , M k − , l − , M k, l − , M k − , l − and M k, l − form an acyclic subcomplex. When k = l − 1, wejust have a subcomplex with the 7 generators c k − , l − , c k, l − , c k − , l − , c k, l − , M k − , l − , M k, l − and M k − , l − , which has homology spannedby c l − , l − in degree n . We therefore obtain N − k = 1 , . . . , N − 1, the generators c k − , N , c k, N , M k − , N and M k, N form an acyclic subcomplex. But the subcomplex spanned by the 3 gen-erators c N − , N , c N, N , M N − , N has homology generated by [ c N, N ] =[ M N − , N ] in degree n . For any k, l = 1 , . . . , N with k ≤ l and k > c l − , k − , c l, k − , c l − , k − and c l, k − form an acyclicsubcomplex. When k = 1, we just have an acyclic subcomplex with the 2generators c l − , and c l, . The total contribution of the chords M i,j with i < j and c i,j to the polynomial P Λ (2 N ) ,ε L ,ε R is therefore the term N t n .The sum of the above two contributions therefore gives P Λ (2 N ) ,ε L ,ε R ( t ) = N (1 + t n ) as announced. (cid:3) The next step is to perform some type of connected sum on the Legendrianlink Λ (2 N ) in order to obtain a Legendrian sphere (cid:101) Λ (2 N ) ⊂ J ( R n ). Moreprecisely, for each i = 1 , . . . , N − 1, we consider the Legendrian link formedby Λ i − , Λ i , Λ i +1 and Λ i +2 as the 2-copy of the Legendrian link formedby Λ i − and Λ i +1 , and we perform the 2-copy of the connected sum ofΛ i − and Λ i +1 .We now describe the connected sum of Λ i − and Λ i +1 in more details.We deform Λ i − by a Legendrian isotopy corresponding to the spinning oftwo iterated first Reidemeister moves on one half of the standard Legendrianunknot in J ( R ). Since this front in J ( R ) has a vertical symmetry axis,we can spin it around this axis to produce a Legendrian surface in J ( R )as in [2, Section 3.2]. The resulting front has vertical symmetry planes andhence is spinnable around such a plane; iterating the spinning construction,we obtain the desired 2-components Legendrian link in J ( R n ) with cuspedges from (the deformation of) Λ i − and Λ i +1 facing each other andhaving the same Maslov potentials. This is illustrated by Figure 7.On this Figure, we consider the short horizontal dashed line: its imagein J ( R + ) ⊂ J ( R n ), i.e. with all spinning angles set to zero, is a lineconnecting two points that are each lying on a cusp locus diffeomorphic to EOGRAPHY OF BILINEARIZED LCH 21 Figure 7. Connected sum of Λ i − and Λ i +1 . S n − and having disjoint interior with the Legendrian front. We perform aconnected sum along this line as in [2, Section 4].Finally, after performing N − i − for i = 1 , . . . , N and the other oneresulting from the connected sum of Λ i for i = 1 , . . . , N . We then performan (ordinary) connected sum between these components in order to obtainthe Legendrian sphere (cid:101) Λ (2 N ) . Proposition 4.10. The augmentations ε L and ε R of Λ (2 N ) induce augmen-tations (cid:101) ε L and (cid:101) ε R of (cid:101) Λ (2 N ) .Proof. Note that it suffices to show that an augmentation induces anotheraugmentation after a single 2-copy of a connected sum. To this end, wedescribe this operation differently, in order to gain a better control on theReeb chords during this process. Before performing the 2-copy connectedsum connecting Λ i − and Λ i to Λ i +1 and Λ i +2 respectively, we deformthese components by a Legendrian isotopy in order to create a pair of can-celing critical points m (cid:48) i − , i of index 0 and s i − , i of index 1 for the Morsefunction f i − f i − and a similar pair m (cid:48) i +1 , i +2 , s i +1 , i +2 for f i +2 − f i +1 near the attaching locus of the connecting double tube. More precisely, thechords m (cid:48) i − , i and m (cid:48) i +1 , i +2 are contained in the small balls that are re-moved during the connected sums, while the chords s i − , i and s i +1 , i +2 are just outside these balls. The connecting double tube is the thickeningof an n − z -direction is minimal in the middle. Weextend the Morse functions f i − f i − and f i +2 − f i +1 by a Morse functionon the connecting tube decreasing towards its middle and having exactlytwo critical points (of index 0 and n − 1) in its middle slice. All Reeb chordsfor the connecting double tube are contained in this middle slice and cor-respond to the generators described in Proposition 4.5 with k = 1 and n replaced with n − grading length c h i − , i − n − (cid:96) (cid:48) < (cid:96)c h i, i n − (cid:96) (cid:48) c h i − , i n (cid:96) (cid:48) + εc h i, i − n − (cid:96) (cid:48) − εm h i − , i ε − δM h i − , i n − ε + δ The last two generators correspond to the critical points of the Morsefunction on the connecting tube mentioned above. The unital subalgebra A h generated by these 6 generators is a subcomplex of the Chekanov-EliashbegDGA, because Morse-flow trees are pushed towards the middle of the doubleconnecting tube due to its shape. By Corollary 4.6, this subcomplex has twoaugmentations such that only m h i − , i is possibly augmented. On the otherhand, we have ∂s i − , i = m i − , i + m (cid:48) i − , i with no other terms becausethe length of s i − , i is very short. Hence, for any augmentation ε , we musthave ε ( m (cid:48) i − , i ) = ε ( m i − , i ) and this forces the choice of the augmentationfor A h .Let us check that the resulting maps (cid:101) ε L , (cid:101) ε R : A ( (cid:101) Λ (2 N ) ) → Z satisfy (cid:101) ε L ◦ ∂ = 0 = (cid:101) ε R ◦ ∂ . We already saw that these relations are satisfied on A h and on s i − , i . Similarly, they are satisfied on s i +1 , i +2 . On the remainingchords, the relation was satisfied before the 2-copy of connected sum. Sincethe attachment of the double connecting tube takes places in a small neigh-borhood, that we can assume to be disjoint from rigid holomorphic curves(or Morse flow trees), the differential of the remaining chords can only bemodified by the appearance of new terms. We just need to check that thesenew terms in the differential of these chords do not destroy this relation.The only possible harmful terms are products of augmented chords, butthese consist only of m h i − , i . If the double connecting tube, as well as thechord s i − , i , is removed, the chord m h i − , i is replaced by m i − , i in theexpression of the differential. But this means that a term from the differen-tial was destroyed by the 2-copy of connected sum, contradicting our earlierassumption. (cid:3) We are now in position to show that these 2-copies of connected sumsdestroy almost all terms in the Poincar´e polynomial for bilinearized LCH. Proposition 4.11. The Poincar´e polynomial P (cid:101) Λ (2 N ) , (cid:101) ε L , (cid:101) ε R is equal to .Proof. Let us show by induction that, after applying k successive 2-copies ofconnected sums on Λ (2 N ) , its Poincar´e polynomial is given by ( N − k )(1+ t n ).Proposition 4.9 corresponds to the case k = 0. When applying a ( k + 1)th2-copy of connected sum, the first step is to remove open neighborhoods ofthe chords m (cid:48) i − , i and m (cid:48) i +1 , i +2 . Since ∂s i − , i = m i − , i + m (cid:48) i − , i as inthe proof of Proposition 4.10, The homology class of m (cid:48) i − , i coincides withthat of m i − , i , which contributes a term 1 in the Poincar´e polynomial. A EOGRAPHY OF BILINEARIZED LCH 23 similar property holds for m (cid:48) i +1 , i +2 . Therefore, removing these two chordscontributes − n − N = 1 