Related papers: Non-Orientable Lagrangian Cobordisms between Legen…
Links in $S^3$ can be encoded by grid diagrams; a grid diagram is a collection of points on a toroidal grid such that each row and column of the grid contains exactly two points. Grid diagrams can be reinterpreted as front projections of…
In this note, we define a new invariant of a Legendrian knot in a contact manifold using an open book decomposition supporting the contact structure. We define the support genus sg(L) of a Legendrian knot L in a contact 3-manifold (M, \xi)…
In this article, we introduce rack invariants of oriented Legendrian knots in the 3-dimensional Euclidean space endowed with the standard contact structure, which we call Legendrian racks. These invariants form a generalization of the…
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…
An exact Lagrangian submanifold $L$ in the symplectization of standard contact $(2n-1)$-space with Legendrian boundary $\Sigma$ can be glued to itself along $\Sigma$. This gives a Legendrian embedding $\Lambda(L,L)$ of the double of $L$…
We introduce and study strongly invertible Legendrian links in the standard contact three-dimensional space. We establish the equivariant analogs of basic results separately well-known for strongly invertible and Legendrian links, i.e. the…
In this short note we discuss certain examples of Legendrian submanifolds, whose linearized Legendrian contact (co)homology groups over integers have non-vanishing algebraic torsion. More precisely, for a given arbitrary finitely generated…
We define relative versions of the classical invariants of Legendrian and transverse knots in contact 3-manifolds for knots that are homologous to a fixed reference knot. We show these invariants are well-defined and give some basic…
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…
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…
This paper discusses the cobordism of Lagrangian submanifolds (in the monotone setting) and structures it as a category that is related in a functorial way to an appropriate (derived) Fukaya category. Are also discussed obstructions to…
We prove that every Legendrian knot in the tight contact structure of the 3-sphere is determined by the contactomorphism type of its exterior. Moreover, by giving counterexamples we show this to be not true for Legendrian links in the tight…
The problem of classification of Legendrian knots (links) up to isotopy in the class of Legendrian embeddings (Legendrian isotopy) naturally leads to the following two subproblems. The first of them is: which combinations of the three…
Ozsvath and Stipsicz showed that the LOSS invariant is natural under +1 contact surgery. We extend their result and prove the naturality of the LOSS invariant of a Legendrian L under any positive integer contact surgery along another…
We classify Legendrian rational unknots with tight complements in the lens spaces L(p,1) up to coarse equivalence. As an example of the general case, this classification is also worked out for L(5,2). The knots are described explicitly in a…
The technique of generating families produces obstructions to the existence of embedded Lagrangian cobordisms between Legendrian submanifolds in the symplectizations of 1-jet bundles. In fact, generating families may be used to construct a…
Using convex surfaces and Kanda's classification theorem, we classify Legendrian isotopy classes of Legendrian linear curves in all tight contact structures on $T^3$. Some of the knot types considered in this article provide new examples of…
We derive constraints on Lagrangian concordances from Legendrian submanifolds of the standard contact sphere admitting exact Lagrangian fillings. More precisely, we show that such a concordance induces an isomorphism on the level of…
In the note we study Legendrian and transverse knots in rationally null-homologous knot types. In particular we generalize the standard definitions of self-linking number, Thurston-Bennequin invariant and rotation number. We then prove a…
We show that a two-dimensional totally real concordance can be approximated by a Lagrangian concordance whose Legendrian boundary has been stabilised both positively and negatively sufficiently many times. The main applications that we…