Contact structures on plumbed 3-manifolds

Abstract

We show that the Ozsváth–Szabó contact invariant $c^{+}\left(\xi \right)\in H{F}^{+}(-Y)$ of a contact $3$-manifold $(Y,\xi )$ can be calculated combinatorially if $Y$ is the boundary of a certain type of plumbing $X$ and if $\xi$ is induced by a Stein structure on $X$. Our technique uses an algorithm of Ozsváth and Szabó 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 Ozsváth–Stipsicz–Szabó obstruction to admitting a planar open book for a certain class of contact structures. We also define a numerical invariant of contact manifolds that respects a partial ordering induced by Stein cobordisms. Using this technique, we do a sample calculation showing that the invariant can get infinitely many distinct values.