and n replaced with n − 1, its homology isgenerated by m h i − , i in degree 0 and by c h i, i in degree n − 1. The proofof Proposition 4.10 shows that m h i − , i is not hit by the differential. Onthe other hand, c h i, i is hit by c i, i and by c i +2 , i +2 , which generate eacha homology class as shown in the proof of Proposition 4.9. Therefore, thesecond step contributes +1 − t n to the Poincar´e polynomial.Overall, each 2-copy of connected sum contributes − (1+ t n ) to the Poincar´epolynomial, leading to the announced result. After these N − t n . The last step in theconstruction of (cid:101) Λ (2 N ) is an ordinary connected sum between the remainingtwo connected components. Since the bilinearized LCH of this link has rank2, both classes are mapped to a nontrivial class in singular homology by map τ in the duality exact sequence (2.2). By Proposition 3.5, this connectedsum therefore modifies the Poincar´e polynomial by − t n − . We are thereforeleft with P (cid:101) Λ (2 N ) , (cid:101) ε L , (cid:101) ε R ( t ) = 1 as announced. (cid:3) The next step in our construction is to add to (cid:101) Λ (2 N ) a standard Legen-drian unknot Λ which forms with the bottom k components Λ , . . . , Λ k aLegendrian link isotopic to the k + 1-copy of the standard Legendrian un-knot, but which is unlinked with the 2 N − k top components Λ k +1 , . . . , Λ N .We fix the Maslov potential of the component Λ to be given by the Maslovpotential of Λ plus m − 1, for some integer m . We can deform this link by aLegendrian isotopy in order to widen the components Λ , . . . , Λ k ⊂ J ( R n )so that their projection to R n becomes much larger than the projectionof the components Λ k +1 , . . . , Λ N . We further narrow the component Λ sothat its projection to R n does not intersect the projection of the componentsΛ k +1 , . . . , Λ N . We denote the resulting Legendrian link by (cid:101) Λ (2 N )( k,m ) .This Legendrian link (cid:101) Λ (2 N )( k,m ) has several additional Reeb chords comparedto (cid:101) Λ (2 N ) . These are easily identified within the k + 1-copy of the standardLegendrian unknot formed by Λ , Λ , . . . , Λ k and are given bygrading length c , n (cid:96)c ,j n + j − m (cid:96) + εjc j, n − j + m (cid:96) − ε ( j − i ) m ,j j − m − εj − δM ,j n + j − m − εj + δ where the index i takes all possible values between 1 and k . We extend the augmentations (cid:101) ε L and (cid:101) ε R by zero on these additionalchords in order to define augmentations, still denoted by (cid:101) ε L and (cid:101) ε R , onthe Chekanov-Eliashberg DGA of (cid:101) Λ (2 N )( k,m ) . Since the mixed chords involvingΛ are not augmented, it follows that the vector space generated by theabove chords is a subcomplex of the bilinearized complex with respect tothe differential ∂ (cid:101) ε L , (cid:101) ε R . Proposition 4.12. The bilinearized differential ∂ (cid:101) ε L , (cid:101) ε R of (cid:101) Λ (2 N )( k,m ) on the sub-complex generated by the chords involving the component Λ is given by ∂ (cid:101) ε L , (cid:101) ε R c , = 0 ,∂ (cid:101) ε L , (cid:101) ε R c ,j = M ,j + j c ,j − ,∂ (cid:101) ε L , (cid:101) ε R c j, = j c j +1 , ,∂ (cid:101) ε L , (cid:101) ε R m ,j = j m ,j − ,∂ (cid:101) ε L , (cid:101) ε R M ,j = j M ,j − , for j = 1 , . . . , k , where j is the modulo 2 reduction of j and where in theright hand sides c k +1 , , c , , m , and M , should be replaced by zero.Proof. This result follows from the same computations as in Proposition 4.8,in which we replace 2 N with k , i with 0 and where all terms obtained bychanging the index i are omitted since the mixed Reeb chords involving Λ are not augmented. (cid:3) Proposition 4.13. Consider the Legendrian link (cid:101) Λ (2 N )( k,m ) ⊂ J ( R n ) . ThePoincar´e polynomial P (cid:101) Λ (2 N )( k,m ) , (cid:101) ε L , (cid:101) ε R is given by t n + t − m + t a , where (4.2) a = (cid:26) k − m − if k is even, n − k + m if k is odd.Proof. Let us compute the homology of the subcomplex generated by allReeb chords involving the component Λ . First note that c , is alwaysa generator in homology, leading to the term t n in the Poincar´e polyno-mial. Moreover, the complex generated by the chords c , , . . . , c ,k and M , , . . . , M ,k is acyclic.If k is even, the complex generated by the chords c , , . . . , c k, is acyclic.On the other hand, the complex generated by the chords m , , . . . , m ,k hasits homology generated by m , and m ,k . These lead to the terms t − m and t k − m − in the Poincar´e polynomial.If k is odd, the complex generated by the chords c , , . . . , c k, has its ho-mology generated by c k, . This leads to the term t n − k + m in the Poincar´epolynomial. On the other hand, the complex generated by the chords m , , . . . , m ,k has its homology generated by m , . This leads to the term t − m in the Poincar´e polynomial. EOGRAPHY OF BILINEARIZED LCH 25 Adding these contributions to the Poincar´e polynomial of (cid:101) Λ (2 N ) fromProposition 4.11, we obtain the announced result. (cid:3) The next step in our construction is to perform a connected sum betweenthe component Λ and the original knot (cid:101) Λ (2 N ) . This can be done after aLegendrian isotopy of Λ similar to the one depicted in Figure 7, so that apiece of cusp in the deformed Λ faces a piece of cusp from the componentΛ . In this case, it will be necessary to use a different number of firstReidemeister moves as in Figure 4 before spinning the resulting front, sothat the Maslov potentials near the facing cusps agree. We denote by Λ (2 N )( k,m ) the resulting Legendrian knot in J ( R n ). We denote by ε L and ε R theaugmentations induced from (cid:101) ε L and (cid:101) ε R via the exact Lagrangian cobordismbetween Λ (2 N )( k,m ) and (cid:101) Λ (2 N )( k,m ) . Proposition 4.14. Consider the Legendrian link Λ (2 N )( k,m ) ⊂ J ( R n ) . Wehave P Λ (2 N )( k,m ) ,ε L ,ε R ( t ) = 1 + t − m + t a , where a is given by (4.2) .Proof. By Proposition 4.13, the generator [ c , ] ∈ LCH (cid:101) ε L , (cid:101) ε R n ( (cid:101) Λ (2 N )( k,m ) ) corre-sponds to the fundamental class [Λ ] of the component Λ of the Legendrianlink (cid:101) Λ (2 N )( k,m ) . By Proposition 3.5, the effect of the connected sum with thiscomponent is to remove the term t n from the Poincar´e polynomial, so thatwe obtain the announced result. (cid:3) Note that, instead of adding a single component Λ to the Legendrianknot (cid:101) Λ (2 N ) , we can add a collection of components Λ , , . . . , Λ ,r ⊂ J ( R n )with similar properties. More precisely, for all i = 1 , . . . , r , Λ ,i forms withthe bottom k i components Λ , . . . , Λ k i a Legendrian link isotopic to the k i + 1-copy of the standard Legendrian unknot, but the projection of Λ ,i to R n is disjoint from the projection of the other components Λ k i +1 , . . . , Λ N .The Maslov potential of Λ ,i is fixed as the Maslov potential of Λ plus m i − 1, for some integer m i . With k = ( k , . . . , k r ) and m = ( m , . . . , m r ),we denote the resulting Legendrian link by (cid:101) Λ (2 N )( k,m ) .Each additional component Λ ,i gives rise to an additional subcomplex inthe bilinearized complex as in Proposition 4.12, hence to additional termsin the Poincar´e polynomial of the form t n + t − m i + t a i with a i given by (4.2).After the connected sum of these components with (cid:101) Λ ( N ) , wo obtain a Leg-endrian knot Λ (2 N )( k,m ) and, arguing as