Embedded Contact Homology (ECH) defines an invariant for contact 3 manifolds. It does this by considering certain J-holomorphic curves in $\mathbb R\times Y$ and "counting" them.

In the symplectic world, there are sum formulas for Gromov-Witten invariants of symplectic manfiolds which can be described a symplectic sum of two symplectic manifolds.

Are there any similar decompositions of contact 3-manifolds which lead to interesting relations in ECH similar to the Gromov-Witten sum formula?