In the previous three papers in this series, [WKO1]-[WKO3],
Z. Dancso and I studied a certain theory of "homomorphic expansions" of
"w-knotted objects", a certain class of knotted objects in 4-dimensional
space. When all layers of interpretation are stripped off, what remains
is a study of a certain number of equations written in a family of spaces
${\mathcal A}^w$, closely related to degree-completed free Lie algebras
and to degree-completed spaces of cyclic words.
The purpose of this paper is to introduce mathematical and computational
tools that enable explicit computations (up to a certain degree)
in these ${\mathcal A}^w$ spaces and to use these tools to solve the
said equations and verify some properties of their solutions, and as
a consequence, to carry out the computation (up to a certain degree)
of certain knot-theoretic invariants discussed in [WKO1]-[WKO3]
and in my related paper [KBH].