  By studying the properties of $q$-series $\widehat Z$-invariants, we develop
a dictionary between 3-manifolds and vertex algebras. In particular, we
generalize previously known entries in this dictionary to Lie groups of higher
rank, to 3-manifolds with toral boundaries, and to BPS partition functions with
line operators. This provides a new physical realization of logarithmic vertex
algebras in the framework of the 3d-3d correspondence and opens new avenues for
their future study. For example, we illustrate how invoking a knot-quiver
correspondence for $\widehat{Z}$-invariants leads to many infinite families of
new fermionic formulae for VOA characters.
