arXiv research

Proves existence of strongly overtwisted contact structures on 3-manifolds.

The study finds algebraically overtwisted tight 3-manifolds via contact surgeries.

Contact surgeries yield algebraically overtwisted manifolds.

Classifies convex disks with Legendrian boundary in overtwisted contact 3-manifolds.

We construct (infinitely many) examples in all dimensions of contactomorphisms of closed overtwisted contact manifolds that are smoothly isotopic but not contact-isotopic to the identity.

Study links in contact manifolds using open books and overtwisted disks.

We give a proof of, for the case of contact structures defined by global contact 1-forms, a Theorem stated by Eliashberg that for any overtwisted contact structure on a closed 3-manifold, its contact homology is 0. A different proof is also outlined in the appendix by Yakov Eliashberg.

In this note, we obtain a new result concluding when contact (+1/n)-surgery is overtwisted. We give a counterexample to a conjecture by James Conway on overtwistedness of manifolds obtained by contact surgery. We list some problems related to the contact surgery.

We establish a parametric extension $h$-principle for overtwisted contact structures on manifolds of all dimensions, which is the direct generalization of the $3$-dimensional result from \cite{Eli89}. It implies, in particular, that any closed manifold admits a contact structure in any given homotopy class of almost co…

The paper deals with topologically trivial Legendrian knots in tight and overtwisted contact 3-manifolds. The first part contains a thorough exposition of the proof of the classification of topologically trivial Legendrian knots (i.e. Legendrian knots bounding embedded 2-disks) in tight contact 3-manifolds. This part w…

In this paper, we explore minimal contact triangulations on contact 3-manifolds. We give many explicit examples of contact triangulations that are close to minimal ones. The main results of this article say that on any closed oriented 3-manifold the number of vertices for minimal contact triangulations for overtwisted …

We show that a null-homologous transverse knot K in the complement of an overtwisted disk in a contact 3-manifold is the boundary of a Legendrian ribbon if and only if it possesses a Seifert surface S such that the self-linking number of K with respect to S satisfies $\sel(K,S)=-χ(S)$. In particular, every null-homolog…

Study confoliations' symplectic fillability, finding obstructions.

We develop a method for preserving pseudoholomorphic curves in contact 3-manifolds under surgery along transverse links. This makes use of a geometrically natural boundary value problem for holomorphic curves in a 3-manifold with stable Hamiltonian structure, where the boundary conditions are defined by 1-parameter fam…

We define invariants of null--homologous Legendrian and transverse knots in contact 3--manifolds. The invariants are determined by elements of the knot Floer homology of the underlying smooth knot. We compute these invariants, and show that they do not vanish for certain non--loose knots in overtwisted 3--spheres. More…

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

We study open books (or open book decompositions) of a closed oriented 3-manifold which support overtwisted contact structures. We focus on a simple closed curve along which one can perform Stallings twist, called ``twisting loop''. We show that the existence of a twisting loop on the fiber surface of an open book is e…

In this note, we use the recent work of Honda-Kazez-Matic [HKM] to prove that a closed contact 3-manifold admitting a compatible open book decomposition with a nontrivial monodromy which can be presented as a product of left handed Dehn twists is overtwisted.

We extend the Heegaard Floer homological definition of spectral order for closed contact 3-manifolds due to Kutluhan, Matić, Van Horn-Morris, and Wand to contact 3-manifolds with convex boundary. We show that the order of a codimension zero contact submanifold bounds the order of the ambient manifold from above. As the…

Authors prove a contact structure result using branched covers and overtwisted disks.

Discusses the tight versus overtwisted dichotomy in 3D contact geometry.

In 1989, Y. Eliashberg proved that two overtwisted contact structures on a closed oriented 3-manifold are isotopic if and only if they are homotopic as 2-plane fields. We provide an alternative proof of this theorem using the convex surface theory and bypasses.

We classify Legendrian unknots in overtwisted contact structures on $S^3$. In particular, we show that up to contact isotopy for every pair $(n,\pm(n-1))$ with $n>0$ there are exactly two oriented non-loose Legendrian unknots in $S^3$ with Thurston-Bennequin invariant $n$ and rotation number $\pm(n-1)$. (Only one overt…

The paper examines conditions for contact surgeries on rational homology 3-spheres.

