Contact Structures on Plumbed 3-Manifolds

TL;DR

This paper presents a combinatorial method to compute the Ozsváth-Szabó contact invariant for certain plumbed 3-manifolds with Stein structures, simplifying calculations and enabling new invariants in contact topology.

Contribution

It introduces a combinatorial approach to calculate contact invariants for plumbed 3-manifolds, facilitating applications in contact topology and defining a new numerical invariant.

Findings

Simplified calculation of contact invariants for specific 3-manifolds.

Demonstrated the invariant can take infinitely many values.

Applied the method to obstruct planar open books.

Abstract

In this paper, we show that the Ozsv\'ath-Szab\'o contact invariant $c^+(\xi)\in HF^+(-Y)$ of a contact 3-manifold $(Y,\xi)$ can be calculated combinatorially if $Y$ is the boundary of a certain type of plumbing $X$, and $\xi$ is induced by a Stein structure on $X$. Our technique uses an algorithm of Ozsv\'ath and Szab\'o to determine the Heegaard-Floer homology of such 3-manifolds. We discuss two important applications of this technique in contact topology. First, we show that it simplifies the calculation of the Ozsv\'ath-Stipsicz-Szab\'o obstruction to admitting a planar open book. Then we define a numerical invariant of contact manifolds that respects a partial ordering induced by Stein cobordisms. We do a sample calculation showing that the invariant can get infinitely many distinct values.