Joshua M Sabloff
The main result of this paper is that, off of a `fundamental class' in degree 1, the linearized Legendrian contact homology obeys a version of Poincare duality between homology groups in degrees k and -k. Not only does the result itself simplify calculations, but its proof also establishes a framework for analyzing cohomology operations on the linearized Legendrian contact homology.
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We show that there exists a Legendrian knot with maximal Thurston-Bennequin invariant whose contact homology is trivial. We also provide another Legendrian knot which has the same knot type and classical invariants but nonvanishing contact homology.
We provide the first example of a Legendrian knot with nonvanishing contact homology whose Thurston-Bennequin invariant is not maximal.
We give a combinatorial description of the Legendrian differential graded algebra associated to a Legendrian knot in PxR, where P is a punctured Riemann surface. As an application we show that for any integer k and any homology class h in H_1(PxR) there are k Legendrian knots all representing h which are pairwise smoothly isotopic through a formal Legendrian isotopy but which lie in mutually distinct Legendrian isotopy classes.
Persistent Legendrian contact homology in $\mathbb{R}^3$
本研究将持久同调思想引入勒让德纽结区分问题，通过作用泛函过滤Chekanov-Eliashberg DGA构建持久勒让德切触同调，提出高度标定算法并分析Reidemeister变换的同调影响，还建立了对应的强Morse不等式。
Using the combinatorial approach to knot Floer homology, we define an invariant for Legendrian knots in the three-sphere, which takes values in link Floer homology. This invariant can be used to also construct an invariant of transverse knots.
Contact homology for Legendrian submanifolds in standard contact $(2n+1)$-space is rigorously defined using moduli spaces of holomorphic disks with Lagrangian boundary conditions in complex $n$-space. It provides new invariants of Legendrian isotopy. Using these invariants the theory of Legendrian isotopy is shown to be very rich. For example, infinite families of pairwise non-isotopic Legendrian $n$-spheres and $n$-tori, which are indistinguishable by means of previously known invariants, are constructed. In a sense, the definition of contact homology presented in this paper is a high dimensi…
Legendrian simple knots not detected by Khovanov homology
本研究构造了Khovanov同调无法检测的Legendrian简单纽结，推翻了Chernov和Maguire的相关猜想。
We give a combinatorial description of the Legendrian contact homology algebra associated to a Legendrian link in $S^1\times S^2$ or any connected sum $\#^k(S^1\times S^2)$, viewed as the contact boundary of the Weinstein manifold obtained by attaching 1-handles to the 4-ball. In view of the surgery formula for symplectic homology, this gives a combinatorial description of the symplectic homology of any 4-dimensional Weinstein manifold (and of the linearized contact homology of its boundary). We also study examples and discuss the invariance of the Legendrian homology algebra under deformation…
We prove that the LOSS and GRID invariants of Legendrian links in knot Floer homology behave in certain functorial ways with respect to decomposable Lagrangian cobordisms in the symplectization of the standard contact structure on $\mathbb{R}^3$. Our results give new, computable, and effective obstructions to the existence of such cobordisms.
We use the spanning tree model for Khovanov homology to study Legendrian links. This leads to an alternative proof for Ng's Khovanov bound for the Thurston-Bennequin number and to both a necessary and a sufficient condition for this bound to be sharp.
We establish an upper bound for the Thurston-Bennequin number of a Legendrian link using the Khovanov homology of the underlying topological link. This bound is sharp in particular for all alternating links, and knots with nine or fewer crossings.
