  We introduce a new approach to universal quantum knot invariants that
emphasizes generating functions instead of generators and relations. All the
relevant generating functions are shown to be perturbed Gaussians of the form
$Pe^G$, where $G$ is quadratic and $P$ is a suitably restricted "perturbation".
After developing a calculus for such Gaussians in general we focus on the rank
one invariant $\mathbf{Z}_\mathbb{D}$ in detail. We discuss how it dominates
the $\mathfrak{sl}_2$-colored Jones polynomials and relates to knot genus and
Whitehead doubling. In addition to being a strong knot invariant that behaves
well under natural operations on tangles $\mathbf{Z}_\mathbb{D}$ is also
computable in polynomial time in the crossing number of the knot. We provide a
full implementation of the invariant and provide a table in an appendix.
