arXiv research

Proves Weinstein conjecture for specific contact manifolds.

The study proves that certain surface singularities are planar only for $A_n$-singularities.

In this note we observe that while all overtwisted contact structures on compact 3--manifolds are supported by planar open book decompositions, not all contact structures are. This has relevance to invariants of contact structures and also to the Weinstein conjecture via work of Abbas Cieliebak and Hofer.

We study fillings of contact structures supported by planar open books by analyzing positive factorizations of their monodromy. Our method is based on Wendl's theorem on symplectic fillings of planar open books. We prove that every virtually overtwisted contact structure on L(p,1) has a unique filling, and describe fil…

We study an explicit construction of planar open books with four binding components on any three-manifold which is given by integral surgery on three component pure braid closures. This construction is general, indeed any planar open book with four binding components is given this way. Using this construction and resul…

We investigate the line between tight and overtwisted for surgeries on fibred transverse knots in contact 3-manifolds. When the contact structure $ξ_K$ is supported by the fibred knot $K \subset M$, we obtain a characterisation of when negative surgeries result in a contact structure with non-vanishing Heegaard Floer c…

In this paper, we focus on contact structures supported by planar open book decompositions. We study right-veering diffeomorphisms to keep track of overtwistedness property of contact structures under some monodromy changes. As an application we give infinitely many examples of overtwisted contact structures supported …

We construct a family of Stein fillable contact homology 3-spheres such that each contact structure of the family is supported by an open book with planar page, and a Stein filling of the contact manifold is of Mazur type.

Planar multilinks prove rational singularities in surface geometry.

In this article, we find the complete list of all contact structures (up to isotopy) on closed three-manifolds which are supported by an open book decomposition having planar pages with three (but not less) boundary components. We distinguish them by computing their first Chern classes and three dimensional invariants …

Study on binding numbers of tight contact structures on lens spaces $L(n,1)$.

Giroux has described a correspondence between open book decompositions on a 3--manifold and contact structures. In this paper we use Heegaard Floer homology to give restrictions on contact structures which correspond to open book decompositions with planar pages, generalizing a recent result of Etnyre.

Study of higher-dimensional contact manifolds and their properties.

The aim of this paper is to use mapping class group relations to approach the `geography' problem for Stein fillings of a contact 3-manifold. In particular, we adapt a formula of Endo and Nagami so as to calculate the signature of such fillings as a sum of the signatures of basic relations in the monodromy of a related…

In this note we introduce the (homologically essential) arc complex of a surface as a tool for studying properties of open book decompositions and contact structures. After characterizing destabilizability in terms of the essential translation distance of the monodromy of an open book we given an application of this re…

Study Stein fillings of planar contact 3-manifolds with relative trisection genus 2.

We use contact fiber sums of open book decompositions to define an infinite hierarchy of filling obstructions for contact 3-manifolds, called planar k-torsion for nonnegative integers k, all of which cause the contact invariant in Embedded Contact Homology to vanish. Planar 0-torsion is equivalent to overtwistedness, w…

Using open book foliations we show that an overtwisted disc in a planar open book can be put in a topologically nice position. As a corollary, we prove that a planar open book whose fractional Dehn twist coefficients grater than one for all the boundary components supports a tight contact structure.

We generalize the familiar notions of overtwistedness and Giroux torsion in 3-dimensional contact manifolds, defining an infinite hierarchy of local filling obstructions called planar torsion, whose integer-valued order $k \ge 0$ can be interpreted as measuring a gradation in "degrees of tightness" of contact manifolds…

In this note, we exhibit infinite families of tight non-fillable contact manifolds supported by planar open books with vanishing Heegaard Floer contact invariants. Moreover, we also exhibit an infinite such family where the supported manifold is hyperbolic.

We exhibit an infinite family of tight contact structures with the property that none of the supporting open books minimizes the genus and maximizes the Euler characteristic of the page simultaneously, answering a question of Baldwin and Etnyre in arXiv:0910.5021 .

Infinitely many 3D shapes have multiple ways to be filled with special surfaces.

