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% Following http://tex.stackexchange.com/a/847/22475:
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% Following http://tex.stackexchange.com/questions/59340/how-to-highlight-an-entire-paragraph
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\def\myurl{http://www.math.toronto.edu/~drorbn}
\def\thistalk{DaNang-1905}
\def\title{Everything around $sl_{2+}^\epsilon$ is {\bf DoPeGDO}. So what?}

\def\navigator{{
  \href{\myurl}{Dror Bar-Natan}:
  \href{\myurl/Talks}{Talks}:
  \href{\myurl/Talks/\thistalk/}{\thistalk}:
}}
\def\thanks{{Thanks for inviting me to \magenta Da Nang!}}
\def\webdef{{{\greektext web}$\coloneqq$\href{http://drorbn.net/v19}{http://drorbn.net/v19/}}}
\def\web#1{{\href{\myurl/Talks/\thistalk/#1}{{\greektext web}/#1}}}
\def\titleA{{\title}}
\def\titleB{{\title}}
\def\titleC{{\title}}
\def\more{More at \web{talks}}

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\def\arXiv#1{{\href{http://front.math.ucdavis.edu/#1}{arXiv:\linebreak[0]#1}}}

\def\qed{{\linebreak[1]\null\hfill\text{$\Box$}}}

\def\act{{\hspace{-1pt}\sslash\hspace{-0.75pt}}}
\def\ad{\operatorname{ad}}
\def\Ad{\operatorname{Ad}}
\def\aft{$\overrightarrow{\text{4T}}$}
\def\AS{\mathit{AS}}
\def\bbZZ{{\mathbb Z\mathbb Z}}
\def\CW{\text{\it CW}}
\def\diag{\operatorname{diag}}
\def\FL{\text{\it FL}}
\def\Hom{\operatorname{Hom}}
\def\IHX{\mathit{IHX}}
\def\mor{\operatorname{mor}}
\def\PvT{{\mathit P\!v\!T}}
\def\remove{\!\setminus\!}
\def\STU{\mathit{STU}}
\def\SW{\text{\it SW}}
\def\TC{\mathit{TC}}
\def\tr{\operatorname{tr}}
\def\vT{{\mathit v\!T}}
\def\wt{\operatorname{wt}}

\def\barT{{\bar T}}
\def\barz{{\bar z}}
\def\bbe{\mathbbm{e}}
\def\bbD{{\mathbb D}}
\def\bbE{{\mathbb E}}
\def\bbF{{\mathbb F}}
\def\bbO{{\mathbb O}}
\def\bbQ{{\mathbb Q}}
\def\bbR{{\mathbb R}}
\def\bbZ{{\mathbb Z}}
\def\bcA{{\bar{\mathcal A}}}
\def\calA{{\mathcal A}}
\def\calD{{\mathcal D}}
\def\calE{{\mathcal E}}
\def\calF{{\mathcal F}}
\def\calG{{\mathcal G}}
\def\calI{{\mathcal I}}
\def\calK{{\mathcal K}}
\def\calO{{\mathcal O}}
\def\calP{{\mathcal P}}
\def\calR{{\mathcal R}}
\def\calS{{\mathcal S}}
\def\calT{{\mathcal T}}
\def\calU{{\mathcal U}}
\def\fraka{{\mathfrak a}}
\def\frakb{{\mathfrak b}}
\def\frakg{{\mathfrak g}}
\def\frakh{{\mathfrak h}}
\def\tilE{\tilde{E}}
\def\tilq{\tilde{q}}

\def\tDelta{\tilde{\Delta}}
\def\tf{\tilde{f}}
\def\tF{\tilde{F}}
\def\tg{\tilde{g}}
\def\tI{\tilde{I}}
\def\tm{\tilde{m}}
\def\tR{\tilde{R}}
\def\tsigma{\tilde{\sigma}}
\def\tS{\tilde{S}}
\def\tSW{\widetilde{\SW}}

% From http://tex.stackexchange.com/questions/154672/how-to-get-a-medium-sized-otimes
\DeclareMathOperator*{\midotimes}{\text{\raisebox{0.25ex}{\scalebox{0.8}{$\bigotimes$}}}}

\def\cellscale{0.645}

%%%

\def\Plan{{\raisebox{2mm}{\parbox[t]{3.95in}{
{\red\bf Talk Plan.} 
\begin{enumerate}[leftmargin=*,labelindent=0pt,itemsep=-4pt,topsep=0pt]
\item Headers and abstract.
\item Ops grid.
\item $sl_{2+}^\epsilon$ and 4D Lie algebras, then the details of $CU$ and $QU$.
\item The {\bf DoPeGDO} box, ``one abstraction level up''.
\item A glance through the {\bf DoPeGDO} footnotes.
\item Naive {\bf DoPeGDO} compositions.
\item The ``debts'' box, and then go through the debts as follows.
\item A quantum algebra example.
\item A knot theory example, followed by the knot table noting affinities with topology.
\end{enumerate}
}}}}

