Researchers found unique exact Lagrangian fillings for a specific type of knot.
problem Tackling the uniqueness of exact Lagrangian fillings for Legendrian (2,n) torus knots. method Computed augmentations induced by exact Lagrangian fillings to distinguish them.
result Identified that these exact Lagrangian fillings are pairwise non-isotopic.
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.
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.
New method fills cluster seeds with exact Lagrangian structures.
problem Surjectivity of map from Lagrangian fillings to cluster seeds.
method Construction of quiver with potential and Lagrangian disk surgeries.
result Trivial deformation space of quiver allows manipulation of fillings.
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.
Positive braids have endless filling possibilities.
problem Understanding exact Lagrangian fillings of positive braid links.
method Analyzing positive braid Legendrian links and their fillings.
result Positive braid Legendrian links have infinitely many exact Lagrangian fillings.
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.
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.
This paper explores the relationship between the existence of an exact embedded Lagrangian filling for a Legendrian knot in the standard contact $\rr^3$ and the hierarchy of positive, strongly quasi-positive, and quasi-positive knots. On one hand, results of Eliashberg and especially Boileau and Orevkov show that every…
The paper connects Legendrian links to cluster theory and exact Lagrangian fillings.
problem Understanding the relationship between Legendrian links and cluster theory.
method Using exact Lagrangian fillings and cluster theory, the paper establishes connections between Legendrian links and cluster varieties.
result The augmentation variety of certain Legendrian 2-bridge links is isomorphic to a product of cluster varieties.
We prove the existence of Lagrangian fillings for Dn-type Legendrian links.
problem Exact Lagrangian fillings of Legendrian links of Dn-type. method Legendrian weave calculus and construction of 1-cycles.
result Existence of a Lagrangian filling represented by a weave.
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.
The study finds many Lagrangian fillings for Legendrian links of specific types.
problem Understanding the number and types of Lagrangian fillings for Legendrian links.
method Proved the existence of at least as many exact embedded Lagrangian fillings as seeds for Legendrian links of finite or affine Dynkin type.
result Found many Lagrangian fillings with rotational and conjugation symmetries for specific types of Legendrian links.
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.
Torsion found in knot homology, challenging augmentation theories.
problem Torsion in linearized contact homology for Legendrian knots.
method Examples of Legendrian knots with non-trivial homology over Z.
result Augmentations not induced by exact fillings, even mod 2.
Study on Legendrian knots and their non-orientable Lagrangian fillings.
problem Conditions for Legendrian knots to have non-orientable exact Lagrangian fillings.
method Developed combinatorial obstructions and classified fillability for various knot families.
result Completely determined decomposably non-orientable fillability for alternating and plus-adequate knots.
New knots found without Lagrangian fillings.
problem Existence of Legendrian knots without Lagrangian fillings.
method Provided a family of Legendrian knots with augmentations not induced by any exact Lagrangian filling.
result Negative answer to the converse of the original statement.
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.
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 $…
The study establishes a lower bound for the number of Reeb chords using stable Morse numbers.
problem Finding a lower bound for the number of Reeb chords on Legendrian submanifolds.
method Using stable Morse numbers and properties of Lagrangian fillings, the study proves a lower bound for the number of Reeb chords.
result The number of Reeb chords is bounded from below by the stable Morse number of the Lagrangian filling.
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.
New examples of Legendrian links with infinitely many fillings.
problem Understanding the structure of Legendrian links and their fillings.
method New combinatorial formula for Legendrian contact DGAs and Floer-theoretic techniques.
result Construction of the first families of Legendrian links with infinitely many Lagrangian fillings.
The study explores Legendrian fillings and augmentations, providing methods to compute induced augmentations.
problem Understanding and computing Legendrian isotopy invariants through augmentations and fillings.
method Developed methods to compute induced augmentations based on Morse complex families and Legendrian cobordisms.
result Established methods to compute Legendrian isotopy invariants using augmentations and fillings.
New surgery operation preserves monotonicity of Lagrangians.
problem Preserving monotonicity of Lagrangians in surgery operations.
method BSP surgery, wall-crossing formula for disk-potentials.
result BSP surgery can preserve monotonicity of Lagrangians.
The paper connects different types of Lagrangian fillings to Legendrian weaves and their sheaf quantizations.
problem Understanding and comparing different types of Lagrangian fillings of Legendrian weaves.
method Establishing new Reidemeister moves and combinatorial isotopies between Lagrangian fillings, comparing sheaf quantizations.
result Legendrian weaves generalize previously known methods to produce infinitely many distinct Lagrangian fillings.
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…
Paper disproves a conjecture with a new family of knots.
problem Proving the existence of decomposable exact Lagrangian fillings.
method Presented an infinite family of Legendrian knots with only 1 normal ruling.
result These knots do not satisfy the even number of clasps condition, disproving a conjecture.
Functor connects sheaves on Lagrangian cobordisms, proving equivalence and action decreasing properties.
problem Understanding sheaf equivalences and actions on Lagrangian cobordisms.
method Analyzing sheaf quantizations and Legendrian lifts, proving functorial properties.
result Lagrangian cobordism functor is action decreasing and Morita equivalent to sheaf categories of Legendrians.
