In algebraic language, an expansion for parenthesized braids is
the same as a "Drinfel'd associator" *Φ*, and an expansion for
w-knotted objects is the same as a solution *V* of the Kashiwara-Vergne
problem [KV] as reformulated by Alekseev
and Torossian [AT]. Hence our
result amounts to a topological re-interpretation of the result of
Alkeseev-Enriquez-Torossian [AET]
that "there is a formula for *V* in terms of *Φ*", along with an
independent topological proof that the said formula indeed works -
that the equations satisfied by *V* follow from the equations satisfied
by *Φ*.