In the previous three papers in this series, \cite{WKO1}--\cite{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 \cite{WKO1}--\cite{WKO3}
and in my related paper \cite{KBH}.