Torsion in linearized contact homology for Legendrian knots
本文构造了 $\mathbb{R}^3$ 中 Legendrian 纽结，其线性化切触同调在整数环上含挠元，从而表明存在非精确填充诱导的增广，但其模 2 约化可诱导。
Legendrian contact homology (LCH) and its associated differential graded algebra are powerful non-classical invariants of Legendrian knots. Linearization makes the LCH computationally tractable at the expense of discarding nonlinear (and noncommutative) information. To recover some of the nonlinear information while preserving computability, we introduce invariant cup and Massey products - and, more generally, an A_\infty structure - on the linearized LCH. We apply the products and A_\infty structure in three ways: to find infinite families of Legendrian knots that are not isotopic to their Le…
Using contact-geometric techniques and sutured Floer homology, we present an alternate formulation of the minus and plus version of knot Floer homology. We further show how natural constructions in the realm of contact geometry give rise to much of the formal structure relating the various versions of Heegaard Floer homology. In addition, to a Legendrian or transverse knot K in a contact manifold (Y,ξ), we associate distinguished classes EHL(K) in the minus-version of knot floer homology and EHIL(K) in the plus version, which are each invariant under Legendrian or transverse isotopies of K. Th…
We use monopole Floer homology for sutured manifolds to construct invariants of Legendrian knots in a contact 3-manifold. These invariants assign to a knot K in Y elements of the monopole knot homology KHM(-Y,K), and they strongly resemble the knot Floer homology invariants of Lisca, Ozsváth, Stipsicz, and Szabó. We prove several vanishing results, investigate their behavior under contact surgeries, and use this to construct many examples of non-loose knots in overtwisted 3-manifolds. We also show that these invariants are functorial with respect to Lagrangian concordance.
Conjectures on the Khovanov Homology of Torus Knots, Twist Knots, and Legendrian Simple Knots
本文基于Khovanov同调的性质，提出涡旋结和扭结的同调能区分所有Legendrian简单结的猜想，并通过大量数据验证其有效性。
In this paper we construct an $\mathcal{A}_\infty$-category associated to a Legendrian submanifold of jet spaces. Objects of the category are augmentations of the Chekanov algebra $\mathcal{A}(Λ)$ and the homology of the morphism spaces forms a new set of invariants of Legendrian submanifolds called the bilinearised Legendrian contact homology. Those are constructed as a generalisation of linearised Legendrian contact homology using two augmentations instead of one. Considering similar constructions with more augmentations leads to the higher order compositions map in the category and generali…
We consider S^1-families of Legendrian knots in the standard contact R^3. We define the monodromy of such a loop, which is an automorphism of the Chekanov-Eliashberg contact homology of the starting (and ending) point. We prove this monodromy is a homotopy invariant of the loop. We also establish techniques to address the issue of Reidemeister moves of Lagrangian projections of Legendrian links. As an application, we exhibit a loop of right-handed Legendrian torus knots which is non-contractible in the space Leg(S^1,R^3) of Legendrian knots, although it is contractible in the space Emb(S^1,R^3…
We define a differential graded algebra associated to Legendrian knots in Seifert fibered spaces with transverse contact structures. This construction is distinguished from other combinatorial realizations of contact homology invariants by the existence of orbifold points in the Reeb orbit space of the contact manifold. These orbifold points are images of the exceptional fibers of the Seifert fibered manifold, and they play a key role in the definitions of the differential and the grading, as well as the proof of invariance. We apply the invariant to distinguish Legendrian knots whose homology…
We use the contact invariant defined in [2] to construct a new invariant of Legendrian knots in Kronheimer and Mrowka's monopole knot homology theory (KHM), following a prescription of Stipsicz and Vértesi. Our Legendrian invariant improves upon an earlier Legendrian invariant in KHM defined by the second author in several important respects. Most notably, ours is preserved by negative stabilization. This fact enables us to define a transverse knot invariant in KHM via Legendrian approximation. It also makes our invariant a more likely candidate for the monopole Floer analogue of the "LOSS" in…
Legendrian contact homology in $\mathbb{R}^3$
本文介绍勒让德接触同调与 Chekanov–Eliashberg DGA，并聚焦 $\mathbb{R}^3$ 中的勒让德纽结，出版版本末尾附有勘误。
Let $L\subset J^1(M)$ be a Legendrian submanifold of the 1-jet space of a Riemannian $n$-manifold $M$. A correspondence is established between rigid flow trees in $M$ determined by $L$ and boundary punctured rigid pseudo-holomorphic disks in $T^\ast M$, with boundary on the projection of $L$ and asymptotic to the double points of this projection at punctures, provided $n\le 2$, or provided $n>2$ and the front of $L$ has only cusp edge singularities. This result, in particular, shows how to compute the Legendrian contact homology of $L$ in terms of Morse theory.
