We simplify singularities of Lagrangian and Legendrian fronts without homotopy obstructions.
problem Simplifying singularities of Lagrangian and Legendrian fronts without homotopy obstructions.
method Establishing a full h−principle for simplification of singularities of Lagrangian and Legendrian fronts. result Simplification of singularities can be achieved by means of a Hamiltonian isotopy if there are no homotopy theoretic obstructions.
The study translates Weinstein structures to Legendrian handlebodies for symplectic topology.
problem Detecting flexibility and rigidity in Weinstein manifolds.
method Systematic recipe for translating Weinstein Lefschetz fibrations to Legendrian handlebodies.
result Verification of Stein deformation equivalence and existence of closed exact Lagrangian submanifolds.
Legendrian knots can be represented by projections with multi-crossings.
problem Representing Legendrian knots with multi-crossings.
method Investigating übercrossing and petal projections in front and Lagrangian projections.
result Legendrian knots with übercrossing projections in front are smoothly isotopic to the unknot.
Cluster structures on moduli spaces are linked to Legendrian knots.
problem Constructing cluster structures on moduli spaces.
method Quantization of exact Lagrangian fillings corresponding to Legendrian knots.
result Cluster structures can be deduced from Legendrian knots.
The paper computes various invariants of Legendrian knots in open book presentations.
problem Computing numerical invariants of Legendrian knots in contact manifolds.
method Front projections, rotation numbers, intersection numbers, Euler class, Lagrangian projections.
result Explicit formulas for computing invariants from front projections.
The paper introduces triple grid diagrams to construct Lagrangian surfaces in complex projective space.
problem Constructing Lagrangian surfaces in complex projective space.
method Defining and analyzing triple grid diagrams to determine Lagrangian caps and surfaces.
result Triple grid diagrams can determine closed Lagrangian surfaces in CP2 under certain conditions. Proves Arnol'd's conjecture about Legendrian curves.
problem Front of Legendrian curves in projectivized cotangent bundle of 2-sphere.
method Microlocal theory of sheaves and derived category of sheaves.
result Proves Arnol'd's three cusps conjecture.
New integer-valued functions for Legendrian knots.
problem Understanding Legendrian knots better.
method Using Legendrian fronts to derive integer-valued linear functions.
result Introduced new invariants similar to Arnold's basic invariant.
Study Legendrian graph invariants via augmentation and ruling polynomials.
problem Equivalence of Legendrian isotopy invariants.
method Use augmentation number and ruling polynomial for front projection.
result Show equivalence between augmentation number and ruling polynomial.
Examples are given of prime Legendrian knots in the standard contact 3-space that have arbitrarily many distinct Chekanov polynomials, refuting a conjecture of Lenny Ng. These are constructed using a new `Legendrian tangle replacement' technique. This technique is then used to show that the phenomenon of multiple Cheka…
We construct a complete, bounded Legendrian immersion in C^3. As direct applications of it, we show the first examples of a weakly complete bounded flat front in hyperbolic 3-space, a weakly complete bounded flat front in de Sitter 3-space, and a weakly complete bounded improper affine front in R^3.
Augmentations and sheaves linked for Legendrian graphs.
problem Understanding categorical Legendrian isotopy invariants.
method Equivalence between augmentation category and DG category of sheaves.
result Proved 'augmentations are sheaves' for Legendrian graphs.
We use parameterized Morse theory on the pages of an open book decomposition to efficiently encode the contact topology in terms of a labelled graph on a disjoint union of tori (one per binding component). This construction allows us to generalize the notion of the front projection of a Legendrian knot from the standar…
Study on hard Legendrian unknots using normal rulings.
problem Understanding the complexity of Legendrian unknots in knot theory.
method Using normal rulings to obstruct and construct hard unknot diagrams.
result Construction of infinitely many smoothly hard max-tb unknot diagrams with bounds on minimum possible writhe.
New methods link Legendrian satellites to Lagrangian cobordisms.
problem Understanding relations between Legendrian and Lagrangian knots.
method Constructing Lagrangian concordances through satellite operations.
result Maximum Thurston-Bennequin number restricts Legendrian satellite Lagrangian sliceness.
Proves existence of exact lagrangian cobordisms between legendrians.
problem Existence of exact lagrangian cobordisms between closed legendrians.
method Proves existence through mathematical proof.
result Existence of non-compact exact lagrangian cobordisms.
