Surgery diagrams for contact 3-manifolds

By Fan Ding, Hansjörg Geiges, A. Stipsicz. 2003

108 citations recorded by the source corpus.

In two previous papers, the two first-named authors introduced a notion of contact r-surgery along Legendrian knots in contact 3-manifolds. They also showed how (at least in principle) to convert any contact r-surgery into a sequence of contact plus or minus 1 surgeries, and used this to prove that any (closed) contact 3-manifold can be obtained from the standard contact structure on the 3-sphere by a sequence of such surgeries. In the present paper, we give a shorter proof of that result and a more explicit algorithm for turning a contact r-surgery into plus or minus 1 surgeries. We use this to give explicit surgery diagrams for all contact structures on the 3-sphere and S^1\times S^2, as well as all overtwisted contact structures on arbitrary closed, orientable 3-manifolds. This amounts to a new proof of the Lutz-Martinet theorem that each homotopy class of 2-plane fields on such a manifold is represented by a contact structure.

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

View Paper: https://www.semanticscholar.org/paper/ea458972584d0d821630f0880eaad4886a9a1834

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Discrete Mathematics and Combinatorics

In Geometry and Topology

In Mathematical Physics

Citations in the corpus

Heegard Floer invariants of Legendrian knots in contact three-manifolds (2008, background): “of [4] computes the Hopf invariant of ξk,l .”
Handle moves in contact surgery diagrams (2008, background): “The resulting contact surgery diagrams were studied more carefully in [6] and since then have led — in particular when combined with Heegaard Floer theory — to a wealth of applications, see e.”
Tight contact structures on some small Seifert fibered 3-manifolds (2005, background): “(For more about contact surgery see [1, 2, 3].”
On the existence of tight contact structures on Seifert fibered 3-manifolds (2007, background): “In [4, 5] it is shown that a rational contact surgery can be replaced by a sequence of contact ( ± 1)–surgeries.”
Right-veering diffeomorphisms of compact surfaces with boundary (2005, background): “)#(S3,ξOT), where ξOT is the unique overtwisted contact structure on S3 which is homotopic to the standard one, and the connected sum is done along a convex sphere S2 with one dividing curve. Now, by [DGS], we may write (S3,ξOT) as a connected sum (S3,ξ′)#(S3,ξ′′), where (S3,ξ′) has an open book dec”
Ozsváth-Szabó invariants and tight contact three-manifolds, II (2004, background): “ar X iv :m at h/ 05 05 49 3v 2 [ m at AMS Classification 57R17; 57R57 Keywords Ozsváth–Szabó invariants, L–spaces, taut foliations, transverse foliations, transverse contact structures, tight contact structures, planar contact structures”
Planar open book decompositions and contact structures (2004, methodology): “Using the notation of [ 2 ] (cf [15]) the homotopy classes of plane”
Ozsvth-Szab invariants and tight contact 3-manifolds, III (2007, methodology): “Now we want to apply the formula from [4] for the d3-invariant of a contact structure defined by a contact (±1)-surgery diagram.”