A rigorous foundation for the contact homology of Legendrian submanifolds in a contact manifold of the form $P\times \R$ where $P$ is an exact symplectic manifold is established. The class of such contact manifolds include 1-jet spaces of smooth manifolds. As an application, contact homology is used to provide (smooth) isotopy invariants of submanifolds of $\R^n$ and, more generally, invariants of self transverse immersions into $\R^n$ up to restricted regular homotopies. When $n=3$, this application is the first step in extending and providing a contact geometric underpinning for the new knot…
An L-infinity structure for Legendrian contact homology
本文为三维空间中的 Legendre 纽结或链环构造了扩展 Chekanov-Eliashberg 微分分次代数的 L-无穷代数，并由此在交换 Legendre 接触同调上建立 Poisson 括号，证明其是合痕不变量。
We establish a long exact sequence for Legendrian submanifolds L in P x R, where P is an exact symplectic manifold, which admit a Hamiltonian isotopy that displaces the projection of L off of itself. In this sequence, the singular homology H_* maps to linearized contact cohomology CH^* which maps to linearized contact homology CH_* which maps to singular homology. In particular, the sequence implies a duality between the kernel of the map (CH_*\to H_*) and the cokernel of the map (H_* \to CH^*). Furthermore, this duality is compatible with Poincare duality in L in the following sense: the Poin…
The Chekanov-Eliashberg differential graded algebra of a Legendrian knot L is a rich source of Legendrian knot invariants, as is the theory of generating families. The set P(L) of homology groups of augmentations of the Chekanov-Eliashberg algebra is an invariant, as is a count of objects from the theory of generating families called graded normal rulings. This article gives two results demonstrating the usefulness of computing the homology group of an augmentation using a combinatorial interpretation of a generating family called a Morse complex sequence. First, we show that if the projection…
We show that if a Legendrian knot in standard contact ${\bb R}^3$ possesses a generating family then there exists an augmentation of the Chekanov-Eliashberg DGA so that the associated linearized contact homology (LCH) is isomorphic to singular homology groups arising from the generating family. In this setting we show Sabloff's duality result for LCH may be viewed as Alexander duality. In addition, we provide an explicit construction of a generating family for a front diagram with graded normal ruling and give a new approach to augmentation $\Rightarrow$ normal ruling.
In this short note, we provide a criterion for DGA-homotopy of augmentations of Chekanov-Eliashberg algebra of disconnected Legendrian submanifolds. We apply the criterion to obtain the extension of geography results of Bourgeois and Galant concerning bilinearized Legendrian contact homology to the case of disconnected Legendrian submanifolds.
This article is a continuation of [15]. For Legendrian surfaces in $1$-jet spaces, we prove that the Cellular DGA defined in [15] is stable tame isomorphic to the Legendrian contact homology DGA.
Using the grid diagram formulation of knot Floer homology, Ozsvath, Szabo and Thurston defined an invariant of transverse knots in the tight contact 3-sphere. Shortly afterwards, Lisca, Ozsvath, Stipsicz and Szabo defined an invariant of transverse knots in arbitrary contact 3-manifolds using open book decompositions. It has been conjectured that these invariants agree where they are both defined. We prove this fact by defining yet another invariant of transverse knots, showing that this third invariant agrees with the two mentioned above.
Loose Legendrian n-submanifolds, for n at least 2, were introduced by Murphy and proved to be flexible in the h-principle sense: any two loose Legendrian submanifolds that are formally Legendrian isotopic are also actually Legendrian isotopic. Legendrian contact homology is a Floer theoretic invariant that associates a differential graded algebra (DGA) to a Legendrian submanifold. The DGA of a loose Legendrian submanifold is trivial.