in Proposition 4.14, its Poincar´e poly-nomial is given by P Λ (2 N )( k,m ) ,ε L ,ε R ( t ) = 1 + r (cid:88) i =1 ( t − m i + t a i ) . Proof of Theorem 1.3. Let P = q + p be a bLCH-admissible polynomial inthe sense of Definition 4.1. If n is even, p ( − 1) = 0 so that the polynomial p can be expressed as a sum of polynomials of the form (cid:80) ri =1 ( t u i + t v i ),where u i and v i have different parities. If n is odd, p ( − 1) is even, so thatthe polynomial p can be expressed as the sum of polynomials of the form (cid:80) ri =1 ( t u i + t v i ), with no parity conditions on u i and v i .In order to realize the polynomial t u i + t v i when n is even, let us choose m i = − u i and k i = v i − u i + 1, which is even. When n is odd, we can take m i = − u i and k i = v i − u i + 1, which is even when u i and v i have differentparities, or k i = n − u i − v i which is odd when u i and v i have the sameparity.Le us define k = ( k , . . . , k r ) and m = ( m , . . . , m r ), and let N be thesmallest even integer such that k i ≤ N for all i = 1 , . . . , r . Then, in viewof our above constructions, the Legendrian knot Λ (2 N )( k,m ) satisfies P Λ (2 N )( k,m ) ,ε L ,ε R ( t ) = 1 + p ( t ) . Consider now the polynomial q . By Definition 4.1, the polynomial q ( t ) + t n − n such that q (0) = 0. By Corollary6.7 in [2], there exists a connected Legendrian submanifold Λ q ⊂ J ( R n )equipped with an augmentation ε such that P Λ q ,ε ( t ) = q ( t ) + t n − . Let Λ (2 N ) P be the disjoint union of the Legendrian knots Λ (2 N )( k,m ) and Λ q , suchthat the projection of these components to R n are disjoint. We denote by (cid:98) ε L and (cid:98) ε R the augmentations for Λ (2 N ) P induced by the augmentation ε forΛ q and the augmentations ε L and ε R for Λ (2 N )( k,m ) . The Poincar´e polynomialof Λ (2 N ) P is given by the sum of the Poincar´e polynomials of its components: P Λ (2 N ) P ,ε L ,ε R ( t ) = P ( t ) + t n . Finally, we perform a connected sum on the Legendrian link Λ (2 N ) P in orderto obtain a Legendrian knot (cid:101) Λ (2 N ) P , equipped with two augmentations stilldenoted by (cid:98) ε L and (cid:98) ε R . Since the augmentations ε L and ε R coincide (with ε )on the component Λ q , by Proposition 3.2 the fundamental class [Λ q ] of thiscomponent is in the image of the map τ n in the duality exact sequence (2.2).By Proposition 3.5, the effect of the connected sum with Λ q is to remove aterm t n from the Poincar´e polynomial. We therefore obtain P Λ (2 N ) P ,ε L ,ε R ( t ) = P ( t )as desired. (cid:3) EOGRAPHY OF BILINEARIZED LCH 27 References [1] Fr´ed´eric Bourgeois, Baptiste Chantraine, Bilinearized Legendrian contact homologyand the augmentation category, J. Symplectic Geom. Algebr. Geom. Topol. Invent. Math. , 150(3):441–483, 2002.[4] Georgios Dimitroglou Rizell, Legendrian ambient surgery and Legendrian contacthomology. J. Symplectic Geom. Geom. Topol. Duke Math. J. R n +1 , J. Differential Geom. , 71(2):177–305, 2005.[8] Tobias Ekholm, John Etnyre, and Michael Sullivan, Non-isotopic Legendrian sub-manifolds in R n +1 , J. Differential Geom. , 71(1):85–128, 2005.[9] Tobias Ekholm, John Etnyre, and Michael Sullivan, Legendrian contact homology in P × R , Trans. Amer. Math. Soc. , 359(7):3301–3335, 2007.[10] Yves F´elix, Stephen Halperin, Jean-Claude Thomas, Rational homotopy theory, Graduate Texts in Mathematics Geom. Topol. Topology preprint arXiv:1502.04939, 2015. Laboratoire de Math´ematiques d’Orsay, Univ. Paris-Sud, CNRS, Universit´eParis-Saclay, 91405 Orsay, France E-mail address : [email protected] URL : D´epartement de Math´ematiques, University of MONS (UMONS), Place duParc 20, 7000 Mons, Belgium E-mail address ::