\def\credits{{\raisebox{0.75mm}{\parbox[t]{1.3in}{
Continues Rozansky \cite{Ro, Rozansky:Burau, Rozansky:U1RCC} and \text{Overbay}
\cite{Overbay:Thesis}, joint with van der Veen \cite{PP1}.
}}}}

\def\AbstractA{{\raisebox{2mm}{\parbox[t]{3.95in}{\begin{hyphenrules}{nohyphenation}
{\red\bf Abstract.} I'll explain what ``everything around'' means:
classical and quantum $m$, $\Delta$, $S$, $tr$, $R$, $C$, and $\theta$,
as well as $P$, $\Phi$, $J$, $\bbD$, and more, and all of their
compositions. What {\bf DoPeGDO} means: the category of
\textcolor{ForestGreen}{Docile Perturbed Gaussian Differential
Operators}. And what $sl_{2+}^\epsilon$ means: a solvable approximation
of the semi-simple Lie algebra $sl_2$.
\end{hyphenrules}}}}}

\def\AbstractB{{\raisebox{-1mm}{\parbox[t]{3.9in}{
Knot theorists should rejoice \text{because} all this leads to very powerful and
well-behaved poly-time-computable knot \text{invariants}. Quantum algebraists
should rejoice because it's a realistic playground for testing complicated
\text{equations} and theories.
}}}}

\def\maximm{{\thickmuskip=0mu $m\colon U \otimes U \to U$}}
\def\maximDelta{{\thickmuskip=0mu $\Delta\colon U \to U \otimes U$}}
\def\maximS{{\thickmuskip=0mu $S\colon U \to U$}}
\def\maximtr{{\thickmuskip=-2mu $\tr\colon U \to U/wx=xw$}}
\def\maximR{{\thickmuskip=0mu $R\in QU\otimes QU$}}
\def\maximC{{\thickmuskip=0mu $C\in QU$}}
\def\maximPhi{{\thickmuskip=0mu $\Phi\in CU^{\otimes 3}$}}
\def\maximJ{{\thickmuskip=0mu $J\in CU\otimes CU$}}

\def\Udef{{\raisebox{2mm}{\parbox[t]{3.95in}{
{\bf\red Our Algebras.} Let
$sl_{2+}^\epsilon \coloneqq L\langle y,b,a,x\rangle$ subject to ${[a,x]=x}$,
$[b,y]=-\epsilon y$, $[a,b]=0$, $[a,y]=-y$, $[b,x]=\epsilon x$, and $[x,y]=\epsilon a+b$. So
$t\coloneqq\epsilon a-b$ is central and if $\exists \epsilon^{-1}$, $sl_{2+}^\epsilon/\langle t\rangle\cong sl_2$.

$U$ is either $CU=\hat\calU(sl_{2+}^\epsilon)$ or
$QU=\calU_\hbar(sl_{2+}^\epsilon) = A\langle y,b,a,x\rangle$ with
$[a,x]=x$, $[b,y]=-\epsilon y$, $[a,b]=0$, $[a,y]=-y$, $[b,x]=\epsilon
x$, and $xy-qyx=(1-AB)/\hbar$, where $q=\bbe^{\hbar\epsilon}$,
$A=\bbe^{-\hbar\epsilon a}$, and $B=\bbe^{-\hbar b}$. Set also
$T=A^{-1}B=\bbe^{\hbar t}$.

{\red The Quantum Leap.} Also decree that in $QU$,
\[ \Delta(y,b,a,x) = (y_1+B_1y_2, b_1+b_2, a_1+a_2, x_1+A_1x_2), \]
\[ S(y,b,a,x) = (-B^{-1}y, -b, -a, -A^{-1}x),\]
\vskip 3pt
and $R=\sum\hbar^{j+k}y^kb^j\otimes a^jx^k/j![k]_q!$.
}}}}

\def\Conventions{{\raisebox{0mm}{\parbox[t]{3.9in}{
{\bf Conventions.} 1.~For a set $A$, let $z_A\coloneqq\{z_i\}_{i\in A}$ and let ${\zeta_A\coloneqq\{z^*_i=\zeta_i\}_{i\in A}}$.$^{\dagger 1}$
2.~Everything converges!
}}}}

