Related papers: On Lagrangian cobordisms and the Chekanov-Eliashbe…
This is the second in a sequence of papers in which we construct Chekanov-Eliashberg algebras for Legendrians in circle-fibered contact manifolds and study the associated augmentation varieties. In this part, we first define the…
We introduce constructions of exact Lagrangian cobordisms with cylindrical Legendrian ends and study their invariants which arise from Symplectic Field Theory. A pair $(X,L)$ consisting of an exact symplectic manifold $X$ and an exact…
For $1$-dimensional Legendrian submanifolds of $1$-jet spaces, we extend the functorality of the Legendrian contact homology DG-algebra (DGA) from embedded exact Lagrangian cobordisms, as in \cite{EHK}, to a class of immersed exact…
We establish tools to facilitate the computation and application of the Chekanov-Eliashberg differential graded algebra (DGA), a Legendrian-isotopy invariant of Legendrian knots in standard contact three-space. More specifically, we…
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…
We define an invariant of Legendrian links in the double-point enhanced grid homology of a link, and prove that it obstructs decomposable Lagrangian cobordisms in the symplectization of the standard contact structure on $\mathbb R^3$.
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.…
This work applies the ideas of persistent homology to the problem of distinguishing Legendrian knots. We develop a persistent version of Legendrian contact homology by filtering the Chekanov-Eliashberg DGA using the action (height)…
In this article, associated to a (bordered) Legendrian graph, we study and show the equivalence between two categorical Legendrian isotopy invariants: the augmentation category, a unital $A_{\infty}$-category, which lifts the set of…
In this article we define intersection Floer homology for exact Lagrangian cobordisms between Legendrian submanifolds in the contactisation of a Liouville manifold, provided that the Chekanov-Eliashberg algebras of the negative ends of the…
To a Legendrian knot, one can associate an $\mathcal{A}_{\infty}$ category, the augmentation category. An exact Lagrangian cobordism between two Legendrian knots gives a functor of the augmentation categories of the two knots. We study the…
We define a differential graded algebra for Legendrian graphs and tangles in the standard contact Euclidean three space. This invariant is defined combinatorially by using ideas from Legendrian contact homology. The construction is…
This is an introduction to Legendrian contact homology and the Chekanov-Eliashberg differential graded algebra, with a focus on the setting of Legendrian knots in $\mathbb{R}^3$. This is the published version of the paper, but with a…
In this paper, we present new obstructions to the existence of Lagrangian cobordisms in $\mathbb{R}^4$ that depend only on the enriched knot diagrams of the boundary knots or links, using holomorphic curve techniques. We define enriched…
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…
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…
We investigate an equivalence relation on Legendrian knots in the standard contact three-space defined by the existence of an interpolating zigzag of Lagrangian cobordisms. We compare this relation, restricted to genus-$0$ surfaces, to…
Embedded Lagrangian cobordisms between Legendrian submanifolds are produced from isotopy, spinning, and handle attachment constructions that employ the technique of generating families. Moreover, any Legendrian with a generating family has…
In this note we construct augmentations of Chekanov-Eliashberg algebras of certain high dimensional Legendrian submanifolds that are not induced by exact Lagrangian fillings. The obstructions to the existence of exact Lagrangian fillings…
In this article, associated to a (bordered) Legendrian graph, we study and show the equivalence between two Legendrian isotopy invariants: augmentation number via point-counting over a finite field, for the augmentation variety of the…