arXiv research

New surgery operation preserves monotonicity of Lagrangians.

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…

The paper proves existence and behavior of Lagrangian tori in complex projective plane.

Study shows links with trivial Lagrangian are slice, useful for surgery problems.

In this note we show that the Lagrangian Luttinger surgery preserves the symplectic Kodaira dimension. Some constraints on Lagrangian tori in symplectic four manifolds with non-positive Kodaira dimension are also derived.

New method fills cluster seeds with exact Lagrangian structures.

The paper solves when surgeries on Legendrian knots yield symplectically fillable contact manifolds.

We discuss the properties of a certain type of Dehn surgery along a Lagrangian torus in a symplectic 4-manifold, known as Luttinger's surgery, and use this construction to provide a purely topological interpretation of a non-isotopy result for symplectic plane curves with cusp and node singularities due to Moishezon.

No Lagrangian Klein bottles found in $S^2 imes S^2$.

Null Lagrangian-preserving surgeries are a generalization of the Garoufalidis and Rozansky null-moves, that these authors introduced to study the Kricker lift of the Kontsevich integral, in the setting of pairs (M,K) composed of a rational homology sphere M and a null-homologous knot K in M. They are defined as replace…

New formulas for knot invariants under specific surgeries.

Researchers develop methods to construct Lagrangian cobordisms between Legendrian knots.

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 …

Study confirms contact cosmetic surgery for most knots, with exceptions.

Study SYZ transforms for immersed Lagrangian multi-sections in symplectic geometry.

We introduce a surgery operation on symplectic manifolds called coisotropic Luttinger surgery, which generalizes Luttinger surgery on Lagrangian tori in symplectic 4-manifolds. We use it to produce infinitely many distinct symplectic non-Kahler 6-manifolds $X$ with $c_1(X)=0$ which are not of the form $M\times F$ for $…

Study extends contact cosmetic surgeries to non-trivial Legendrian knots in L-spaces.

We define an simple invariant of an embedded nullhomologous Lagrangian torus and use this invariant to show that many symplectic 4-manifolds have infinitely many pairwise symplectically inequivalent nullhomologous Lagrangian tori. We further show that for a large class of examples that lambda(T) is actually a C-infinit…

The study connects knot Floer homology and Lagrangian cobordisms.

Constructs invariants from Lagrangian tori in symplectic 4-manifolds.

Study finite type invariants for knots in rational homology 3-spheres.

We construct many self-similar and translating solitons for Lagrangian mean curvature flow, including self-expanders and translating solitons with arbitrarily small oscillation on the Lagrangian angle. Our translating solitons play the same role as cigar solitons in Ricci flow, and are important in studying the regular…

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…

The paper extends Gompf's invariant to all compact 3-manifolds and studies its variations.

Let $π: E\to M$ be a smooth fiber bundle whose total space is a symplectic manifold and whose fibers are Lagrangian. Let $L$ be an embedded Lagrangian submanifold of $E$. In the paper we address the following question: how can one simplify the singularities of the projection $π: L \to M$ by a Hamiltonian isotopy of $L$…

Introduces integer-valued Heegaard Floer theory with canonical orientations.

New symplectic instanton homology constructed via traceless character varieties.

The abstract discusses constructing 3d N=2 gauge theories using surgeries and M5-branes.

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

New proof of Heegaard Floer surgery formulas using Fukaya category of the torus.

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

In this article we use the technique of Luttinger surgery to produce small examples of simply connected and non-simply connected minimal symplectic 4-manifolds. In particular, we construct: (1) An example of a minimal symplectic 4-manifold that is homeomorphic but not diffeomorphic to CP^2#3(-CP^2) which contains a sym…

Given a mapping class f of an oriented surface Sigma and a lagrangian lambda in the first homology of Sigma, we define an integer n_{lambda}(f). We use n_{lambda}(f) (mod 4) to describe a universal central extension of the mapping class group of Sigma as an index-four subgroup of the extension constructed from the Masl…

We prove that the stable moduli space of $(n-1)$-connected, $n$-parallelizable, $(2n+1)$-dimensional manifolds is homology equivalent to an infinite loopspace for $n \geq 4, n \neq 7$. The main novel ingredient is a version of the cobordism category incorporating surgery data in the form of Lagrangian subspaces.

In a previous article, we constructed an invariant Z for null-homologous knots in rational homology spheres, from equivariant intersections in configuration spaces. Here we present an equivalent definition of Z in terms of configuration space integrals, we prove that Z is multiplicative under connected sum, and we prov…

The paper develops a new homology theory for 3-manifolds and proves its functoriality.

Almost toric manifolds form a class of singular Lagrangian fibered symplectic manifolds that is a natural generalization of toric manifolds. Notable examples include the K3 surface, the phase space of the spherical pendulum and rational balls useful for symplectic surgeries. The main result of the paper is a complete c…

Let $M$ be a Calabi-Yau $m$-fold, and consider compact, graded Lagrangians $L$ in $M$. Thomas and Yau math.DG/0104196, math.DG/0104197 conjectured that there should be a notion of "stability" for such $L$, and that if $L$ is stable then Lagrangian mean curvature flow $\{L^t:t\in[0,\infty)\}$ with $L^0=L$ should exist f…

We use monopole Floer homology for sutured manifolds to construct invariants of Legendrian knots in a contact 3-manifold. These invariants assign to a knot K in Y elements of the monopole knot homology KHM(-Y,K), and they strongly resemble the knot Floer homology invariants of Lisca, Ozsváth, Stipsicz, and Szabó. We pr…

We present an alternative definition for the Goussarov--Habiro filtration of the Z-module freely generated by oriented integral homology 3-spheres, by means of Lagrangian-preserving homology handlebody replacements (LP-surgeries). Garoufalidis, Goussarov and Polyak proved that the graded space (G_n)_n associated to thi…

Given a null-homologous knot $K$ in a rational homology 3-sphere $M$, and the standard infinite cyclic covering $\tilde{X}$ of $(M,K)$, we define an invariant of triples of curves in $\tilde{X}$, by means of equivariant triple intersections of surfaces. We prove that this invariant provides a map $φ$ on $\Al^{\otimes 3…

Knot Floer homology is reinterpreted as immersed curves.

Geometrically generates Fukaya categories of Weinstein manifolds.

We investigate the Dolbeault operator on a pair of pants, i.e., an elementary cobordism between a circle and the disjoint union of two circles. This operator induces a canonical selfadjoint Dirac operator $D_t$ on each regular level set $C_t$ of a fixed Morse function defining this cobordism. We show that as we approac…

Formula connects surgeries to Seiberg-Witten invariants.

Let K be a knot in the 3--sphere. An r-surgery on K is left-orderable if the resulting 3--manifold K(r) of the surgery has left-orderable fundamental group, and an r-surgery on K is called an L-space surgery if K(r) is an L-space. A conjecture of Boyer, Gordon and Watson says that non-reducing surgeries on K can be cla…

Study cosmetic surgeries on knots in homology spheres using Casson-Walker invariant.

Cosmetic surgeries on pretzel knots are unique.