\def\DoPeGDO{{\raisebox{1mm}{\parbox[t]{2.96in}{
$\text{\bf DoPeGDO} \coloneqq$ The category with objects finite sets$^{\dagger 2}$ and
$\mor(A\to B)$:
\[ \left\{\calF =\omega\exp(Q+P) \right\} \subset \bbQ\llbracket\zeta_A,z_B\rrbracket \]
Where:
$\bullet$~$\omega$ is a scalar.$^{\dagger 3}$
$\bullet$~$Q$ is a ``small'' quadratic in $\zeta_A\cup z_B$.$^{\dagger 4}$
$\bullet$~$P$ is a ``docile perturbation'': $P=\sum_{k\geq 1}\epsilon^kP^{(k)}$, where
  $\deg P^{(k)}\leq 2k+2$.$^{\dagger 5}$
$\bullet$~Compositions:$^{\dagger 6}$
\[ \calF\act \calG = \calG\circ \calF \coloneqq
  \left( \calG|_{\zeta_i\to\partial_{z_i}}\calF \right)_{z_i=0}
  \!\! = \left( \calF|_{z_i\to\partial_{\zeta_i}}\calG \right)_{\zeta_i=0}.
\]
\vskip 1mm
{\bf\red Cool!} $(V^*)^{\otimes\Sigma}\otimes V^{\otimes S}$ explodes; the ranks of quadratics and
bounded-degree polynomials grow slowly!$^{\dagger 7}$

\vskip 1mm

\centering{\bf\red Representation theory is over-rated!}
}}}}

\def\Footnotes{{\raisebox{2mm}{\parbox[t]{3.96in}{\small\hangindent=5mm
{\bf\red DoPeGDO Footnotes.}
$\dagger 1$.~Each variable has a ``weight''$\in\{0,1,2\}$, and always $\wt z_i+\wt\zeta_i=2$.
\par$\dagger 2$.~Really, ``weight-graded finite sets'' $A=A_0\sqcup A_1\sqcup A_2$.
\par$\dagger 3$.~Really, a power series in the weight-0 variables$^{\dagger 9}$.
\par\hangindent=5mm$\dagger 4$.~The weight of $Q$ must be 2, so it
  decomposes as $Q=Q_{20}+Q_{11}$. The coefficients of $Q_{20}$ are rational
  numbers while the coefficients of $Q_{11}$ may be weight-0 power series$^{\dagger 9}$.
\par\hangindent=5mm$\dagger 5$.~Setting $\wt\epsilon=-2$, the weight of $P$
  is $\leq 2$ (so the powers of the weight-0 variables are not
  constrained$^{\dagger 9}$).
\par\hangindent=5mm$\dagger 6$.~There's also an obvious product
\newline\null\hfill$
  \mor(A_1\to B_1)\times\mor(A_2\to B_2)\to\mor(A_1\sqcup A_2\to B_1\sqcup B_2).
$\hfill\null
\par\hangindent=5mm$\dagger 7$.~That is, if the weight-0 variables are
  ignored. Otherwise more care is needed yet the conclusion remains.
\par\hangindent=5mm$\dagger 8$.~$\Hom(U^{\otimes\Sigma}\to U^{\otimes S})
  \leadsto\mor(\{\eta_i,\beta_i,\tau_i,\alpha_i,\xi_i\}_{i\in\Sigma}\to
    \{y_i,b_i,t_i,a_i,x_i\}_{i\in S})$,
  where $\wt(\eta_i,\xi_i,y_i,x_i)=1$ and $\wt(\beta_i,\tau_i,\alpha_i;b_i,t_i,a_i)=(2,2,0;0,0,2)$.
\par\hangindent=5mm$\dagger 9$.~For tangle invariants the weight-0 power
  series are always rational functions in the exponentials of the weight-0
  variables (for knots: just one variable).
}}}}

\def\QEFG{{In $\mor(A\!\to\!B)$, $Q\!= \hspace{-2mm}\sum\limits_{i\in A,j\in B}\hspace{-2mm}E_{ij}\zeta_iz_j \!+\! \frac12\hspace{-1mm}\sum\limits_{i,j\in
A}\hspace{-1mm}F_{ij}\zeta_i\zeta_j \!+\! \frac12\hspace{-1mm}\sum\limits_{i,j\in B}\hspace{-1mm}G_{ij}z_iz_j $}}

\def\Compositions{{\raisebox{1mm}{\parbox[t]{2.65in}{
Where
$\bullet$\quad $\omega=\omega_1\omega_2\det(I-F_2G_1)^{-1}$.
\newline$\bullet$\quad $E=E_1(I-F_2G_1)^{-1}E_2$.
\newline$\bullet$\quad $F=F_1+E_1F_2(I-G_1F_2)^{-1}E_1^T$.
\newline$\bullet$\quad $G=G_2+E_2^TG_1(I-F_2G_1)^{-1}E_2$.
\newline$\bullet$\quad $P$ is computed using ``connected Feynman diagrams'' or as the solution of a
messy PDE (yet we're still in algebra!).
%\newline\null\hfill\text{\mpink \ldots a presentation / implementation dead end.}
}}}}

