In this paper we utilize a certain "double tree construction"
to show that every "expansion", namely "universal finite type
invariant (UFTI)" of parenthesized braids extends uniquely first to an
expansion/UFTI of knotted trivalent graphs (a well known result), and
then on to an expansion/UFTI of w-knotted objects, namely to knottings of
"2-dimensional foams" and various associated objects in four-dimensioanl
space.
<p>
In algebraic language, an expansion for parenthesized braids is
the same as a "Drinfel'd associator" <em>&Phi;</em>, and an expansion for
w-knotted objects is the same as a solution <em>V</em> of the Kashiwara-Vergne
problem&nbsp;<a href=#KV>[KV]</a> as reformulated by Alekseev
and Torossian&nbsp;<a href=#AT>[AT]</a>. Hence our
result amounts to a topological re-interpretation of the result of
Alkeseev-Enriquez-Torossian&nbsp;<a href=#AET>[AET]</a>
that "there is a formula for <em>V</em> in terms of <em>&Phi;</em>", along with an
independent topological proof that the said formula indeed works -
that the equations satisfied by <em>V</em> follow from the equations satisfied
by <em>&Phi;</em>.