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\begin{document} \thispagestyle{first}

\par\noindent{\Large\bf WKO3: Executive Summary}
\vskip 2mm

{\footnotesize This section is followed by a more traditional introduction.}

A ``homomorphic expansion'' for a certain class of topological objects $\calK$ is an invariant $Z\colon\calK\to\calA$ whose target space $\calA$ is canonically associated with $\calK$ (its ``associated graded''), and which satisfies a certain universality property, and which respects a certain collection of operations which exist on $\calK$, and therefore also on $\calA$. Homomorphic expansions are often hard to find, and when they are found, they often correspond to some deep mathematics:

\begin{itemize}

\item Many classes of knotted objects in 3-dimensional spaces do not have homomorphic expansions --- one would have loved ordinary tangles to have homomorphic expansions, for example, but they don't.

\item Yet a certain class $\calK^u$ of knotted objects in 3-space, the class of ``parenthesized tangles'', or nearly-equivalently, the class of knotted trivalent graphs (which we adopt in this paper) does have homomorphic expansions. A homomorphic expansion $Z^u\colon\calK^u\to\calA^u$ is defined by its values on a couple of elements of $\calK^u$ which generate $\calK^u$ using the operations $\calK^u$ is equipped with. The most interesting of these generators is the tetrahedron $\tetrahedron$, and $\Phi=Z^u(\tetrahedron)$ turns out to be equivalent to ``a Drinfel'd associator''.

\item A certain class $\calK^w$ of graphs, which is conjectured to be equivalent to a certain class of 2-dimensional knotted objects in 4-space, also has homomorphic expansions. The most interesting generator of $\calK^w$ is the ``vertex'' $\YGraph$, and if $Z^w\colon\calK^w\to\calA^w$ is a homomorphic expansion, then it turns out that $V=Z^w(\YGraph)$ is equivalent to ``a solution of the Kashiwara-Vergne problem''.

\end{itemize}

\parpic[r]{\input{uw.pdf_t}}
%\parpic[r]{\includegraphics[width=8cm]{Rough1.png}}
Roughly speaking, $\calK^u$ is a part of $\calK^w$ and $\calA^u$ is a part of $\calA^w$, as in the figure on the right (more precisely, there are natural maps $a\colon\calK^u\to\calK^w$ and $\alpha\colon\calA^u\to\calA^w$). The main purpose of this paper is to prove the following theorem:

\par\noindent{\bf Theorem} (precise version in Theorem~\ref{thm:main}). Any homomorphic expansion $Z^u$ for $\calK^u$ extends uniquely to a homomorphic expansion $Z^w$ for $\calK^w$, and therefore, any Drinfel'd associator $\Phi$ yields a solution $V$ of the Kashiwara-Vergne problem.

The proof of this theorem is almost banal. We simply show that the generators of $\calK^w$ can be explicitly expressed using the generators of $\calK^u$ and the operations of $\calK^w$, and that the resulting explicit formulas for $Z^w(\YGraph)$ (and for $Z^w$ of the other generators) satisfies all the required relations.

The devil is in the details. It is in fact impossible to express the generators of $\calK^w$ in terms of the generators of $\calK^u$ --- to do that, one first has to pass to a larger space $\tilde{\calK}^w$ that has more objects and more operations, and in which the desired explicit expressions do exist. But even in $\tilde{\calK}^w$ these expressions are complicated, and are best described within a certain ``double tree construction'' which also provides the framework for the verification of relations. Here's an unexplained summary; the explanations make the bulk of this paper:

\[ \input{QuickDT.pdf_t} \]
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