\def\oneup{{\raisebox{1mm}{\parbox[t]{1.08in}{
One abstraction level up from tangles!
\par $\{\text{tangles}\}\to\left\{\rule{0pt}{10pt}\hspace{11pt}\right\}$
\par with compositions:
}}}}

\def\OverratedOne{{\raisebox{2mm}{\parbox[t]{3.96in}{
{\bf\red Categories are overrated (1)!} $\bullet$~Tangles are artificially made to have a
``top'' and a ``bottom''. $\bullet$~Tangles are accessed by their ends and not by their strands;
crossings are named by their position and not by the strands involved:
}}}}

\def\TwoReidemeister{{\raisebox{1mm}{\parbox[t]{2.15in}{
Is this $\sigma_1\sigma_2\sigma_1=\sigma_2\sigma_1\sigma_2$
\newline or $\sigma_{12}\sigma_{13}\sigma_{23}=\sigma_{23}\sigma_{13}\sigma_{12}$?

\vskip 3mm
$\bullet$~Easier to talk about ``skein theory''.
\newline $\bullet$~Harder to talk about ``universal quantum invariants''.
}}}}

\def\Debts{{\raisebox{2mm}{\parbox[t]{3.95in}{
{\bf\red Mid-Talk Debts.} $\bullet$~What is this good for in quantum algebra?
\par$\bullet$~In knot theory?
\par\hangindent=5mm$\bullet$~How does the ``inclusion''
  $\calD\colon\Hom(U^{\otimes\Sigma}\to U^{\otimes S}) \leadsto$ \text{\bf
  DoPeGDO} work?
\par$\bullet$~Proofs that everything around $sl_{2+}^\epsilon$ really
is {\bf DoPeGDO}.
\par$\bullet$~Relations with prior art.
\par$\bullet$~The rest of the ``compositions'' story.
}}}}

\def\mmr{{\raisebox{2mm}{\parbox[t]{3.95in}{
\parshape 3 0in 2.45in 0in 2.45in 0in 3.95in
\begin{hyphenrules}{nohyphenation}{\red\bf Theorem}
(\cite{Bar-NatanGaroufalidis:MMR}, conjectured \cite{MM}, elucidated
\cite{Ro}). Let $J_d(K)$ be the coloured Jones polynomial of $K$, in
the $d$-dimensional representation of $sl_2$. Writing
\end{hyphenrules}
\[ \left.
  \frac{(q^{1/2}-q^{-1/2})J_d(K)}{q^{d/2}-q^{-d/2}}
  \right|_{q=e^\hbar} =
  \sum_{j,m\geq 0} a_{jm}(K)d^j\hbar^m,
\]

\parshape 5 0in 3in 0in 3in 0in 3in 0in 3in 0in 3.95in
``below diagonal'' coefficients vanish, $a_{jm}(K)=0$ if $j>m$, and
``on diagonal'' coefficients give the inverse of the
Alexander polynomial:
$\left(\sum_{m=0}^\infty a_{mm}(K)\hbar^m\right)\cdot \omega(K)(e^\hbar)=1$.

``Above diagonal'' we have {\red Rozansky's Theorem}
\cite[(1.2)]{Rozansky:U1RCC}:
\[ J_d(K)(q) = \frac{q^d-q^{-d}}{(q-q^{-1})\omega(K)(q^d)}
  \left(1+
    \sum_{k=1}^\infty \frac{(q-1)^k\rho_k(K)(q^d)}{\omega^{2k}(K)(q^d)}
  \right).
\]
}}}}

\def\MMG{\parbox{0.6in}{\scriptsize\raggedright
  Melvin, Morton, Garoufalidis
}}

\pagestyle{empty}

\begin{document} \latintext
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\begin{center}
\null\vfill\input{DoPeGDO1.pdftex_t}\vfill\null
%\null\vfill\input{DoPeGDO2.pdftex_t}\vfill\null
\end{center}