Vanishing of higher Legendrian homology for rainbow closures
该研究证明正辫子的彩虹闭包给出的勒让德链环，其高阶勒让德同调在任意股数、交叉数及积分、单基点情形下均消失。
In this paper, sufficient conditions for contact $(+1)$-surgeries along Legendrian knots in contact rational homology 3-spheres to have vanishing contact invariants or to be overtwisted are given. They can be applied to study contact $(\pm1)$-surgeries along Legendrian links in the standard contact 3-sphere. We also obtain a sufficient condition for contact $(+1)$-surgeries along Legendrian two-component links in the standard contact 3-sphere to be overtwisted via their front projections.
This research announcement continues the study of the symplectic homology of Weinstein manifolds undertaken in [BEE1] where the symplectic homology, as a vector space, was expressed in terms of the Legendrian homology algebra of the attaching spheres of critical handles. Here we express the product and $\BV$-operator of symplectic homology in that context.
Weak Relative Calabi-Yau Structures for Legendrian Contact Homology
本研究将切触纽结线性化Legendrian Contact Homology的对偶同构与长正合序列推广为正增广范畴上$A_\infty$双模的弱相对Calabi-Yau结构，深化了对该同调代数结构的认知。
If a Legendrian knot $Λ$ in the standard contact 3-sphere bounds an orientable exact Lagrangian surface $Σ$ in the standard symplectic 4-ball, then the genus of $Σ$ is equal to the slice genus of (the smooth knot underlying) $Λ$, the sum of the Thurston-Bennequin number of L and the Euler characteristic of $Σ$ is zero as well as the rotation number of $Λ$, and moreover, the linearized contact homology of $Λ$ with respect to the augmentation induced by $Σ$ is isomorphic to the (singular) homology of $Σ$. It was asked in arXiv:1212.1519 whether the converse of this statement is true. We give a n…
Let $Λ$ be a link of Legendrian spheres in the boundary of a subcritical $2n$-dimensional Weinstein manifold $X$. We show that, under some geometrical assumptions, the computation of the Legendrian contact homology of $Λ$ can be reduced to a computation of Legendrian contact homology in 1--jet spaces. Since the Legendrian contact homology in 1--jet spaces is well studied, this gives a simplified way to compute the Legendrian contact homology of $Λ$. We restrict to the case when the attaching spheres of the subcritical handles of $X$ do not interact with each other, and we assume that there are…
On torsion in (bi)linearized Legendrian contact homology in dimension 3
本文研究三维勒让德纽结的线性化与双线性化接触同调在整数系数下的挠元性质，并完整刻画了双线性化LCH的地理分布。
Toward a topological description of Legendrian contact homology of unit conormal bundles
本文为 Legendrian 切触同调提出一种基于弦拓扑的新代数不变量，证明其在光滑同伦下不变，并猜想与原有同调同构，且对特定例子及平凡法丛情形给出计算。
Let E be a circle bundle over a Riemann surface that supports a contact structure transverse to the fibers. This paper presents a combinatorial definition of a differential graded algebra (DGA) that is an invariant of Legendrian knots in E. The invariant generalizes Chekanov's combinatorial DGA invariant of Legendrian knots in the standard contact 3-space using ideas from Eliashberg, Givental, and Hofer's contact homology. The main difficulty lies in dealing with what are ostensibly 1-parameter families of generators for the DGA; these are solved using ``Morse-Bott'' techniques. As an applicat…
On a generalization of Jones polynomial and its categorification for Legendrian Knots
本文通过新型 skein 关系推广 Jones 多项式，并构造相应的 Khovanov 型范畴化，用于刻画 Legendrian 纽结。
Let L be a Legendrian knot in R^3 with the standard contact structure. In [10], a map was constructed from equivalence classes of Morse complex sequences for L, which are combinatorial objects motivated by generating families, to homotopy classes of augmentations of the Legendrian contact homology algebra of L. Moreover, this map was shown to be a surjection. We show that this correspondence is, in fact, a bijection. As a corollary, homotopic augmentations determine the same graded normal ruling of L and have isomorphic linearized contact homology groups. A second corollary states that the cou…
