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 $\Pic_{\C}(M)$.

The construction associates to a contact structure $\xi$ and a supporting open book decomposition an embedding into $\C^3$, where the contact structure becomes the holomorphic line field along the binding.

Using Stein theory, the induced holomorphic line bundle extends to all of $\C^3$ 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 $\mu_M(\xi) \in \Pic_{\C}(M)$ over the manifold.

As an application, we distinguish two tight contact structures on the 3-torus $\T^3$ by showing their first Chern classes are different.