\eject
\newgeometry{textwidth=8in,textheight=10.5in}
\begin{multicols}{2}

{\bf\red $\calD\colon\Hom(U^{\otimes\Sigma}\to U^{\otimes S}) \to \bbQ\llbracket
  \eta_\Sigma,\beta_\Sigma,\alpha_\Sigma,\xi_\Sigma,y_S,b_S,a_S,x_S
\rrbracket$.}
%$ \text{\bf DoPeGDO}.}
The PBW theorem for $CU$ (always in the $ybax$ order), or its
quantum analog for $QU$, say that if $U=CU\text{ or }QU$
then $U^{\otimes S}$ is isomorphic as a vector space to $\bbQ\llbracket
y_i,b_i,a_i,x_i\rrbracket_{i\in S}$; so it is enough to understand
$\Hom(\bbQ\llbracket z_A\rrbracket\to\bbQ\llbracket z_B\rrbracket)$
for finite sets $A$ and $B$. Using the pairing
\[
  \langle z_i^m,\zeta_j^n \rangle
  = \left.\partial_{\zeta_i}^m\zeta_j^n\right|_{\zeta_A\to 0}
  = \delta_{ij}\delta_{mn}n!,
\]
we get a map
\begin{multline*}
   \calD\colon\Hom(\bbQ\llbracket z_A\rrbracket\to\bbQ\llbracket z_B\rrbracket)
  \cong\bbQ\llbracket z_A\rrbracket^*\otimes\bbQ\llbracket z_B\rrbracket \\
  \cong\bbQ\llbracket \zeta_A\rrbracket\otimes\bbQ\llbracket z_B\rrbracket
  \cong\bbQ\llbracket \zeta_A,z_B \rrbracket
\end{multline*}

{\bf Example.} $\calD(id\colon\bbQ\llbracket z\rrbracket\to\bbQ\llbracket
z\rrbracket)=\bbe^{\zeta z}$. Indeed,
\[ \langle z^n, \bbe^{\zeta z}\rangle
  = \left\langle z^n, \sum_m\frac{(\zeta z)^m}{m!}\right\rangle
  = \sum_m\frac{z^m}{m!}\delta_{mn}n! = z^n.
\]
{\bf Example.} $\calD(id\colon U\to U) = \bbe^{\eta y+\beta b+\alpha a+\xi x}$.

{\bf Claim.} Assuming convergence, if $F\in\Hom(\bbQ\llbracket
z_A\rrbracket\to\bbQ\llbracket z_B\rrbracket)$, $G\in\Hom(\bbQ\llbracket
z_B\rrbracket\to\bbQ\llbracket z_C\rrbracket)$, $\calF=\calD(F)$, and $\calG=\calD(G)$, then
\[ \calD(F\act G) = \left( \calF|_{z_i\to\partial_{\zeta_i}}\calG \right)_{\zeta_i=0}. \]

And so the title of the talk finally makes sense!

\rule{\linewidth}{1pt}

{\bf\red Other GDOs.} {\bf Claim.} If $L\colon\bbQ\llbracket
z_A\rrbracket\to\bbQ\llbracket z_B\rrbracket$ is linear, then
$\calD(L)=L\left(\bbe^{\sum_{i\in A}\zeta_i z_i}\right)$.\hfill\text{{\bf
Proof.} Exercise.}

{\bf Example.} Let $c\Delta^i_{jk}\colon CU^{\otimes\{i\}}\to CU^{\otimes\{j,k\}}$ be the
standard co-product, given by $c\Delta^i_{jk}(y_i,b_i,a_i,x_i) = (y_j+y_k, b_j+b_k, a_j+a_k,
x_j+x_k)$. Then
\begin{multline*}
   \calD(c\Delta^i_{jk}) = c\Delta^i_{jk}(\bbe^{\eta_iy_i+\beta_ib_i+\alpha_ia_i+\xi_ix_i}) \\
  = \bbe^{\eta_i(y_j+y_k)+\beta_i(b_j+b_k)+\alpha_i(a_j+a_k)+\xi_i(x_j+x_k)}.
\end{multline*}

\needspace{1.25cm}
\parpic[r]{$\xymatrix@C=5mm@R=5mm{
  \bbQ\llbracket z\rrbracket_i\otimes\bbQ\llbracket z\rrbracket_j
    \ar[r]^<>(0.5){m^{ij}_k} \ar@{=}[d] &
  \bbQ\llbracket z\rrbracket_k \ar@{=}[d] \\
  \bbQ\llbracket z_i,z_j\rrbracket \ar[r]^<>(0.5){m^{ij}_k} &
  \bbQ\llbracket z_k\rrbracket
}$}
{\bf Example.} The standard commutative product  $m^{ij}_k$ of polynomials 
is given by $z_i,z_j\to z_k$. Hence $\calD(m^{ij}_k) = m^{ij}_k(\bbe^{\zeta_i z_i+\zeta_j z_j})
= \bbe^{(\zeta_i+\zeta_j)z_k}$.

