A Legendrian surgery presentation of contact 3-manifolds

By Fan Ding, Hansjörg Geiges. Mathematical Proceedings of the Cambridge Philosophical Society, 2001

125 citations recorded by the source corpus.

We prove that every closed, connected contact 3-manifold can be obtained from $S^3$ with its standard contact structure by contact (${\pm}1$)-surgery along a Legendrian link. As a corollary, we derive a result of Etnyre and Honda about symplectic cobordisms (in a slightly stronger form).

Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/458f2172eb2ba2b17707dd4a929c4e5ed92a193f

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Citations in the corpus

Tight contact structures on some small Seifert fibered 3-manifolds (2005, unknown): “(For more about contact surgery see [1, 2, 3].”
On the existence of tight contact structures on Seifert fibered 3-manifolds (2007, methodology): “In [4, 5] it is shown that a rational contact surgery can be replaced by a sequence of contact ( ± 1)–surgeries.”
Heegard Floer invariants of Legendrian knots in contact three-manifolds (2008, background): “(For contact (±1)-surgeries and surgery presentations see [3, 30].”
Handle moves in contact surgery diagrams (2008, background): “By [4] we can represent (M, ξ) by a surgery diagram L = L− L+ in (S3 , ξst), that is, a Legendrian link L such that contact (−1)-surgery along the components of”
The vanishing of the contact invariant in the presence of torsion (2007, methodology): “ has twisting number −1 with respect to the framing coming from Tε. THE VANISHING OF THE CONTACT INVARIANT IN THE PRESENCE OF TORSION 3 Now apply a contact (+1)-surgery to N along L. (See for example [DG].) As the surgery coeﬃcient is 0 with respect to the framing induced by the torus Tε, the result”
Ozsvth-Szab invariants and tight contact 3-manifolds, III (2007, background): “In fact, the contact structure obtained by performing contact (−1)-surgeries along the knots L1, L2, L3 is tight, due to the cancellation of (±1)-surgeries of opposite signs along Legendrian push-offs [3].”
Ozsváth-Szabó invariants and tight contact three-manifolds, II (2004, background): “For example, by [39, Theorem 1.1] the Thurston semi–norm is determined by the evaluation of the first Chern classes of spinc structures with nontrivial Ozsváth–Szabó homology groups.”