Characterizes Legendrian knots with exact Lagrangian fillings.
problem Identifying Legendrian knots with exact Lagrangian fillings.
method Characterization through exact orientable Lagrangian fillings.
result Underlying smooth knot types of fillable Legendrian 4-plats are positive.
A first order Vassiliev invariant of an oriented knot in an S1-fibration and a Seifert fibration over a surface is constructed. It takes values in a quotient of the group ring of the first homology group of the total space of the fibration. It gives rise to an invariant of wave fronts on surfaces and orbifolds relat…
In the symplectization of standard contact 3-space, R×R3, it is known that an orientable Lagrangian cobordism between a Legendrian knot and itself, also known as an orientable Lagrangian endocobordism for the Legendrian knot, must have genus 0. We show that any Legendrian knot has a non-or…
Study uses Newton polytopes to distinguish Lagrangian fillings of Legendrian submanifolds.
problem Distinguishing Lagrangian fillings of Legendrian submanifolds.
method Utilizes Newton polytopes associated with augmented values of Reeb chords.
result Newton polytopes can distinguish infinitely many distinct Lagrangian fillings.
Study on exact Lagrangian submanifolds in unit ball with Legendrian boundary.
problem Exact Lagrangian submanifolds with Legendrian boundary in unit ball.
method Uses Liouville form and boundary unique continuation for differential forms.
result Equatorial n-disk rigidity for compact exact Lagrangian self-similar submanifolds. Computes bounds on mosaic number of Legendrian knots.
problem Determining the mosaic number of Legendrian knots.
method Using modified mosaic tiles and classical invariants, computed lower bounds and provided examples for sharpness.
result Sharp bounds on mosaic number of Legendrian knots in certain cases.
Holomorphic curves in SL2(C) embed open surfaces.
problem Embedding open Riemann surfaces in SL2(C). method Proving existence of holomorphic Legendrian curves.
result Existence of proper, weakly complete, flat fronts in H3. Study confirms conjecture: minimal Lagrangian surfaces with Legendrian boundary are rigid.
problem Characterize minimal Lagrangian surfaces with Legendrian capillary boundary.
method Analyzes surfaces in B4 with Legendrian boundary on S3. result Equatorial plane disks and catenoids are the only minimal Lagrangian surfaces with Legendrian capillary boundary.
New algebraic structures help distinguish Legendrian knots.
problem Distinguishing Legendrian knots from smooth knots.
method Defined Legendrian racks and used them to define invariants.
result Distinguished certain Legendrian knots.
Upper bounds for Legendrian links in tight contact 3-manifolds.
problem Bounding the complexity of Legendrian links in tight contact 3-manifolds.
method Constructing exact Lagrangian cobordisms and defining minimal Lagrangian genus.
result Established upper bounds for Legendrian links with a common rotation number.
The paper studies how Lagrangian cobordisms affect DGAs of Legendrian ends.
problem Understanding how Lagrangian cobordisms impact DGAs of Legendrian ends.
method Adapting the map induced by cobordisms on DGAs to linearizations using augmentations, and showing invariance under Lagrangian isotopy.
result The induced map on linearized Legendrian contact homology is invariant under Lagrangian isotopy under mild hypotheses.
Study finds many Lagrangian fillings for certain Legendrian links.
problem Understanding Lagrangian fillings for Legendrian links of finite type.
method Use of N-graphs and combinatorics of seed patterns.
result Proves existence of at least seeds many exact embedded Lagrangian fillings for Legendrian links of type ADE.
Study of Legendrian links using Floer theory and cluster varieties.
problem Understanding exact Lagrangian fillings of positive braid Legendrian links.
method Floer-theoretic approach and exact Lagrangian cobordisms.
result Proves that positive braid Legendrian links admit infinitely many exact Lagrangian fillings.
The paper proves there are many Lagrangian fillings for Legendrian links of affine type.
problem Proving the existence of many Lagrangian fillings for Legendrian links of affine type.
method Using cluster structures and Coxeter mutation to prove the existence of fillings.
result There are at least as many exact embedded Lagrangian fillings as seeds for Legendrian links of affine type.
New disks fill Legendrian knots without being smoothly isotopic.
problem Finding non-isotopic Lagrangian fillings of Legendrian knots.
method Constructing distinct Lagrangian ribbon disks with same boundary.
result Found non-isotopic Lagrangian disks with same boundary.