\rule{\linewidth}{1pt}

{\red\bf A real DoPeGDO Example.} Let $cm^{ij}_k\colon CU_i\otimes CU_j\to CU_k$
be ``classical multiplication'' for $sl_{2+}^\epsilon$, and let
$\bbO_i\colon\bbQ\llbracket y_i,b_i,a_i,x_i\rrbracket\to CU_i$ be the PBW
ordering map.
\[ \xymatrix@C=5mm@R=5mm{
  CU_i\otimes CU_j \ar[r]^<>(0.5){cm^{ij}_k} &
  CU_k \\
  \bbQ\llbracket y_i,b_i,a_i,x_i,y_j,b_j,a_j,x_j \rrbracket
    \ar[u]_<>(0.4){\bbO_{i,j}} &
  \bbQ\llbracket y_k,b_k,a_k,x_k \rrbracket
    \ar[u]_<>(0.4){\bbO_k} &
} \]

{\bf Claim.} Let
\begin{multline*}
  \Lambda = \left(\eta _i+\frac{e^{-\alpha _i-\epsilon  \beta _i} \eta _j}{1+\epsilon  \eta _j \xi
_i}\right) y_k+\left(\beta _i+\beta
   _j+\frac{\log \left(1+\epsilon  \eta _j \xi _i\right)}{\epsilon }\right) b_k+ \\
  \left(\alpha
_i+\alpha _j+\log \left(1+\epsilon  \eta
   _j \xi _i\right)\right) a_k+\left(\frac{e^{-\alpha _j-\epsilon  \beta _j} \xi _i}{1+\epsilon
\eta _j \xi _i}+\xi _j\right) x_k
\end{multline*}
Then
$\bbe^{\eta_iy_i+\beta_ib_i+\alpha_ia_i+\xi_ix_i +
\eta_jy_j+\beta_jb_j+\alpha_ja_j+\xi_jx_j} \act \bbO_{i,j} \act cm^{ij}_k
= \bbe^\Lambda \act \bbO_k$, and hence $\calD(cm^{ij}_k)=\bbe^\Lambda$ and $cm^{ij}_k$ is 
DoPeGDO.

\needspace{3\baselineskip}
{\bf Proof.} We compute in a faithful 2D representation $\rho$ of $CU$:
\newline\null\hfill\text{(\web{cm})}

\newcount\snip
\snip=0\loop
  \advance \snip 1
  \par\needspace{0mm}\includegraphics[scale=\cellscale]{Snips/Examples/rho-\the\snip.pdf}
\ifnum \snip<6 \repeat

(Shame, but this technique fails for $QU$).

\rule{\linewidth}{1pt}

{\red\bf Claim. In $QU$, $R$ is {\bf DoPeGDO}.}

{\bf Proof.} Recall that with $q=\bbe^{\hbar\epsilon}$,
\[
  R = \sum\hbar^{j+k}y^kb^j\otimes a^jx^k/j![k]_q!
  = \bbO\left(\bbe^{\hbar b_1a_2}\bbe_q^{\hbar y_1x_2}\right).
\]
Now expand $\bbe_q^{\hbar y_1x_2}$ in powers of $\epsilon$ using:

{\red Faddeev's Formula} (In as much as we can tell, first
appeared without proof in Faddeev~\cite{Faddeev:ModularDouble},
rediscovered and proven in Quesne~\cite{Quesne:Jackson}, and
again with easier proof, in Zagier~\cite{Zagier:Dilogarithm}).
\text{With} ${[n]_q\coloneqq\frac{q^n-1}{q-1}}$, with
$[n]_q!\coloneqq[1]_q[2]_q\cdots[n]_q$ and with
$\bbe_q^x\coloneqq\sum_{n\geq 0}\frac{x^n}{[n]_q!}$, we have

\[ \log\bbe_q^x = \sum_{k\geq 1}\frac{(1-q)^kx^k}{k(1-q^k)}
  = x + \frac{(1-q)^2x^2}{2(1-q^2)} + \ldots .
\]

{\bf Proof.} We have that $\bbe_q^x = \frac{\bbe_q^{qx}-\bbe_q^x}{qx-x}$ (``the $q$-derivative
of
$\bbe_q^x$ is \text{itself}''), and hence $\bbe_q^{qx} = (1+(1-q)x)\bbe_q^x$, and
\[ \log\bbe_q^{qx} = \log(1+(1-q)x) + \log\bbe_q^x. \]
Writing $\log\bbe_q^x=\sum_{k\geq 1}a_kx^k$ and comparing powers of $x$, we get
$q^ka_k=-(1-q)^k/k+a_k$, or $a_k=\frac{(1-q)^k}{k(1-q^k)}$. \qed

