2003/07/17 by Fan Ding, Hansjörg Geiges, András I. Stipsicz · 2 citations
Mathematics · #math.SG #math.GT #msc:53D35 #msc:57M25 #msc:57R65
published as Turkish J. Math. 28 (2004), 41-74 · 32 pages, 14 figures
arxiv created 2003/07/17 · arxiv updated 2009/12/01
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 S1× S2, 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.