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Register a new userContact (+1)-surgeries along Legendrian Two-component Links
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In this paper, we study contact surgeries along Legendrian links in the standard contact 3-sphere. On one hand, we use algebraic methods to prove the vanishing of the contact Ozsv{a}th-Szab{o} invariant for contact $(+1)$-surgery along certain Legendrian two-component links. The main tool is a link surgery formula for Heegaard Floer homology developed by Manolescu and Ozsv{a}th. On the other hand, we use contact-geometric argument to show the overtwistedness of the contact 3-manifolds obtained by contact $(+1)$-surgeries along Legendrian two-component links whose two components are linked in some special configurations.
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In this note, we classify Stein fillings of an infinite family of contact 3-manifolds up to diffeomorphism. Some contact 3-manifolds in this family can be obtained by Legendrian surgeries on $(S^3,xi_{std})$ along certain Legendrian 2-bridge knots. We also classify Stein fillings, up to symplectic deformation, of an infinite family of contact 3-manifolds which can be obtained by Legendrian surgeries on $(S^3,xi_{std})$ along certain Legendrian twist knots. As a corollary, we obtain a classification of Stein fillings of an infinite family of contact hyperbolic 3-manifolds up to symplectic deformation.
In this paper, sufficient conditions for contact $(+1)$-surgeries along Legendrian knots in contact rational homology 3-spheres to have vanishing contact invariants or to be overtwisted are given. They can be applied to study contact $(pm1)$-surgeries along Legendrian links in the standard contact 3-sphere. We also obtain a sufficient condition for contact $(+1)$-surgeries along Legendrian two-component links in the standard contact 3-sphere to be overtwisted via their front projections.
An elementary stabilization of a Legendrian link $L$ in the spherical cotangent bundle $ST^*M$ of a surface $M$ is a surgery that results in attaching a handle to $M$ along two discs away from the image in $M$ of the projection of the link $L$. A virtual Legendrian isotopy is a composition of stabilizations, destabilizations and Legendrian isotopies. In contrast to Legendrian knots, virtual Legendrian knots enjoy the property that there is a bijective correspondence between the virtual Legendrian knots and the equivalence classes of Gauss diagrams. We study virtual Legendrian isotopy classes of Legendrian links and show that every such class contains a unique irreducible representative. In particular we get a solution to the following conjecture of Cahn, Levi and the first author: two Legendrian knots in $ST^*S^2$ that are isotopic as virtual Legendrian knots must be Legendrian isotopic in $ST^*S^2.$
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 Matics, Ozsvath and Szabos Floer-theoretic contact invariants. We compute the Ozsvath and Szabos invariant of positive contact surgeries along Legendrian knots in the 3-sphere in terms of the classical invariants of the knot. We also combine a Legendrian cabling construction with contact surgeries to get results about rational contact surgeries.
We classify the Legendrian torus knots in S^1times S^2 with its standard tight contact structure up to Legendrian isotopy.
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