A correspondence is studied by H. Matsuda between front projections of Legendrian links in the standard contact structure for 3-space and rectangular diagrams. In this paper, we introduce braided rectangular diagrams, and study a relationship with Legendrian links in the standard contact structure for 3-space. We show …
Constructs Lagrangian skeleta for curve singularities.
problem Understanding Lagrangian skeleta of curve singularities.
method Constructs closed arboreal Lagrangian skeleta associated to links of isolated plane curve singularities.
result Provides computations of Legendrian and Weinstein invariants.
New invariants for Legendrian graphs defined and proven.
problem Defining and proving invariants for Legendrian graphs.
method Defined ruling invariants and proved their properties.
result Ruling invariants are compatible with vertex-identifying operations and vertical cuts/gluings.
In this article we define Lagrangian concordance of Legendrian knots, the analogue of smooth concordance of knots in the Legendrian category. In particular we study the relation of Lagrangian concordance under Legendrian isotopy. The focus is primarily on the algebraic aspects of the problem. We study the behavior of t…
The study finds infinitely many Lagrangian fillings for most Legendrian torus links.
problem Infinitely many Lagrangian fillings for Legendrian torus links except for a few.
method Constructing infinite order Lagrangian concordances and using actions of modular and mapping class groups.
result There exist infinitely many Lagrangian fillings for most Legendrian torus links.
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 reformulate the DGA in terms of front projection, and introduce the characteristic alge…
Researchers develop methods to construct Lagrangian cobordisms between Legendrian knots.
problem Understanding the relationship between Legendrian knots through Lagrangian cobordisms.
method Combinatorial and geometric methods, including Heegaard Floer Homology and contact surgery.
result Construction of nondecomposable Lagrangian cobordisms between Legendrian knots.
Study Lagrangian zigzag cobordisms for Legendrian knots, comparing to smooth concordance.
problem Understanding Legendrian knots through Lagrangian cobordisms.
method Defined an equivalence relation on Legendrian knots using interpolating zigzag Lagrangian cobordisms, studied metric monoid, and compared to other concordance types.
result Proved structural results on torsion and satellite operators in the Lagrangian zigzag concordance classes.
The study finds an obstruction for decomposable exact Lagrangian fillings of certain Legendrian knots.
problem Properties of decomposable exact Lagrangian cobordisms between Legendrian links.
method Analysis of normal rulings associated with decomposable exact Lagrangian fillings.
result A Legendrian knot has a decomposable exact Lagrangian filling if and only if it has an even number of normal rulings.
This is mainly a survey article on the recent development of the theory of graph-like Legendrian unfoldings and its applications. The notion of big Legendrian submanifolds was introduced by Zakalyukin for describing the wave front propagations. Graph-like Legendrian unfoldings belong to a special class of big Legendria…
Proves conditions for generating families on Lagrangian cobordisms.
problem Existence of generating families on Lagrangian cobordisms.
method Analyzes exact Lagrangian cobordisms and Legendrian submanifolds using generating families.
result Conditions for extending generating families linear at infinity.
Sphere theorems for submanifolds in Kähler and Sasaki spaces.
problem Proving sphere theorems for Lagrangian and Legendrian submanifolds.
method Differentiable and topological sphere theorems for submanifolds in Kähler and Sasaki spaces.
result Proved sphere theorems for Lagrangian and Legendrian submanifolds.
Flexible Lagrangians in Weinstein domains lead to new constructions and exotic structures.
problem Understanding regularity and flexibility of Lagrangian manifolds.
method Introducing and discussing regularity and flexibility for Lagrangian manifolds with Legendrian boundary in Weinstein domains.
result Many closed n-manifolds can be exact Lagrangian submanifolds of T∗Sn with exotic Weinstein structures. Functor connects sheaf categories of Legendrian submanifolds.
problem Connecting sheaf categories of Legendrian submanifolds.
method Constructs a functor between sheaf categories using Nadler-Shende's work.
result Provides a sheaf theory description of Lagrangian cobordism.
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 Lagrangian cobordism L⊂X which agrees with cylinders over Legendrian links $…
Projection maps virtual Legendrian knots to classical ones.
problem Classifying virtual Legendrian knots.
method Developed a projection operation from virtual to classical Legendrian knots.
result Virtual crossing number equals classical crossing number.
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 TQFT-like theory that, in addition to giving the aforementioned obstructions, yi…