Contact surgery and supporting open books

TL;DR

This paper introduces algorithms to convert between open book decompositions and contact surgery diagrams for contact 3-manifolds, refining existing results and providing bounds on support invariants, with computational examples.

Contribution

It presents two algorithms for converting between open books and contact surgery diagrams, and introduces ribbon moves for diagram modification, enhancing the understanding of contact 3-manifold representations.

Findings

Algorithms for converting open books to surgery diagrams and vice versa.

Refinement of Ding-Geiges result on contact surgery from standard 3-sphere.

Introduction of ribbon moves relating different surgery diagrams.

Abstract

Let $(M,\xi)$ be a contact 3-manifold. We present two new algorithms, the first of which converts an open book $(\Sigma,\Phi)$ supporting $(M,\xi)$ with connected binding into a contact surgery diagram. The second turns a contact surgery diagram for $(M,\xi)$ into a supporting open book decomposition. These constructions lead to a refinement of a result of Ding-Geiges, which states that every such $(M,\xi)$ may be obtained by contact surgery from $(S^{3},\xi_{std})$, as well as bounds on the support norm and genus of contact manifolds obtained by surgery in terms of classical link data. We then introduce Kirby moves called ribbon moves which use mapping class relations to modify contact surgery diagrams. Any two surgery diagrams of the same contact 3-manifold are related by a sequence of Legendrian isotopies and ribbon moves. As most of our results are computational in nature, a number of examples are analyzed.