相关论文: Positive Legendrian isotopies and Floer Theory

Given a cooriented contact manifold $(M,\xi)$, it is possible to define a notion of positivity on the group $\mathrm{Diff}(M)$ of diffeomorphisms of $M$, by looking at paths of diffeomorphisms that are positively transverse to the contact…

We prove that any positive braid Legendrian link not isotopic to a standard finite type link admits infinitely many exact Lagrangian fillings.

We give a construction of embedded contact homology (ECH) for a contact $3$-manifold $Y$ with convex sutured boundary and a pair of Legendrians $\Lambda_+$ and $\Lambda_-$ contained in $\partial Y$ satisfying an exactness condition. The…

We define combinatorial Floer homology of a transverse pair of noncontractibe nonisotopic embedded loops in an oriented 2-manifold without boundary, prove that it is invariant under isotopy, and prove that it is isomorphic to the original…

We give a quantitative refinement of the invariance of the Legendrian contact homology algebra in general contact manifolds. We show that in this general case, the Lagrangian cobordism trace of a Legendrian isotopy defines a DGA stable tame…

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 $\mathcal{A}(\Lambda)$ and the homology of the morphism…

Let $\Lambda$ be a closed, connected, spin Legendrian submanifold of the 1-jet space of a smooth $n$-dimensional manifold. We give a coherent orientation scheme for the moduli space of rigid Morse flow trees of $\Lambda$, implying that the…

We construct infinite families of non-simple isotopy classes of links in overtwisted contact structures on $S^1$-bundles over surfaces. These examples include: (1) a pair of Legendrian links that are not Legendrian isotopic, but which are…

We collect some observations about Legendrian links with non-vanishing contact invariants, mostly concerning the non-loose realizations of links and the addition of boundary-parallel half Giroux torsion. In particular, we show that every…

Floer theory for Lagrangian cobordisms was developed by Biran and Cornea to study the triangulated structure of the derived Fukaya category of monotone symplectic manifolds. This paper explains how to use the language of stops to study…

We investigate families of Legendrian submanifolds of 1-jet spaces by developing and applying a theory of families of generating family homologies. This theory allows us to detect an infinite family of loops of Legendrian n-spheres embedded…

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…

We study Legendrian singularities arising in complex contact geometry. We define a one-parameter family of bases in the ring of Legendrian characteristic classes such that any Legendrian Thom polynomial has nonnegative coefficients when…

Let $\lambda$ be a Legendrian link in standard contact $\mathbb{R}^3$, such that $L_1$, $L_2$ are two exact fillings of $\lambda$ and $\varphi$ is a Legendrian loop of $\lambda$. We study fillability and isotopy characterizations of…

Real Legendrian subvarieties are classical objects of differential geometry and classical mechanics and they have been studied since antiquity. However, complex Legendrian subvarieties are much more rigid and have more exceptional…

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…

We establish a new version of Floer homology for monotone Lagrangian submanifolds and apply it to prove the following (generalized) version of Audin's conjecture : if $L$ is an aspherical manifold which admits a monotone Lagrangian…

We consider Lagrangian Floer cohomology for a pair of Lagrangian submanifolds in a symplectic manifold M. Suppose that M carries a symplectic involution, which preserves both submanifolds. Under various topological hypotheses, we prove a…

We study homotopically non-trivial spheres of Legendrians in the standard contact R3 and S3. We prove that there is a homotopy injection of the contactomorphism group of S3 into some connected components of the space of Legendrians induced…

We present examples of Legendrian knots in $\mathbb{R}^3$ that have linearized Legendrian contact homology over $\mathbb{Z}$ containing torsion. As a consequence, we show that there exist augmentations of Legendrian knots over $\mathbb{Z}$…