Related papers: Legendrian singular links and singular connected s…
We study the behavior of Legendrian and transverse knots under the operation of connected sums. As a consequence we show that there exist Legendrian knots that are not distinguished by any known invariant. Moreover, we classify Legendrian…
In this paper, we give a complete coarse classification of strongly exceptional Legendrian realizations of connected sum of two Hopf links in contact 3-spheres. This is the first classification result about exceptional Legendrian…
In this paper, we define the set of singular grid diagrams $\mathcal{SG}$ which provides a unified description for singular links, singular Legendrian links, singular transverse links, and singular braids. We also classify the complete set…
In this note we study several aspects of coisotropic submanifolds of a contact manifold. In particular we give a structure theorem for the singularity of the characteristic foliation of a coisotropic submanifold. Moreover we establish the…
A Legendrian link is called a d\'ej\`a vu link if its components can be connected by a positive Legendrian isotopy but this isotopy cannot be embedded. This is the contact geometric analogue of a pair of events in a spacetime such that…
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…
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 study the singularities of Legendrian subvarieties of contact manifolds in the complex-analytic category and prove two rigidity results. The first one is that Legendrian singularities with reduced tangent cones are contactomorphically…
We present the Legendre transformation in a geometric way based on the procedure of the Legendrian lift. This approach allows us to understand some interesting properties of it, in particular, the reason for the appearance of singularities…
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…
An elementary stabilization of a Legendrian link $L$ in the spherical cotangent bundle $ST^*M$ of a surface $M$ is a surgery that results in attaching a handle to $M$ along two discs away from the image in $M$ of the projection of the link…
Associated to Legendrian links in the standard contact three-space, Ruling polynomials are Legendrian isotopy invariants, which also compute augmentation numbers, that is, the points-counting of augmentation varieties for Legendrian links…
We construct infinitely many Legendrian links in the standard contact $\mathbb{R}^3$ with arbitrarily many topologically distinct Lagrangian fillings. The construction is used to find links in $S^3$ that bound topologically distinct pieces…
Arboreal singularities are an important class of Lagrangian singularities. They are conical, meaning that they can be understood by studying their links, which are singular Legendrian spaces in $S^{2n-1}_{\text{std}}$. Loose Legendrians are…
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…
We define a grid presentation for singular links i.e. links with a finite number of rigid transverse double points. Then we use it to generalize link Floer homology to singular links. Besides the consistency of its definition, we prove that…
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…
We formulate conjectures generalizing some known results to the category of virtual Legendrian knots. This includes statements relating virtual Legendrian knots to ordinary Legendrian knots, non-existence of positive virtual Legendrian self…
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 give a classification of Legendrian torus links. Along the way, we give the first classification of infinite families of Legendrian links where some smooth symmetries of the link cannot be realized by Legendrian isotopies. We also give…