Surgery diagrams for contact 3-manifolds

Abstract

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.

32 pages, 14 figures

References in corpus (3)

• On the classification of tight contact structures I
• A Legendrian surgery presentation of contact 3-manifolds
• Heegaard Floer Invariants and Tight Contact Three--Manifolds

Cited by in corpus (12)

• Ozsvath-Szabo invariants and tight contact three-manifolds, I
• On the existence of tight contact structures on Seifert fibered 3-manifolds
• Handle moves in contact surgery diagrams
• Seifert fibered contact three-manifolds via surgery
• Dehn surgery, rational open books and knot Floer homology
• Dehn Twists in Heegaard Floer Homology
• Contact surgery and symplectic caps
• Bott-integrable Reeb flows on 3-manifolds
• On the 3-dimensional invariant for cyclic contact branched coverings
• Contact surgery graphs
• Legendrian Hopf links in L(p,1)
• Strongly exceptional Legendrian connected sum of two Hopf links