Surgery diagrams for contact 3-manifolds

TL;DR

This paper provides explicit surgery diagrams for all contact structures on certain 3-manifolds, simplifying the process of representing contact structures via contact surgeries and offering a new proof of the Lutz-Martinet theorem.

Contribution

It introduces a shorter proof and an explicit algorithm for converting contact r-surgeries into plus or minus 1 surgeries, enabling explicit diagrams for all contact structures on key 3-manifolds.

Findings

Explicit surgery diagrams for all contact structures on S^3 and S^1×S^2.

Explicit diagrams for all overtwisted contact structures on closed 3-manifolds.

A new proof of the Lutz-Martinet theorem using surgery diagrams.

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.