In this note we observe that one can contact embed all contact 3-manifolds into a Stein fillable contact structure on the twisted $S^3$-bundle over $S^2$ and also into a unique overtwisted contact structure on $S^3\times S^2$. These results are proven using "spun embeddings" and Lefschetz fibrations.

We study open books on three manifolds which are compatible with an overtwisted contact structure. We show that the existence of certain arcs, called sobering arcs, is a sufficient condition for an open book to be overtwisted, and is necessary up to stabilization by positive Hopf-bands. Using these techniques we prove …

We study a coverings of open books and virtually overtwisted contact manifolds using open book foliations. We show that open book coverings produces interesting examples such as transverse knots with depth grater than 1. We also demonstrate explicit examples of virtually overtwisted open books.

In this note we make several observations concerning symplectic cobordisms. Among other things we show that every contact 3-manifold has infinitely many concave symplectic fillings and that all overtwisted contact 3-manifolds are ``symplectic cobordism equivalent.''

Study contact structures on specific manifolds, proving infinite non-isotopic structures and tight/overtwisted classifications.

Twists of contact structures in dimension 3 and higher are studied in this paper from a viewpoint of contact round surgery. Three kinds of new modifications of contact structures which are higher-dimensional generalizations of the $3$-dimensional Lutz twists are introduced. One of the operations makes a contact manifol…

We prove every oriented compact cyclic $3$-orbifold has a contact structure. There is another proof in the web by Daniel Herr in his uploaded thesis which depends on open book decompositions, ours is independent of that. We define overtwisted contact structures, tight contact structures and Lutz twist on oriented compa…

Given a contact structure on a manifold $V$ together with a supporting open book decomposition, Bourgeois gave an explicit construction for a contact structure on $V \times \mathbb{T}^2$. We prove that all such structures are universally tight in dimension $5$, independent on whether the original contact manifold is ti…

Real algebraic structures help classify overtwisted contact 3-spheres.

A Legendrian or transverse knot in an overtwisted contact 3-manifold is non-loose if its complement is tight and loose if its complement is overtwisted. We define three measures of the extent of non-looseness of a non-loose knot and show they are distinct.

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 prove that each overtwisted contact structure has knot types that are represented by infinitely many distinct transverse knots all with the same self-linking number. In some cases, we can even classify all such knots. We also show similar results for Legendrian knots and prove a "folk" result concerning loose transv…

The study examines Morse diagrams and their behavior under Murasugi sums, leading to contact structure classifications.

In this paper we prove that the closed $4$-ball admits non-Kähler complex structures with strictly pseudoconcave boundary. Moreover, the induced contact structure on the boundary $3$-sphere is overtwisted.

In this note we show that $+1$-contact surgery on distinct Legendrian knots frequently produces contactomorphic manifolds. We also give examples where this happens for $-1$-contact surgery. As an amusing corollary we find overtwisted contact structures that contain a large number of distinct Legendrian knots with the s…

Plastikstufes and overtwistedness for higher-dimensional contact manifolds are studied in this paper. It is proved that a contact structure is overtwisted if and only if there exists a small plastikstufe with toric core that has trivial rotation.

Classifies tight contact structures on surgeries of the Whitehead link.

Stabilization operation for high-dimensional contact manifolds, proving many links are non-simple.

We introduce a modification procedure for Engel structures that is reminiscent of the Lutz twist in 3-dimensional Contact Topology. This notion allows us to define what an Engel overtwisted disc is, and to prove a complete h-principle for overtwisted Engel structures with fixed overtwisted disc.

Given a contact structure on a closed, oriented three-manifold $Y$, we describe an invariant which takes values in the three-manifold's Floer homology $\HFa$. This invariant vanishes for overtwisted contact structures and is non-zero for Stein fillable ones. The construction uses of Giroux's interpretation of contact s…

We obtain some obstructions to existence of Legendrian surgeries between tight lens spaces. We also study Legendrian surgeries between overtwisted contact manifolds.

We define an invariant of contact structures in dimension three from Heegaard Floer homology. This invariant takes values in the set $\mathbb{Z}_{\geq0}\cup\{\infty\}$. It is zero for overtwisted contact structures, $\infty$ for Stein fillable contact structures, non-decreasing under Legendrian surgery, and computable …

Introduction to contact invariant in bordered Floer homology.

Study contact structures on four-punctured spheres, finding infinitely many overtwisted monodromies.