On tight contact structures with negative maximal twisting number on small Seifert manifolds

Ghiggini, Paolo

doi:10.2140/agt.2008.8.381

Cited by 25 publications

(41 citation statements)

References 19 publications

(33 reference statements)

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“…In particular, we used the fact that the unique tight contact structure on this manifold is not supported by a planar open book. A similar argument can be made for k D 0; 1; 2; 3 to obtain rightveering, not destabilizable monodromies which support overtwisted contact structures (using the classification result in [11], which in particular says that all the tight contact structures on these manifolds are Stein fillable). However, we do not know if is overtwisted or tight for k > 4.…”

Section: An Example 41 Poincaré Homology Spherementioning

confidence: 72%

Planar open books with four binding components

Lekili

2011

Algebr. Geom. Topol.

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We study an explicit construction of planar open books with four binding components on any three-manifold which is given by integral surgery on three component pure braid closures. This construction is general, indeed any planar open book with four binding components is given this way. Using this construction and results on exceptional surgeries on hyperbolic links, we show that any contact structure of S^3 supports a planar open book with four binding components, determining the minimal number of binding components needed for planar open books supporting these contact structures. In addition, we study a class of monodromies of a planar open book with four binding components in detail. We characterize all the symplectically fillable contact structures in this class and we determine when the Ozsvath-Szabo contact invariant vanishes. As an application, we give an example of a right-veering diffeomorphism on the four-holed sphere which is not destabilizable and yet supports an overtwisted contact structure. This provides a counterexample to a conjecture of Honda, Kazez, Matic from arXiv:0609734 .Comment: 19 pages, 10 figures. Two cancelling sign errors remove

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Section: An Example 41 Poincaré Homology Spherementioning

confidence: 72%

Planar open books with four binding components

Lekili

2011

Algebr. Geom. Topol.

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“…The proof in [29] combines the Eliashberg-Thurston perturbation theorem with the more analytical adjunction inequality in symplectic geometry. In the case g D 0, r D 3 and b D 2, a topological proof of Theorem B by P Ghiggini appeared in [11] while this paper was in preparation.…”

Section: Introductionmentioning

confidence: 99%

Geodesible contact structures on 3–manifolds

Massot

2008

Geom. Topol.

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“…The main invariant in the classification of tight Seifert fibered manifolds is the maximal twisting number (that is, the difference between the contact framing and the fibration framing, maximized in the smooth isotopy class of a regular fiber)applied in convex surface theory, it allows one to give upper bounds on the number of tight structures. By the results of Wu [20] all tight contact structures when e 0 ≤ −2 have negative maximal twisting, while for e 0 ≥ 0 they are all zero-twisting; in the work of Ghiggini [3] the negative maximal twisting is further related to the existence of transverse contact structures. This, in the case of L-spaces, results in a simple division: maximal twisting is equal to zero when e 0 ≥ −1, and has value −1 when e 0 ≤ −2.…”

Section: Introductionmentioning

confidence: 92%