Tight contact structures on Seifert surface complements

The aim of this paper is to show that the Seifert fibered space Σ(-½, ⅓, ⅓) over S2does not admit any tight contact structures. As a consequence, we can conclude that Legendrian surgery is not category-preserving for tight contact structures on closed 3-manifolds.

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LEGENDRIAN VERTICAL CIRCLES IN SMALL SEIFERT SPACES

Communications in Contemporary Mathematics ◽

10.1142/s0219199706002106 ◽

2006 ◽

Vol 08 (02) ◽

pp. 219-246 ◽

Cited By ~ 7

Author(s):

HAO WU

Keyword(s):

Contact Structures ◽

Tight Contact

We discuss the relations between the e0 invariant of a tight contact small Seifert space and the twisting numbers of Legendrian vertical circles in it, and apply these relations to classify up to isotopy tight contact structures on small Seifert spaces with e0 ≠ 0, -1, -2.

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CONTACT STRUCTURES, σ-CONFOLIATIONS AND CONTAMINATIONS IN 3-MANIFOLDS

Communications in Contemporary Mathematics ◽

10.1142/s021919970900334x ◽

2009 ◽

Vol 11 (02) ◽

pp. 201-264 ◽

Cited By ~ 1

Author(s):

ULRICH OERTEL ◽

JACEK ŚWIATKOWSKI

Keyword(s):

Contact Structure ◽

Contact Structures ◽

Tight Contact ◽

Branched Surface ◽

Definition Of ◽

Branched Surfaces ◽

Natural Way

We propose in this paper a method for studying contact structures in 3-manifolds by means of branched surfaces. We explain what it means for a contact structure to be carried by a branched surface embedded in a 3-manifold. To make the transition from contact structures to branched surfaces, we first define auxiliary objects called σ-confoliations and pure contaminations, both generalizing contact structures. We study various deformations of these objects and show that the σ-confoliations and pure contaminations obtained by suitably modifying a contact structure remember the contact structure up to isotopy. After defining tightness for all pure contaminations in a natural way, generalizing the definition of tightness for contact structures, we obtain some conditions on (the embedding of) a branched surface in a 3-manifold sufficient to guarantee that any pure contamination carried by the branched surface is tight. We also find conditions sufficient to prove that a branched surface carries only overtwisted (non-tight) contact structures. Our long-term goal in developing these methods is twofold: Not only do we want to study tight contact structures and pure contaminations, but we also wish to use them as tools for studying 3-manifold topology.

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Gluing tight contact structures

Duke Mathematical Journal ◽

10.1215/s0012-7094-02-11532-4 ◽

2002 ◽

Vol 115 (3) ◽

pp. 435-478 ◽

Cited By ~ 26

Author(s):

Ko Honda

Keyword(s):

Contact Structures ◽

Tight Contact

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EXPLICIT HORIZONTAL OPEN BOOKS ON SOME PLUMBINGS

International Journal of Mathematics ◽

10.1142/s0129167x06003801 ◽

2006 ◽

Vol 17 (09) ◽

pp. 1013-1031 ◽

Cited By ~ 11

Author(s):

TOLGA ETGÜ ◽

BURAK OZBAGCI

Keyword(s):

Contact Structure ◽

Complex Surface ◽

Contact Structures ◽

Open Book ◽

Open Books ◽

Tight Contact ◽

Weinstein Conjecture ◽

Circle Bundles ◽

Surface Singularities

We describe explicit open books on arbitrary plumbings of oriented circle bundles over closed oriented surfaces. We show that, for a non-positive plumbing, the open book we construct is horizontal and the corresponding compatible contact structure is also horizontal and Stein fillable. In particular, on some Seifert fibered 3-manifolds we describe open books which are horizontal with respect to their plumbing description. As another application we describe horizontal open books isomorphic to Milnor open books for some complex surface singularities. Moreover we give examples of tight contact 3-manifolds supported by planar open books. As a consequence, the Weinstein conjecture holds for these tight contact structures [1].

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