The paper develops spectral networks in symplectic topology and their relation to Lagrangian fillings.
problem Understanding spectral networks in symplectic topology and their role in Lagrangian fillings.
method Analytic results on adiabatic degeneration of Floer trajectories and explicit computation of continuation strips.
result Established equivalence between Family Floer functor and non-abelianization functor for Lagrangian fillings with spectral networks.
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.
Study on exact and Stein fillings of unit cotangent bundles of surfaces.
problem Topology of exact and Stein fillings of unit cotangent bundles.
method Proved uniqueness theorem and showed homology results.
result Uniqueness theorem for Stein fillings and homology results for exact fillings.
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.
We construct infinitely many Legendrian links in the standard contact R3 with arbitrarily many topologically distinct Lagrangian fillings. The construction is used to find links in S3 that bound topologically distinct pieces of algebraic curves in B4⊂C2, is applied to find contact 3-…
Develops a diagrammatic method for symplectic filling classifications.
problem Classifying exact/weak symplectic fillings of 3D contact manifolds.
method Symplectic JSJ decomposition applied to contact surgery diagrams.
result Recover symplectic fillings for certain lens spaces and torus bundles, and classify fillings for a large class of plumbed 3-manifolds.
The abstract discusses applications of Menke's JSJ decomposition to symplectic fillings of various 3-manifolds.
problem Classifying symplectic fillings of contact 3-manifolds.
method Application of Menke's JSJ decomposition to families of contact 3-manifolds.
result Unique exact fillings for virtually overtwisted circle bundles over surfaces with genus > 1 and negative twisting number.
The study explores surgeries on Lagrangian skeleta in 4-manifolds, connecting geometric and algebraic structures.
problem Understanding the combinatorics of exact Lagrangian surfaces in 4-manifolds.
method Analyzes surgeries on Lagrangian skeleta and their impact on the Fukaya category and cluster transformations.
result Induces cluster transformations on spaces of local systems and higher rank local systems.
New constraints found for Lagrangian embeddings in symplectic fillings.
problem Understanding Lagrangian embeddings in symplectic fillings with semisimple cohomology.
method Deriving constraints through Lagrangian embeddings and symplectic cohomology analysis.
result Existence of many non-toric monotone symplectic manifolds with proper wrapped Fukaya categories.
The paper solves when surgeries on Legendrian knots yield symplectically fillable contact manifolds.
problem When contact surgeries on Legendrian knots yield symplectically fillable contact manifolds.
method Analyzes contact (r)-surgeries on Legendrian knots and investigates Lagrangian fillings.
result Completely answers the question of symplectically fillable contact manifolds after contact surgeries.
We provide in this note two relevant examples of Lagrangian cobordisms. The first one gives an example of two exact Lagrangian submanifolds which cannot be composed in an exact fashion. The second one is an example of an exact Lagrangian cobordism on which all primitive of the Liouville form is not constant on the nega…
Study exact Lagrangian cobordisms in cotangent bundles, proving bounds on sheaf interleaving distance and shadow distance.
problem Understanding Lagrangian cobordisms and their properties in cotangent bundles.
method Use microlocal theory of sheaves, sheaf quantization, and cone decompositions.
result Interleaving distance of sheaves is bounded by the shadow distance of the cobordism.
New augmentations of twist knots found that can't be filled.
problem Finding augmentations of twist knots that cannot be filled by orientable Lagrangian fillings.
method Using a Floer-theoretic version of a result from microlocal sheaf theory, showing augmentations cannot be induced by algebraic tori.
result Established new examples of augmentations of Legendrian twist knots that cannot be induced by orientable Lagrangian fillings.
We introduce symplectic Calabi-Yau caps to obtain new obstructions to exact fillings. In particular, it implies that any exact filling of the standard unit cotangent bundle of a hyperbolic surface has vanishing first Chern class and has the same integral homology and intersection form as its disk cotangent bundle. This…
Study exact Lagrangian cobordisms between Legendrian knots using functorial properties of augmentation categories.
problem Understanding exact Lagrangian cobordisms between Legendrian knots.
method Study the functor between augmentation categories induced by exact Lagrangian cobordisms and establish a long exact sequence.
result Prove the functor between augmentation categories is injective on the level of equivalence classes of objects and find new obstructions to exact Lagrangian cobordisms.
The paper proves almost positive links can be filled with Lagrangian structures.
problem Understanding Lagrangian fillability of specific link types.
method Analyzing almost positive diagrams with a specific condition.
result Almost positive links are Lagrangian fillable under certain conditions.
Paper constructs exact discrete Lagrangian for variational integrators.
problem Error analysis between exact and discrete trajectories.
method Construction of exact discrete Lagrangian in Lie groupoid and algebroid setting.
result Rigorous construction of exact discrete Lagrangian for variational integrators.
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. New method bounds intersections of exact Lagrangians in cotangent bundles.
problem Counting intersections of exact Lagrangian submanifolds in cotangent bundles.
method Sheaf quantization in Tamarkin's category for clean and degenerate intersections.
result Cardinality of intersections is bounded by sheaf Hom spaces.
Classifies symplectic fillings of specific torus bundles.
problem Classifying strong and exact symplectic fillings of virtually overtwisted torus bundles.
method Using Menke's JSJ-type decomposition theorem and round symplectic 1-handle attachment.
result Conditions for distinct tight lens space fillings to yield the same torus bundle filling.