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 (\pm 1)-surgeries, and used this to prove that any (closed) contact 3-manifold can be obtained from the standard contact structure on S^3 by a sequence of such contact (\pm 1)-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 (\pm 1)-surgeries. We use this to give explicit surgery diagrams for all contact structures on S^3 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.
DING, FAN; GEIGES, HANSJÖRG; and STIPSICZ, ANDRAS I. (2004) "Surgery Diagrams for Contact 3-Manifolds," Turkish Journal of Mathematics: Vol. 28: No. 1, Article 4. Available at: https://journals.tubitak.gov.tr/math/vol28/iss1/4