arXiv research

New method fills cluster seeds with exact Lagrangian structures.

In this paper, we construct the first families of distinct Lagrangian ribbon disks in the standard symplectic 4-ball which have the same boundary Legendrian knots, and are not smoothly isotopic or have non-homeomorphic exteriors.

New surgery operation preserves monotonicity of Lagrangians.

This paper completely answers the question of when contact (r)-surgery on a Legendrian knot in the standard contact structure on the 3-sphere yields a symplectically fillable contact manifold for r in (0,1]. We also give obstructions for other positive r and investigate Lagrangian fillings of Legendrian knots.

Characterizes symplectic rational homology ball fillings of Seifert fibered spaces.

We prove the existence of Lagrangian fillings for $D_n$-type Legendrian links.

The paper introduces triple grid diagrams to construct Lagrangian surfaces in complex projective space.

Worldsheet skein D-module for Hopf link conormal uniquely determines partition functions.

New disks found with similar outer shapes.

Symplectic fillings of prequantization bundles are shown to be disk bundles under certain conditions.

Study finds many Lagrangian fillings for certain Legendrian links.

The paper connects different types of Lagrangian fillings to Legendrian weaves and their sheaf quantizations.

Study uses Newton polytopes to distinguish Lagrangian fillings of Legendrian submanifolds.

For a Legendrian $(2,n)$ torus knot or link with maximal Thurston-Bennequin number, Ekholm, Honda, and Kálmán constructed $C_n$ exact Lagrangian fillings, where $C_n$ is the $n$-th Catalan number. We show that these exact Lagrangian fillings are pairwise non-isotopic through exact Lagrangian isotopy. To do that, we com…

We prove a relative isoperimetric inequalities for Lagrangian half disks in $\mathbb{C}^2$ with respect to a Lagrangian plane, or a complex plane, or a union of any two of Lagrangian or complex planes that intersect transversally at the origin.

This paper classifies and determines the length of the shortest filling pairs on a specific type of surface.

Holomorphic disks on compact Lagrangian surfaces are shown to exist.

The study finds infinitely many Lagrangian fillings for most Legendrian torus links.

The paper proves there are many Lagrangian fillings for Legendrian links of affine type.

We study Weinstein 4-manifolds which admit Lagrangian skeleta given by attaching disks to a surface along a collection of simple closed curves. In terms of the curves describing one such skeleton, we describe surgeries that preserve the ambient Weinstein manifold, but change the skeleton. The surgeries can be iterated …

Positive braids have endless filling possibilities.

Study on Legendrian knots and their non-orientable Lagrangian fillings.

Study of Legendrian links using Floer theory and cluster varieties.

Study on the minimum length of curves on once-punctured hyperbolic surfaces.

We study the topology of exact and Stein fillings of the canonical contact structure on the unit cotangent bundle of a closed surface $Σ_g$, where $g$ is at least 2. In particular, we prove a uniqueness theorem asserting that any Stein filling must be s-cobordant rel boundary to the disk cotangent bundle of $Σ_g$. For …

The study finds many Lagrangian fillings for Legendrian links of specific types.

The paper develops spectral networks in symplectic topology and their relation to Lagrangian fillings.

Unbraided wiring diagrams for Stein fillings of lens spaces are described.

Positive surgery on knots in weakly fillable 3-manifolds yields weakly fillable contact structures.

We considered a surgery, called Lagrangian attaching disk surgery, that can be applied to a Lagrangian surface L at the presence of a Lagrangian attaching disk D, to obtain a new Lagrangian surface L' which is always smoothly isotopic to L. We showed that this type of surgery includes all even generalized Dehn twists a…

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…

We study some properties of decomposable exact Lagrangian cobordisms between Legendrian links in $\mathbb{R}^3$ with the standard contact structure. In particular, for any decomposable exact Lagrangian filling $L$ of a Legendrian link $K$, we may obtain a normal ruling of $K$ associated with $L$. We prove that the asso…

The paper connects Legendrian links to cluster theory and exact Lagrangian fillings.

We characterize which Legendrian $4$-plat knots in the standard contact $3$-space have exact orientable Lagrangian fillings. As a corollary, we show that the underlying smooth knot types of fillable Legendrian $4$-plats are positive.

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 of algebraic curves in $B^4 \subset \mathbb{C}^2$, is applied to find contact 3-…

The article finds conditions for separating filling pairs on surfaces and constructs a Morse function.

We use holomorphic disks to describe the formation of singularities in the mean curvature flow of monotone Lagrangian submanifolds in $\mathbb C^{n}$.

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…

We classify the resolution graphs of weighted homogeneous surface singularities which admit rational homology disk smoothings. The nonexistence of rational homology disk smoothings is shown by symplectic geometric methods, while the existence is verified via smoothings of negative weights. In particular, it is shown th…

We give an algorithm which produces infinitely many pairwise exotic Stein fillings of the same contact 3-manifolds, applying positive allowable Lefschetz fibrations over the disk. As a corollary, for a large class of Stein fillings, we realize the topological invariants (i.e. fundamental group, homology group, homology…

New augmentations of twist knots found that can't be filled.

Study shows infinite Hofer diameter for Lagrangian orbits in cotangent bundles.

The paper classifies symplectic fillings of lens spaces and constructs cobordisms.

We prove that any minimal weak symplectic filling of the canonical contact structure on the unit cotangent bundle of a nonorientable closed surface other than the real projective plane is s-cobordant rel boundary to the disk cotangent bundle of the surface. If the nonorientable surface is the Klein bottle, then we show…

We give simple examples of elements of SL(2,Z) admitting inequivalent factorizations into products of Dehn twists. This can be interpreted in terms of inequivalent Stein fillings of a same contact 3-manifold by genus 1 Lefschetz fibrations over the disk.

We construct a positive allowable Lefschetz fibration over the disk on any minimal weak symplectic filling of the canonical contact structure on a lens space. Using this construction we prove that any minimal symplectic filling of the canonical contact structure on a lens space is obtained by a sequence of rational blo…

We resolve parts (A) and (B) of Problem 1.100 from Kirby's list by showing that many nontrivial links arise as cross-sections of unknotted holomorphic disks in the four-ball. The techniques can be used to produce unknotted ribbon surfaces with prescribed cross-sections, including unknotted Lagrangian disks with nontriv…

The study explores Legendrian fillings and augmentations, providing methods to compute induced augmentations.