In this paper we study the growth rate of a version of Legendrian contact homology, which we call strip Legendrian contact homology, in 3-dimensional contact manifolds and its relation to the topological entropy of Reeb flows. We show that: if for a pair of Legendrian knots in a contact 3-manifold $(M,ξ)$ the strip Legendrian contact homology is defined and has exponential homotopical growth with respect to the action, then every Reeb flow on $(M,ξ)$ has positive topological entropy. This has the following dynamical consequence: for all Reeb flows (even degenerate ones) on $(M,ξ)$ the number o…
Geography of bilinearized Legendrian contact homology
本文研究闭连通 Legendrian 子流形的双线性化 Legendrian 接触同调，证明其可检测增广的 DGA 同伦性，并刻画了所有可能取值的分次向量空间及其可实现性。
On torsion in linearized Legendrian contact homology
本文构造标准接触向量空间中的 Legendrian 子流形，证明其整数系数的线性化 Legendrian 接触（上）同调可实现任意给定的有限生成阿贝尔群，从而展示该同调中代数挠元的存在性。
We extend the sutured framework to the case of Legendrians with boundary. Using ideas from Lagrangian Floer theory, we define the cylindrical and the wrapped sutured Legendrian homologies of a pair of sutured Legendrians. They fit together into an exact sequence, and the exact triangle is invariant along an Legendrian isotopy fixed at the boundary. For a single Legendrian, we also define a wrapped version of its Chekanov-Eliashberg dga. Our main example of sutured Legendrian is obtained via the unit conormal construction : a submanifold $N \subset M$, such that $\partial N \subset \partial M$ …
We relate the version of rational Symplectic Field Theory for exact Lagrangian cobordisms introduced in [5] with linearized Legendrian contact homology. More precisely, if $L\subset X$ is an exact Lagrangian submanifold of an exact symplectic manifold with convex end $Λ\subset Y$, where $Y$ is a contact manifold and $Λ$ is a Legendrian submanifold, and if $L$ has empty concave end, then the linearized Legendrian contact cohomology of $Λ$, linearized with respect to the augmentation induced by $L$, equals the rational SFT of $(X,L)$. Following ideas of P. Seidel, this equality in combination wi…
We show that all positive contact surgeries on every Legendrian figure-eight knot in $(S^3, ξ_{\rm{std}})$ result in an overtwisted contact structure. The proof uses convex surface theory and invariants from Heegaard Floer homology.
We give a computation of the Legendrian contact homology (LCH) DGA for an arbitrary generic Legendrian surface $L$ in the $1$-jet space of a surface. As input we require a suitable cellular decomposition of the base projection of $L$. A collection of generators is associated to each cell, and the differential is given by explicit matrix formulas. In the present article, we prove that the equivalence class of this cellular DGA does not depend on the choice of decomposition, and in the sequel [35] we use this result to show that the cellular DGA is equivalent to the usual Legendrian contact homo…
We show how to orient moduli spaces of holomorphic disks with boundary on an exact Lagrangian immersion of a spin manifold into complex n-space in a coherent manner. This allows us to lift the coefficients of the contact homology of Legendrian spin submanifolds of standard contact (2n+1)-space from Z_2 to Z. We demonstrate how the Z-lift provides a more refined invariant of Legendrian isotopy. We also apply contact homology to produce lower bounds on double points of certain exact Lagrangian immersions into C^n and again including orientations strengthens the results. More precisely, we prove …
We apply Heegaard-Floer homology theory to establish generalized slicing Bennequin inequalities closely related to a recent result of T. Mrowka and Y. Rollin proved using Seiberg-Witten monopoles.
