arXiv research

New method fills cluster seeds with exact Lagrangian structures.

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…

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.

Positive braids have endless filling possibilities.

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

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

Study of Legendrian links using Floer theory and cluster varieties.

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.

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-…

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.

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

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

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

New examples of Legendrian links with infinitely many fillings.

Characterizes symplectic rational homology ball fillings of Seifert fibered spaces.

Torsion found in knot homology, challenging augmentation theories.

The study shows knots from 3-braids cannot be concordant to a specific Legendrian unknot.

Proves conditions for generating families on Lagrangian cobordisms.

The paper connects Legendrian links to cluster algebras via microlocal methods.

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.

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

Book teaches how Lagrangian torus fibration base geometry can be read off.

In this article we study complex properties of minimal Lagrangian submanifolds in Kaehler ambient spaces, and how they depend on the ambient curvature. In particular, we prove that, in the negative curvature case, minimal Lagrangians do not admit fillings by holomorphic discs. The proof relies on a mix of holomorphic c…

Constructs Lagrangian skeleta for curve singularities.

Many interesting spaces --- including all positroid strata and wild character varieties --- are moduli of constructible sheaves on a surface with microsupport in a Legendrian link. We show that the existence of cluster structures on these spaces may be deduced in a uniform, systematic fashion by constructing and taking…

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…

Assume that we are given a closed chord-generic Legendrian submanifold $Λ\subset P \times \mathbb R$ of the contactisation of a Liouville manifold, where $Λ$ moreover admits an exact Lagrangian filling $L_Λ \subset \mathbb R \times P \times \mathbb R$ inside the symplectisation. Under the further assumptions that this …

New surgery operation preserves monotonicity of Lagrangians.

We derive constraints on Lagrangian embeddings in completions of certain stable symplectic fillings with semisimple symplectic cohomologies. Manifolds with these properties can be constructed by generalizing the boundary connected sum operation to our setting, and are related to certain birational surgeries like blow-d…

Functor connects sheaves on Lagrangian cobordisms, proving equivalence and action decreasing properties.

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 $L\subset X$ which agrees with cylinders over Legendrian links $…

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 …

BALLAST optimizes Lagrangian observer placement for ocean vector fields.

In the scientific literature there are basically two schools of formulating Lagrangian (or Hamiltonian) mechanics in the (Lie) algebroid setting: in terms of prolongations and in terms of Tulczyjew triples. Despite the fact that in both approaches we describe the same phenomena, so far no comparison between prolongatio…

Study Legendrian surfaces using N-graphs and flag moduli.

If a Legendrian knot $Λ$ in the standard contact 3-sphere bounds an orientable exact Lagrangian surface $Σ$ in the standard symplectic 4-ball, then the genus of $Σ$ is equal to the slice genus of (the smooth knot underlying) $Λ$, the sum of the Thurston-Bennequin number of L and the Euler characteristic of $Σ$ is zero …

New method constructs small symplectic 4-manifolds via contact gluing.

We study the connection between topological strings and contact homology recently proposed in the context of knot invariants. In particular, we establish the proposed relation between the Gromov-Witten disk amplitudes of a Lagrangian associated to a knot and augmentations of its contact homology algebra. This also impl…

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