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We study relative symplectic cobordisms between contact submanifolds, and in particular relative symplectic cobordisms to the empty set, that we call hats. While we make some observations in higher dimensions, we focus on the case of transverse knots in the standard 3-sphere, and hats in blow-ups of the (punctured) complex projective planes.We apply the construction to give constraints on the algebraic topology of fillings of double covers of the 3-sphere branched over certain transverse quasipositive knots. [ABSTRACT FROM AUTHOR]

We give several criteria on a closed, oriented 3-manifold that will imply that it is the boundary of a (simply connected) 4-manifold that admits infinitely many distinct smooth structures. We also show that any weakly fillable contact 3-manifold, or contact 3-manifold with non-vanishing Heegaard Floer invariant, is the boundary of a simply connected 4-manifold that admits infinitely many distinct smooth structures each of which supports a symplectic structure with concave boundary, that is there are infinitely many exotic caps for any such contact manifold. [ABSTRACT FROM AUTHOR]

We complete the classification of symplectic fillings of tight contact structures on lens spaces. In particular, we show that any symplectic filling X of a virtually overtwisted contact structure on L(p,q) has another symplectic structure that fills the universally tight contact structure on L(p,q). Moreover, we show that the Stein filling of L(p,q) with maximal second homology is given by the plumbing of disk bundles. We also consider the question of constructing symplectic cobordisms between lens spaces and report some partial results. [ABSTRACT FROM AUTHOR]

This paper completely answers the question of when contact |$(r)$| –surgery on a Legendrian knot in the standard contact structure on |$S^3$| yields a symplectically fillable contact manifold for |$r\in (0,1]$|⁠. We also give obstructions for other positive |$r$| and investigate Lagrangian fillings of Legendrian knots. [ABSTRACT FROM AUTHOR]

In this note, we show that a closed oriented contact manifold is obtained from the standard contact sphere of the same dimension by contact surgeries on isotropic and coisotropic spheres. In addition, we observe that all closed oriented contact manifolds admit symplectic caps. [ABSTRACT FROM AUTHOR]

In this note we observe that one can contact embed all contact 3‐manifolds into a Stein fillable contact structure on the twisted S3‐bundle over S2 and also into a unique overtwisted contact structure on S3×S2. These results are proven using 'spun embeddings' and Lefschetz fibrations. [ABSTRACT FROM AUTHOR]

In this paper we study embeddings of contact manifolds using braidings of one manifold about another. In particular, we show how to embed many contact 3-manifolds into the standard contact 5-sphere. We also show how to obstruct braidings and branched covers of one manifold over another using contact geometry. [ABSTRACT FROM AUTHOR]

In this note we point out an error in our article. We show how to repair the proof in dimension 5. The results are true in general as can easily be seen from recent work of Borman et al. (Acta Math. 215 (2015) 281–361). [ABSTRACT FROM AUTHOR]

In 1997, Chekanov gave the first example of a Legendrian nonsimple knot type: the m.52 knot. Epstein, Fuchs, and Meyer extended his result by showing that there are at least n different Legendrian representatives with maximal Thurston-Bennequin number of the twist knot K2n with crossing number 2n C 1. In this paper we give a complete classification of Legendrian and transverse representatives of twist knots. In particular, we show that K2n has exactly dn2=2e Legendrian representatives with maximal Thurston-Bennequin number, and dn=2e transverse representatives with maximal self-linking number. Our techniques include convex surface theory, Legendrian ruling invariants, and Heegaard Floer homology. [ABSTRACT FROM AUTHOR]

We give a possible generalization of a Lutz twist to all dimensions. This re-proves the fact that every contact manifold can be given a non-fillable contact structure and also shows great flexibility in the manifolds that can be realized as cores of overtwisted families. We, moreover, show that ℝ2n+1 has at least three distinct contact structures. [ABSTRACT FROM AUTHOR]

We prove that a nicely fibered link (by which we mean the binding of an open book) in a tight contact manifold (M, ξ) with zero Giroux torsion has a transverse representative realizing the Bennequin bound if and only if the contact structure it supports (since it is also the binding of an open book) is ξ. This gives a geometric reason for the non-sharpness of the Bennequin bound for fibered links. We also give the classification, up to contactomorphism, of maximal self-linking number links in these knot types. Moreover, in the standard tight contact structure on S3 we classify, up to transverse isotopy, transverse knots with maximal self-linking number in the knot types given as closures of positive braids and given as fibered strongly quasi-positive knots. We derive several braid theoretic corollaries from this. In particular, we give conditions under which quasi-positive braids are related by positive Markov stabilizations and when a minimal braid index representative of a knot is quasi-positive. Finally, we observe that our main result can be used to show, and make rigorous the statement, that contact structures on a given manifold are in a strong sense classified by the transverse knot theory they support. [ABSTRACT FROM PUBLISHER]

We show that if (B, π) is an open book decomposition of a contact 3-manifold (Y, ξ), then the complement of the binding B has no Giroux torsion and no virtual Giroux torsion. We also prove that the sutured Heegaard-Floer c-bar invariant of the binding of an open book is nonzero. [ABSTRACT FROM PUBLISHER]

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 the Weinstein conjecture and invariants of contact structures. [ABSTRACT FROM AUTHOR]

Provides a corrected reprint of the article 'Tight Contact Structures on Lens Spaces,' published in the November 2001 issue of the periodical 'Communications in Contemporary Mathematics.'

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. [ABSTRACT FROM AUTHOR]

The article discusses papers which proposes to generalize the notion of overtwisted disk to higher dimensions. It states that overtwisted contact structures are classified and are determined by algebraic topological data and main theorem is that if contact structure admits plastikstufe then it cannot be filled by semi-positive symplectic manifold. It mentions that detailed analysis of this moduli space leads to conclusion that the homology of plastikstufe removed is trivial in dimension.