\documentclass[11pt]{article}
\usepackage{amsmath,amssymb,amscd,dbnsymb,graphicx,stmaryrd,../picins,fancyhdr,dbnsymb}
\usepackage{txfonts} % for the likes of \coloneqq.
\usepackage[usenames,dvipsnames]{xcolor}
% Following http://tex.stackexchange.com/a/847/22475:
\usepackage[setpagesize=false]{hyperref}
\hypersetup{colorlinks,
linkcolor={blue!50!black},
citecolor={blue!50!black},
urlcolor=blue
}
% Following http://tex.stackexchange.com/questions/69901/how-to-typeset-greek-letters:
\usepackage[LGR,T1]{fontenc}
\newcommand{\textgreek}[1]{\begingroup\fontencoding{LGR}\selectfont#1\endgroup}
\usepackage[all]{xy}
\usepackage[textwidth=7in,textheight=9.5in,headsep=0.15in]{geometry}
\usepackage{pdfpages}
\usepackage{xr}
\externaldocument{WKO3}
\def\myurl{http://www.math.toronto.edu/~drorbn}
\def\pensieve{http://drorbn.net/AcademicPensieve}
%\paperwidth 8.5in
%\paperheight 11in
%\textwidth 7in
%\textheight 9.5in
%\oddsidemargin -0.25in
%\evensidemargin \oddsidemargin
%\topmargin -0.25in
%\headheight 0in
%\headsep 0in
%\footskip 0.25in
%%\parindent 0in
%\setlength{\topsep}{0pt}
\def\bbQ{{\mathbb Q}}
\def\bbR{{\mathbb R}}
\def\calA{{\mathcal A}}
\def\calI{{\mathcal I}}
\def\calK{{\mathcal K}}
\def\calP{{\mathcal P}}
\def\calR{{\mathcal R}}
\def\calT{{\mathcal T}}
\def\calU{{\mathcal U}}
\def\fraka{{\mathfrak a}}
\def\frakg{{\mathfrak g}}
\def\AS{\mathit{AS}}
\def\gr{\operatorname{gr}}
\def\IHX{\mathit{IHX}}
\def\STU{\mathit{STU}}
\def\TC{\mathit{TC}}
\def\arXiv#1{{\href{http://front.math.ucdavis.edu/#1}{arXiv:#1}}}
\renewcommand{\headrulewidth}{0pt}
\fancypagestyle{first}{
\lhead{\footnotesize\href{\myurl}{Dror Bar-Natan}: \href{\pensieve}{Academic Pensieve}: \href{\pensieve/Projects}{Projects}: \href{\pensieve/Projects/WKO3}{WKO3}:}
\rhead{\footnotesize\textgreek{web}$\coloneqq$\url{\pensieve/Projects/WKO3}}
}
\fancypagestyle{rest}{
\lhead{\footnotesize\href{\myurl}{Dror Bar-yNatan}}
\rhead{\footnotesize\textgreek{web}$\coloneqq$\url{\pensieve/Projects/WKO3}}
}
\pagestyle{rest}
\def\red{\color{red}}
\catcode`\@=11
\long\def\@makecaption#1#2{%
\vskip 10pt
\setbox\@tempboxa\hbox{%\ifvoid\tinybox\else\box\tinybox\fi
\small\sf{\bfcaptionfont #1. }\ignorespaces #2}%
\ifdim \wd\@tempboxa >\captionwidth {%
\rightskip=\@captionmargin\leftskip=\@captionmargin
\unhbox\@tempboxa\par}%
\else
\hbox to\hsize{\hfil\box\@tempboxa\hfil}%
\fi}
\font\bfcaptionfont=cmssbx10 scaled \magstephalf
\newdimen\@captionmargin\@captionmargin=2\parindent
\newdimen\captionwidth\captionwidth=\hsize
\catcode`\@=12
\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:
\[ \input{QuickDT.pdf_t} \]
%\[ \includegraphics{Rough2.png} \]
\end{document}