  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.
