  This paper provides a connection between two distinct branches of research in
CR geometry -- namely, analytic and geometric conditions that suffice to
establish the closed range of the Cauchy-Riemann operator and CR invariants on
CR manifolds. Specifically, we work on not necessarily pseudoconvex domains
$\Omega\subset\mathbb{C}^n$ and define third and fourth order CR invariants on
$\bd\Om$ and show that these invariants provide enough information to establish
closed range for the $\bar\partial$-Laplacian in $L^2_{(0,q)}(\Omega)$ for a
given, fixed $q$. The closed range estimates follow from our previously defined
weak $Z(q)$ condition. We also develop powerful linear algebra machinery to
translate the information from the invariants into information about the Levi
form and its eigenvalues. We conclude with several examples that demonstrate
the usefulness and ease of use of the new conditions.
