arXiv research

Abstract: Counterexamples found for lifting Hamiltonian and contact isotopies.

New examples show contact structures can be isotopic but not contact-isotopic.

The paper extends knot contact homology to tangles and proves a gluing formula.

Enhanced knot contact homology uniquely identifies knots.

A new inequality linking submanifold's topology to its Reeb chords count.

We prove that a topological contact isotopy uniquely defines a topological contact Hamiltonian. Combined with previous results from [MS11], this generalizes the classical one-to-one correspondence between smooth contact isotopies and their generating smooth contact Hamiltonians and conformal factors to the group of top…

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

The study constructs and shows isotopy of high-dimensional Legendrian spheres.

We classify Legendrian torus knots and figure eight knots in the tight contact structure on the 3-sphere up to Legendrian isotopy. As a corollary to this we also obtain the classification of transversal torus knots and figure eight knots up to transversal isotopy.

This paper proves a map from flow-spines to contact structures is surjective.

Legendrian Lavrentiev links are shown to be equivalent to smooth links.

Fractional Dehn twists give a measure of the difference between the relative isotopy class of a homeomorphism of a bordered surface and the Thurston representative of its free isotopy class. We show how to estimate and compute these invariants. We discuss the the relationship of our work to stabilization problems in cl…

In this paper, we study the global behaviour of contact structures on oriented manifolds V which are circle bundles over a closed orientable surface S of genus g>0. We establish in particular contact analogs of a number of classical results about foliations due to Milnor, Wood, Thurston, Matsumoto, and Ghys. In Section…

This is the less official, English version of the proof of the fact that every closed atoroidal 3-manifold carries finitely many isotopy classes of tight contact structures.

Defines Lorentzian distance on contactomorphisms, proving continuity and finite conditions.

Contact homology for Legendrian submanifolds in standard contact $(2n+1)$-space is rigorously defined using moduli spaces of holomorphic disks with Lagrangian boundary conditions in complex $n$-space. It provides new invariants of Legendrian isotopy. Using these invariants the theory of Legendrian isotopy is shown to b…

We prove a neighbourhood theorem for arbitrary knots in contact 3-manifolds. As an application we show that two topologically isotopic Legendrian knots in a contact 3-manifold become Legendrian isotopic after suitable stabilisations.

New invariant distinguishes Legendrian surfaces in 5-manifolds.

In this article we classify up to isotopy tight contact structures on Seifert manifolds over the torus with one singular fibre.

We present a sketch of the proof of the following theorems: (1) Every 3-manifold has only finitely many homotopy classes of 2-plane fields which carry tight contact structures. (2) Every closed atoroidal 3-manifold carries finitely many isotopy classes of tight contact structures.

New Legendrian knots found with equivalent Stein traces.

We classify up to isotopy the tight contact structures on small Seifert spaces with $e_0\neq0,-1,-2$. (The first version contains on the $e_0<-2$ case.)

We classify positive transversal torus knots in tight contact structures up to transversal isotopy.

Study finds all Brieskorn spheres with at most two fillable contact structures.

Let V be a closed 3-manifold. In this paper we prove that the homotopy classes of plane fields on V that contain tight contact structures are in finite number and that, if V is atoroidal, the isotopy classes of tight contact structures are also in finite number.

Common positive stabilisation found for isotopic contact structures.

A real 3-manifold is a smooth 3-manifold together with an orientation preserving smooth involution, called a real structure. In this article we study open book decompositions on smooth real 3-manifolds that are compatible with the real structure. We call them real open book decompositions. We show that each real open b…

Study contact structures on lens spaces, classifying rational knots.

We prove that there is a unique real tight contact structure on the 3-ball with convex boundary up to isotopy through real tight contact structures. We also give a partial classification of the real tight solid tori with the real structure being antipodal map along longitudinal and the identity along meridional directi…

Projection maps virtual Legendrian knots to classical ones.

This is an introductory text on the more topological aspects of contact geometry, written for the Handbook of Differential Geometry vol. 2. After discussing (and proving) some of the fundamental results of contact topology (neighbourhood theorems, isotopy extension theorems, approximation theorems), I move on to a deta…

The paper studies how Lagrangian cobordisms affect DGAs of Legendrian ends.

We classify transverse Hopf links in the standard contact 3-space up to transverse isotopy in terms of their components' self-linking number.

The study finds non-simple isotopy classes of links in 3-manifolds, including Legendrian and pseudo-Legendrian examples.

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 …

In this paper, we determine the group of contact transformations modulo contact isotopies for Legendrian circle bundles over closed surfaces of nonpositive Euler characteristic. These results extend and correct those presented by the first author in a former work. The main ingredient we use is connectedness of certain …

A rigorous foundation for the contact homology of Legendrian submanifolds in a contact manifold of the form $P\times \R$ where $P$ is an exact symplectic manifold is established. The class of such contact manifolds include 1-jet spaces of smooth manifolds. As an application, contact homology is used to provide (smooth)…

We classify positive tight contact structures, up to isotopy fixing the boundary, on the manifolds $N=M(D^{2}; r_1, r_2)$ with minimal convex boundary of slope $s$ and Giroux torsion 0 along $\partial N$, where $r_1,r_2\in (0,1)\cap\mathbb{Q}$, in the following cases: (1) $s\in(-\infty, 0)\cup[2, +\infty)$; (2) $s\in[0…

We classify the Legendrian torus knots in S^1\times S^2 with its standard tight contact structure up to Legendrian isotopy.

Authors create déjà vu links in Legendrian geometry.

Ozsvath-Szabo contact invariants are a powerful way to prove tightness of contact structures but they are known to vanish in the presence of Giroux torsion. In this paper we construct, on infinitely many manifolds, infinitely many isotopy classes of universally tight torsion free contact structures whose Ozsvath-Szabo …

We define symplectic fractional twists, which generalize Dehn twists, and use these in open books to investigate contact structures. The resulting contact structures are invariant under a circle action, and share several similarities with the invariant contact structures that were studied by Lutz and Giroux. We show th…

New theorem allows transverse links to be braided with rational book structure.

We define an integer-valued non-degenerate bi-invariant metric (the discriminant metric) on the universal cover of the identity component of the contactomorphism group of any contact manifold. This metric has a very simple geometric definition, based on the notion of discriminant points of contactomorphisms. Using gene…

New algebra structure for Legendrian knots preserves contact homology invariants.

We give a combinatorial description of the Legendrian differential graded algebra associated to a Legendrian knot in PxR, where P is a punctured Riemann surface. As an application we show that for any integer k and any homology class h in H_1(PxR) there are k Legendrian knots all representing h which are pairwise smoot…

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

We resolve a question of Fuchs and Tabachnikov by showing that there is a Legendrian knot in standard contact three-space with zero Maslov number which is not Legendrian isotopic to its mirror. The proof uses the differential graded algebras of Chekanov.