A New CR Invariant for Contact 3-Manifolds and Classes of Open Books

This paper introduces a new CR invariant for co-oriented contact structures on closed, orientable 3-manifolds. The invariant, which we denote as ${\mu }_M(\xi )$, takes values in the Picard group of complex line bundles ${\text{Pic}}_{\text{C}}(M)$. The construction associates to a contact structure $\xi$ and a supporting open book decomposition an embedding into ${\text{C}}^3$, where the contact structure becomes the holomorphic line field along the binding. Using Stein theory, the induced holomorphic line bundle extends to all of ${\text{C}}^3$ but we consider only its restriction to $M$. By Giroux’s correspondence, we prove this construction is independent of the choice of open book, yielding a well-defined invariant ${\mu }_M(\xi )\in {\text{Pic}}_{\text{C}}(M)$ over the manifold. As an application, we distinguish two tight contact structures on the 3-torus ${\text{T}}^3$ by showing their first Chern classes are different.