The paper is a summary of the results of the authors concerning computations of symplectic invariants of Weinstein manifolds and contains some examples and applications. Proofs are sketched. The detailed proofs will appear in our forthcoming paper. In the Appendix written by S. Ganatra and M. Maydanskiy it is shown that the results of this paper imply P. Seidel's conjecture equating symplectic homology with Hochschild homology of a certain Fukaya category.
We give a construction of embedded contact homology (ECH) for a contact $3$-manifold $Y$ with convex sutured boundary and a pair of Legendrians $Λ_+$ and $Λ_-$ contained in $\partial Y$ satisfying an exactness condition. The chain complex is generated by certain configurations of closed Reeb orbits of $Y$ and Reeb chords of $Λ_+$ to $Λ_-$. The main ingredients include: a general Legendrian adjunction formula for curves in $\mathbb{R} \times Y$ with boundary on $\mathbb{R} \times Λ$; a relative writhe bound for curves in contact $3$-manifolds asymptotic to Reeb chords; and a Legendrian ECH inde…
We provide a translation between Chekanov's combinatorial theory for invariants of Legendrian knots in the standard contact R^3 and a relative version of Eliashberg and Hofer's Contact Homology. We use this translation to transport the idea of ``coherent orientations'' from the Contact Homology world to Chekanov's combinatorial setting. As a result, we obtain a lifting of Chekanov's differential graded algebra invariant to an algebra over Z[t,t^{-1}] with a full Z grading.
By examining knot Floer homology, we extend a result of Ozsváth and Stipsicz and show further infinitely many Legendrian and transversely non-simple knot types among two-bridge knots. We give sufficient conditions of Legendrian and transverse non-simplicity on the continued fraction expansion of the corresponding rational number.
We construct a Legendrian 2-torus in the 1-jet space of $S^1\times\R$ (or of $\R^2$) from a loop of Legendrian knots in the 1-jet space of $\R$. The differential graded algebra (DGA) for the Legendrian contact homology of the torus is explicitly computed in terms of the DGA of the knot and the monodromy operator of the loop. The contact homology of the torus is shown to depend only on the chain homotopy type of the monodromy operator. The construction leads to many new examples of Legendrian knotted tori. In particular, it allows us to construct a Legendrian torus with DGA which does not admit…
Let $L \subset Y$ be a Legendrian submanifold of a contact manifold, $S\subset L$ a framed embedded sphere bounding an isotropic disc $D_S \subset Y \setminus L$, and use $L_S$ to denote the manifold obtained from $L$ by a surgery on $S$. Given some additional conditions on $D_S$ we describe how to obtain a Legendrian embedding of $L_S$ into an arbitrarily small neighbourhood of $L \cup D_S \subset Y$ by a construction that we call Legendrian ambient surgery. In the case when the disc is subcritical, we produce an isomorphism of the Chekanov-Eliashberg algebra of $L_S$ with a version of the Ch…
The Chekanov-Eliashberg dg-algebra is an algebraic invariant of Legendrian submanifolds of contact manifolds, whose definition recently has been extended to singular Legendrians. We describe a way of constructing simpler models of this dg-algebra for singular Legendrian knots in $\mathbb{R}^{3}$, and give the first examples of singular knots for which the full cohomology can be computed. This includes the $A_{n}$-Legendrians. An important question in the study of the Chekanov-Eliashberg dg-algebra is to understand how its quasi-isomorphism type changes when the Legendrian undergoes Weinstein r…
Positive loops of Legendrian embeddings are examined from the point of view of Floer homology of Lagrangian cobordisms. This leads to new obstructions to the existence of a positive loop containing a given Legendrian, expressed in terms of the Legendrian contact homology of the Legendrian submanifold. As applications, old and new examples of orderable contact manifolds are obtained and discussed. We also show that contact manifolds admitting a filling of a Liouville domain with non-zero symplectic homology is strongly orderable in the sense of Liu.
We construct a combinatorial invariant of Legendrian knots in standard contact three-space. This invariant, which encodes rational relative Symplectic Field Theory and extends contact homology, counts holomorphic disks with an arbitrary number of positive punctures. The construction uses ideas from string topology.
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