We characterize L-spaces which are Seifert fibered over the 2-sphere in terms of taut foliations, transverse foliations and transverse contact structures. We give a sufficient condition for certain contact Seifert fibered 3-manifolds with e_0=-1 to have nonzero contact Ozsvath--Szabo invariants. This yields an algorith…

Higher-dimensional contact manifolds lack certain properties.

We show that an immersed minimal annulus, with two planar boundary curves along which the surface meets these planes with constant contact angle, is part of the catenoid.

In this note, we define a new invariant of a Legendrian knot in a contact manifold using an open book decomposition supporting the contact structure. We define the support genus sg(L) of a Legendrian knot L in a contact 3-manifold (M, ξ) as the minimal genus of a page of an open book of M supporting the contact structu…

We prove that if a contact manifold $(M,ξ)$ is supported by a planar open book, then Euler characteristic and signature of any Stein filling of $(M,ξ)$ is bounded. We also prove a similar finiteness result for contact manifolds supported by spinal open books with planar pages. Moving beyond the geography of Stein filli…

Real algebraic structures help classify overtwisted contact 3-spheres.

The paper derives a signature formula for Lefschetz fibrations with planar fibers.

The contact invariant in monopole Floer homology behaves naturally under strong symplectic cobordisms.

We construct a Seifert surface for a given null-homologous transverse link in a contact manifold that is compatible with a planar open book decomposition, then obtain a formula of the self-linking number. It extends Bennequin's self-linking number formula for braids in the standard contact 3-sphere.

New open books solve a long-standing surface mapping class group question.

On small Seifert fibered spaces $M(e_0;r_1,r_2,r_3)$ with $e_0\neq-1,-2,$ all tight contact structures are Stein fillable. This is not the case for $e_0=-1$ or $-2$. However, for negative twisting structures it is expected that they are all symplectically fillable. Here, we characterize fillable structures among zero-t…

New open books defy positive factorisation in genus one.

We describe explicit open books on arbitrary plumbings of oriented circle bundles over closed oriented surfaces. We show that, for a non-positive plumbing, the open book we construct is horizontal and the corresponding compatible contact structure is also horizontal and Stein fillable. In particular, we describe horizo…

In this paper, we show that the Ozsváth-Szabó contact invariant $c^+(ξ)\in HF^+(-Y)$ of a contact 3-manifold $(Y,ξ)$ can be calculated combinatorially if $Y$ is the boundary of a certain type of plumbing $X$, and $ξ$ is induced by a Stein structure on $X$. Our technique uses an algorithm of Ozsváth and Szabó to determi…

New proof shows unique symplectic fillings for certain surface singularity links.

Study symplectic fillings of lens spaces, focusing on virtually overtwisted contact structures.

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…

A relatively common sight in graphic designs is a planar arrangement of three gears in contact. However, since neighboring gears must rotate in opposite directions, none of the gears can move. We give a non-planar, and non-frozen, arrangement of three linked gears.

This paper classifies symplectic and Stein fillings of contact 3-manifolds with spinal open book decompositions.

Thanks to a result of Lisca and Matic and a refinement by Plamenevskaya, it is known that on a 4-manifold with boundary Stein structures with non-isomorphic Spinc structures induce contact structures with distinct Ozsvath-Szabo invariants. Here we give an infinite family of examples showing that converse of Lisca-Matic…

Study Legendrian surfaces using N-graphs and flag moduli.

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

We show that the planar circular restricted three body problem is of restricted contact type for all energies below the first critical value (action of the first Lagrange point) and for energies slightly above it. This opens up the possibility of using the technology of Contact Topology to understand this particular dy…

Suppose that S is a surface with boundary and that g and h are diffeomorphisms of S which restrict to the identity on the boundary. Let Y_g, Y_h, and Y_{hg} be the three-manifolds with open book decompositions given by (S,g), (S,h), and (S,hg), respectively. We show that the Ozsvath-Szabo contact invariant is natural u…

New geometric constructions for contact manifolds, not symplectically fillable.

New tools classify symplectic fillings of contact 3-manifolds.