arXiv research

Sharp systolic inequality for invariant tight contact forms on S1-bundles over S2.

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

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

We consider complements of standard Seifert surfaces of special alternating links. On these handlebodies, we use Honda's method to enumerate those tight contact structures whose dividing sets are isotopic to the link, and find their number to be the leading coefficient of the Alexander polynomial. The Euler classes of …

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

Let $M$ be a compact orientable Seifered fibered 3-manifold without a boundary, and $α$ an $S^1$-invariant contact form on $M$. In a suitable adapted Riemannian metric to $α$, we provide a bound for the volume $\text{Vol}(M)$ and the curvature, which implies the universal tightness of the contact structure $ξ=\kerα$.

The paper simplifies proofs and characterizes contact structures in 3D.

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

Surgery on knots always admits a tight contact structure.

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

New techniques reveal tight contact manifolds with vanishing contact homology.

Quantizes contact structures using dynamical methods.

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.

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

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…

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

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

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

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.

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…

We prove gluing theorems for tight contact structures. In particular, we rederive (as special cases) gluing theorems due to Colin and Makar-Limanov, and present an algorithm for determining whether a given contact structure on a handlebody is tight. As applications, we construct a tight contact structure on a genus 4 h…

We show the equivalence of several notions in the theory of taut foliations and the theory of tight contact structures. We prove equivalence, in certain cases, of existence of tight contact structures and taut foliations.

New evidence supports the Euler class one conjecture for tight contact structures.

Two of the basic questions in contact topology are which manifolds admit tight contact structures, and on those that do, can we classify such structures. We present the first such classification on an infinite family of (mostly) hyperbolic 3-manifolds: surgeries on the figure-eight knot. We also determine which of the …

We consider the problem of realizing tight contact structures on closed orientable three-manifolds. By applying the theorems of Hofer et al., one may deduce tightness from dynamical properties of (Reeb) flows transverse to the contact structure. We detail how two classical constructions, Dehn surgery and branched cover…

Classifies real tight contact structures on lens spaces and solid tori.

New proof of Giroux Correspondence for tight contact 3-manifolds.

We compute the Ozsváth--Szabó contact invariants for all tight contact structures on the manifolds -Σ(2,3,6n-1).

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…

We employ cut and paste contact topological techniques to classify some tight contact structures on the closed, oriented genus-2 surface times the interval. A boundary condition is specified so that the Euler class of the of the contact structure vanishes when evaluated on each boundary component. We prove that there e…

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.

We exhibit a 3-manifold which admits no tight contact structure.

In this article we present infinitely many 3-manifolds admitting infinitely many universally tight contact structures each with trivial Ozsvath-Szabo contact invariants. By known properties of these invariants the contact structures constructed here are non weakly symplectically fillable.

We study weak versus strong symplectic fillability of some tight contact structures on torus bundles over the circle. In particular, we prove that almost all of these tight contact structures are weakly, but not strongly symplectically fillable. For the 3-torus this theorem was established by Eliashberg.

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…

Let Y(r) be the closed, oriented three-manifold obtained by performing rational r-surgery on the right-handed trefoil knot in the three-sphere. Using contact surgery and the Heegaard Floer contact invariants we construct positive, tight contact structures on Y(r) for every r not equal to 1. This implies, in particular,…

The Ozsvath-Szabo contact invariant is a complete classification invariant for tight contact structures on small Seifert fibered 3-manifolds which are L-spaces.

Classifies tight contact structures with special symmetries.

The paper classifies tight contact structures on Seifert fiber spaces.

A contact foliation is a foliation endowed with a leafwise contact structure. In this remark we explain a turbulisation procedure that allows us to prove that tightness is not a homotopy invariant property for contact foliations.

Let p and n be positive integers with p>1, and let E(p,n) be the oriented 3-manifold obtained by performing pn(p-1)-1 surgery on a positive torus knot of type (p, pn+1). We prove that E(2,n) does not carry tight contact structures for any n, while E(p,n) carries tight contact structures for any n and any odd p. In part…