arXiv research

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

New disks fill Legendrian knots without being smoothly isotopic.

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

New method fills cluster seeds with exact Lagrangian structures.

Connected sums of quasipositive links are quasipositive.

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 …

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

Characterizes symplectic rational homology ball fillings of Seifert fibered spaces.

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

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…

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…

New Stein fillings found for rational surface singularities.

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.

Sharp bounds on Euler characteristics for lens space fillings.

New surgery operation preserves monotonicity of Lagrangians.

Optimal inequalities for metric surfaces derived from filling minimality.

A manifold M is simple if it contains no essential disk, sphere, annulus or torus. If M is simple and two Dehn fillings M(r_1), M(r_2) are nonsimple, then there is an upper bound on Δ(r_1,r_2), the geometric intersection number between r_1 and r_2. There are 10 possibilities, depending on the types of M(r_i). In this p…

New Stein fillings found for non-weighted homogeneous singularities.

The homological and homotopical Dehn functions are different ways of measuring the difficulty of filling a closed curve inside a group or a space. The homological Dehn function measures fillings of cycles by chains, while the homotopical Dehn function measures fillings of curves by disks. Since the two definitions invo…

The study describes Lefschetz-Bott fibrations on symplectic manifolds and their applications.

Algorithm decides if geodesic curves are filling on surfaces.

The exceptional Dehn filling conjecture of the second author concerning the relationship between exceptional slopes $α, β$ on the boundary of a hyperbolic knot manifold $M$ has been verified in all cases other than small Seifert filling slopes. In this paper we verify it when $α$ is a small Seifert filling slope and $β…

We answer a question of Liokumovich-Nabutovsky-Rotman showing that if D is a Riemannian 2-disc with boundary length L, diameter d and area A << d then D can be filled by a homotopy where the lengths of the intermediate curves are bounded by $L+2d+O(\sqrt A)$.

An important class of contact 3--manifolds are those that arise as links of rational surface singularities with reduced fundamental cycle. We explicitly describe symplectic caps (concave fillings) of such contact 3--manifolds. As an application, we present a new obstruction for such singularities to admit rational homo…

We show that there are vast families of contact 3-manifolds each member of which admits infinitely many Stein fillings with arbitrarily big euler characteristics and arbitrarily small signatures ---which disproves a conjecture of Stipsicz and Ozbagci. To produce our examples, we set a framework which generalizes the co…

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

RODMAN improves ML-based disk failure prediction accuracy in cloud environments.

Study symplectic fillings of sandwiched singularities.

P. Papasoglu asked in [Pap13] whether for any Riemannian 3-disk $M$ with diameter $d$, boundary area $A$ and volume $V$, there exists a homotopy $S_t$ contracting the boundary to a point so that the area of $S_t$ is bounded by $f(d,A,V)$ for some function $f$. He further asks whether it is possible to subdivide $M$ by …

This paper tackles Gromov's filling area conjecture using discrete graph theory.

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

It is shown that given any link-manifold, there is an algorithm to decide if the manifold contains an embedded, essential planar surface; if it does, the algorithm will construct one. If a slope on the boundary of the link-manifold is given, there is an algorithm to determine if the slope bounds an embedded punctured-d…

The paper proposes a new way to approximate Riemannian metrics using discrete wall systems.

Unique CaTherine wheel found for LQG geodesic tree.

Geometrically, Legendrian surfaces related by surgery have related skein-valued cluster spaces.

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

This paper reveals hidden hyperbolic structures in divide links.

Fixed points found in Teichmüller space via anti-de Sitter geometry.

Reconstruct flows and manifolds from their boundary actions on circles.

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

Let $S$ be a Riemann surface of type $(p,n)$ with $3p-3+n>0$. Let $ω$ be a pseudo-Anosov map of $S$ that is obtained from Dehn twists along two families $\{A,B\}$ of simple closed geodesics that fill $S$. Then $ω$ can be realized as an extremal Teichmüller mapping on a surface of type $(p,n)$ which is also denoted by $…

It is known that for coprime integers $p>q\geq 1$, the lens space $L(p^2,pq-1)$ bounds a rational ball, $B_{p,q}$, arising as the 2-fold branched cover of a (smooth) slice disk in $B^4$ bounding the associated 2-bridge knot. Lekilli and Maydanskiy give handle decompositions for each $B_{p,q}$. Whereas, Yamada gives an …