  In the previous three papers in this series, [WKO1]-[WKO3] (arXiv:1405.1956,
arXiv:1405.1955, and to appear), 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 $A^w$, 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 $A^w$
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] (arXiv:1308.1721).