\rule{\linewidth}{1pt}

\par{\bf\red Compositions (2).} Recall that with all indices $i$ running in some set $B$,
\[ \calF\act \calG
  = \left(\calF|_{z_i\to\partial_{\zeta_i}}\calG \right)_{\zeta_i=0}
  = \left.\bbe^{\sum\partial_{z_i}\partial_{\zeta_i}}(\calF\calG)\right|_{z_i=\zeta_i=0},
\]
so in general we wish to understand 
\[ [F\colon\calE]_B\coloneqq \bbe^{\frac12\sum_{i,j\in B} F_{ij}\partial_{z_i}\partial_{z_j}}\calE
  \quad\text{and}\quad
  \langle F\colon\calE\rangle_B\coloneqq \left.[F\colon\calE]_B\right|_{z_B\to 0},
\]
where $\calE$ is a docile perturbed Gaussian. The following lemma allows us to restrict to the
case where $\calE$ has no $B$-$B$ quadratic part:

{\bf Lemma 1.} With convergences left to the reader,
\[ \left\langle F\colon\calE\,\bbe^{\frac12\sum_{i,j\in B}G_{ij}z_iz_j}\right\rangle_B
  = \det(1-GF)^{-1/2}\left\langle F(1-GF)^{-1} \colon \calE \right\rangle_B.
\]

The next lemma dispatches the case where $\calE$ has a $B$-linear part:

{\bf Lemma 2.} $\displaystyle \left\langle F\colon\calE\,\bbe^{\sum_{i\in B}y_iz_i}\right\rangle_B
  = \bbe^{\frac12\sum_{i,j\in B}F_{ij}y_iy_j}
    \left\langle F\colon \left.\calE\right|_{z_B\to z_B+Fy_B}\right\rangle_B$.

\needspace{4\baselineskip}
Finally, we deal with the docile perturbation case:

{\bf Lemma 3.} With an extra variable $\lambda$,
$Z_\lambda\coloneqq\log[\lambda F\colon \bbe^P]_B$ satisfies and is
determined by the following PDE / IVP:
\[ Z_0=P
  \quad\text{and}\quad
  \partial_\lambda Z_\lambda = \frac12 \sum_{i,j\in B}F_{ij}\left(
    \partial_{z_i}\partial_{z_j}Z_\lambda + (\partial_{z_i}Z_\lambda)(\partial_{z_j}Z_\lambda)
  \right).
\]

\resizebox{\linewidth}{!}{\input{Lemmas.pdftex_t}}

\rule{\linewidth}{1pt}

{\bf Warning.} Some implementation details match earlier versions of the theory.

\null\hfill\web{engine}
\vskip -9mm

\newcount\snip
\snip=0\loop
  \advance \snip 1
  \par\needspace{0mm}\includegraphics[scale=\cellscale]{Snips/Engine-Speedy/Program-\the\snip.pdf}
\ifnum \snip<27 \repeat

\rule{\linewidth}{1pt}

\null\hfill\web{objects}
\vskip -6mm

\newcount\snip
\snip=0\loop
  \advance \snip 1
  \par\needspace{0mm}\includegraphics[scale=\cellscale]{Snips/Objects/Program-\the\snip.pdf}
\ifnum \snip<20 \repeat

\rule{\linewidth}{1pt}
\par{\red\bf A Quantum Algebra Example.}\hfill\text{\web{qa}}

{\bf Proto-Theorem}$^{\dagger 0}$ (with Jesse Frohlich and Roland van der Veen). 
Let $H$ be a finite dimensional Hopf algebra and let $U=H^{\ast\text{\it cop}}\otimes H$ be its Drinfel'd
double, with $R$-matrix $R\in H^\ast\otimes H\subset U\otimes U$. Write $R^{\dagger 1}=\sum\rho_a\otimes r_a$, and let
$\langle\cdot\mid\cdot\rangle\colon H^\ast\otimes H\to\bbF$ be the duality pairing. Then the functional $\int\in U^\ast$ defined
by
\[ \int\phi\otimes x\coloneqq\sum\langle\phi\rho_a^{\dagger 2}\mid xr_a^{\dagger 3}\rangle \]
is a right$^{\dagger 4}$ integral in $U^\ast$. (Meaning $\Delta^i_{jk}\act\int_j = \int_i\act\epsilon_k$ in
$\Hom(U^{\otimes\{i\}}\to U^{\otimes\{k\}})$).

