arXiv research

The study finds many tight contact structures on hyperbolic 3-spheres.

Contractibility of tight contact structures on $S^3$ proven.

New contact structures on spheres and tori with weakly compatible metrics.

Computes knot filtered ECH for torus knots on tight 3-sphere.

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

We study compatible contact structures of fibered Seifert multilinks in homology 3-spheres and especially give a necessary and sufficient condition for the contact structure to be tight in the case where the Seifert fibration is positively twisted. As a corollary we determine the strongly quasipositivity of fibered Sei…

Quantizes contact structures using dynamical methods.

In this short note, we exhibit an infinite family of hyperbolic rational homology $3$--spheres which do not admit any fillable contact structures. We also note that most of these manifolds do admit tight contact structures.

We give new tightness criteria for positive surgeries along knots in the 3-sphere, generalising results of Lisca and Stipsicz, and Sahamie. The main tools will be Honda, Kazez and Matic's, Ozsvath and Szabo's Floer-theoretic contact invariants. We compute the Ozsvath and Szabo's invariant of positive contact surgeries …

We show that every tight contact structure on any of the lens spaces $L(ns^2-s+1,s^2)$ with $n\geq 2$, $s\geq 1$, can be obtained by a single Legendrian surgery along a suitable Legendrian realisation of the negative torus knot $T(s,-(sn-1))$ in the tight or an overtwisted contact structure on the 3-sphere.

We prove that every Legendrian knot in the tight contact structure of the 3-sphere is determined by the contactomorphism type of its exterior. Moreover, by giving counterexamples we show this to be not true for Legendrian links in the tight 3-sphere. On the way a new user-friendly formula for computing the Thurston-Ben…

Study on Legendrian and transverse realizations of negative torus knots.

Let S^3_r(K) be the oriented 3--manifold obtained by rational r-surgery on a knot K in S^3. Using the contact Ozsvath-Szabo invariants we prove, for a class of knots K containing all the algebraic knots, that S^3_r(K) carries positive, tight contact structures for every r not= 2g_s(K)-1, where g_s(K) is the slice genus…

The paper defines and studies contact surgery numbers for contact 3-manifolds.

We study compatible contact structures of fibered, positively-twisted graph multilinks in the 3-sphere and prove that the contact structure of such a multilink is tight if and only if the orientations of its link components are all consistent with or all opposite to the orientation of the fibers of the Seifert fibratio…

Contact group retracts to unitary subgroup.

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.

Given any smooth fibration of the unit 3-sphere by great circles, we show that the distribution of 2-planes orthogonal to the great circle fibres is a tight contact structure, a fact well known in the special case of the Hopf fibrations. The proof expresses hypothesis and conclusion as differential inequalities involvi…

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

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…

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…

It is shown that Legendrian (resp. transverse) cable links in the 3-sphere with its standard tight contact structure, i.e. links consisting of an unknot and a cable of that unknot, are classified by their oriented link type and the classical invariants (Thurston-Bennequin invariant and rotation number in the Legendrian…

We prove that a version of the Thurston-Bennequin inequality holds for Legendrian and transverse links in a rational homology contact 3-sphere $(M,ξ)$, whenever $ξ$ is tight. More specifically, we show that the self-linking number of a transverse link $T$ in $(M,ξ)$, such that the boundary of its tubular neighbourhood …

It is known that for every smooth great circle fibration of the 3-sphere, the distribution of tangent 2-planes orthogonal to the fibres is a contact structure, in fact a tight one, but we show here that, beginning with the 5-sphere, there exist smooth great circle fibrations of all odd-dimensional spheres for which the…

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

Study contact structures on projective spaces, proving infinite non-isotopic structures.

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

Using the grid diagram formulation of knot Floer homology, Ozsvath, Szabo and Thurston defined an invariant of transverse knots in the tight contact 3-sphere. Shortly afterwards, Lisca, Ozsvath, Stipsicz and Szabo defined an invariant of transverse knots in arbitrary contact 3-manifolds using open book decompositions. …

We construct a simple topological invariant of certain 3-manifolds, including quotients of the 3-sphere by finite groups, based on the fact that the tangent bundle of an orientable 3-manifold is trivialisable. This invariant is strong enough to yield the classification of lens spaces of odd, prime order. We also use pr…

We study cosmetic contact surgeries along transverse knots in the standard contact 3-sphere, i.e. contact surgeries that yield again the standard contact 3-sphere. The main result is that we can exclude non-trivial cosmetic contact surgeries along all transverse knots not isotopic to the transverse unknot with self-lin…

We show that the pre-order defined on the category of contact manifolds by arbitrary symplectic cobordisms is considerably less rigid than its counterparts for exact or Stein cobordisms: in particular, we exhibit large new classes of contact 3-manifolds which are symplectically cobordant to something overtwisted, or to…

Surgery on knots always admits a tight contact structure.

Real algebraic structures help classify overtwisted contact 3-spheres.

The study finds tight contact structures without fillings in high dimensions.

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

Classifies tight contact structures on surgeries of the Whitehead link.

In this paper we develop a method for studying tight contact structures on lens spaces. We then derive uniqueness and non-existence statements for tight contact structures with certain (half) Euler classes on lens spaces. We also prove that any lens space admits only finitely many tight contact structures.

Study tight contact structures on specific 3-manifolds.

In the present paper we determine the Thurston-Bennequin invariant of graph divide links, which include all closed positive braids, all divide links and certain negative twist knots. As a corollary of this and a result of P. Lisca and A.I. Stipsicz, we prove that the 3-manifold obtained from the 3-sphere by Dehn surger…

Classifies tight contact structures on specific Seifert fibered manifolds.

Whether every hyperbolic 3-manifold admits a tight contact structure or not is an open question. Many hyperbolic 3-manifolds contain taut foliations and taut foliations can be perturbed to tight contact structures. The first examples of hyperbolic 3-manifolds without taut foliations were constructed by Roberts, Sharesh…

Every real 3-manifold can be turned into a real contact structure.

The paper conjectures a 4D characterization of tight contact structures and proves it for certain cases.

New contact structures on folded sums of contact mapping tori are tight under certain conditions.

Study proves all left-invariant contact structures on 3D Lie groups are tight.

Study non-fibered links' relation to tight contact structures.

We use the Ozsváth-Szabó contact invariants to distinguish between tight contact structures obtained by Legendrian surgeries on stabilized Legendrian links in tight contact 3-manifolds. We also discuss the implication of our result on the tight contact structures on the Brieskon homology spheres $-Σ(2,3,6n-1)$.

The paper classifies all tight contact structures on a solid torus.