{\footnotesize
$\dagger 0$~A ``proto-theorem'' is something that will become a theorem once you figure out the correct statement.
$\dagger 1$~Or did we want it to be $R\act S_1^2$? Or $R\act S_2^2$?
$\dagger 2$~Or is it $\rho_a\phi$?
$\dagger 3$~Or is it $r_ax$?
$\dagger 4$~Or maybe ``left''?
}

\newcount\snip
\snip=0\loop
  \advance \snip 1
  \par\needspace{0mm}\includegraphics[scale=\cellscale]{Snips/Examples/Integrals-\the\snip.pdf}
\ifnum \snip<2 \repeat

\rule{\linewidth}{1pt}
\par{\red\bf A Knot Theory Example.}\hfill\text{\web{kt}}

\newcount\snip
\snip=0\loop
  \advance \snip 1
  \par\needspace{0mm}\includegraphics[scale=\cellscale]{Snips/Examples/Trefoil-\the\snip.pdf}
\ifnum \snip<3 \repeat

\rule{\linewidth}{1pt}
\par{\red\bf References.}
%\par\vspace{-3mm}
\renewcommand{\section}[2]{}%
\begin{thebibliography}{}
\setlength{\parskip}{0pt}
\setlength{\itemsep}{0pt plus 0.3ex}

\bibitem[BG]{Bar-NatanGaroufalidis:MMR} D.~Bar-Natan and S.~Garoufalidis,
  {\em On the Melvin-Morton-Rozansky conjecture,}
  Invent.\ Math.\ {\bf 125} (1996) 103--133.

\bibitem[BV]{PP1} D.~Bar-Natan and R.~van~der~Veen,
  {\em A Polynomial Time Knot Polynomial,}
  \arXiv{1708.04853}.

\bibitem[Fa]{Faddeev:ModularDouble} L.~Faddeev,
  {\em Modular Double of a Quantum Group,}
  \arXiv{math/9912078}.

%\bibitem[GR]{GaroufalidisRozansky:LoopExpansion} S.~Garoufalidis and L.~Rozansky,
%  {\em The Loop Exapnsion of the Kontsevich Integral, the Null-Move, and $S$-Equivalence,}
%  \arXiv{math.GT/0003187}.

\bibitem[MM]{MM} P.~M.~Melvin and H.~R.~Morton,
  {\em The coloured Jones function,}
  Commun.\ Math.\ Phys.\ {\bf 169} (1995) 501--520.

\bibitem[Ov]{Overbay:Thesis} A.~Overbay,
  {\em Perturbative Expansion of the Colored Jones Polynomial,}
  University of North Carolina PhD thesis, \web{Ov}.

\bibitem[Qu]{Quesne:Jackson} C.~Quesne,
  {\em Jackson's $q$-Exponential as the Exponential of a Series,}
  \arXiv{math-ph/0305003}.

\bibitem[Ro1]{Ro} L.~Rozansky,
  {\em A contribution of the trivial flat connection to the Jones
polynomial and Witten's invariant of 3d manifolds, I,}
  Comm.\ Math.\ Phys.\ {\bf 175-2} (1996) 275--296, \arXiv{hep-th/9401061}.

\bibitem[Ro2]{Rozansky:Burau} L.~Rozansky,
  {\em The Universal $R$-Matrix, Burau Representation and the Melvin-Morton
    Expansion of the Colored Jones Polynomial,}
  Adv.\ Math.\ {\bf 134-1} (1998) 1--31, \arXiv{q-alg/9604005}.

\bibitem[Ro3]{Rozansky:U1RCC} L.~Rozansky,
  {\em A Universal $U(1)$-RCC Invariant of Links and Rationality Conjecture,}
  \arXiv{math/0201139}.

\bibitem[Za]{Zagier:Dilogarithm} D.~Zagier,
  {\em The Dilogarithm Function,}
  in Cartier, Moussa, Julia, and Vanhove (eds) {\em Frontiers in Number
  Theory, Physics, and Geometry II.} Springer, Berlin, Heidelberg,
  and \web{Za}.

\end{thebibliography}

\end{multicols}

\rule{\linewidth}{1pt}

\newpage

\begin{multicols}{2}

{\bf\red KiW 43 Abstract} (\web{kiw}). Whether or not you like the formulas
on this page, they describe the strongest truly computable knot invariant
we know.%\hfill\text{\footnotesize(experimental analysis @\web{kiw})}

{\bf\red Observations.} $\bullet$~Separates the Rolfsen table; does better than Khovanov plus
HOMFLY-PT on knots with up to 12 crossings (not tested
beyond). $\bullet$~The degrees are bounded by the genus!
$\bullet$~$\rho_1$ vanishes for amphichiral knots. $\bullet$~Has a chance
of detecting non-ribbonness (\web{ind})!

\end{multicols}

%\vskip -7mm

\def\N{\ding{56}}
\def\gY{\textcolor{ForestGreen}{\ding{52}}}
\def\oY{\textcolor{Orange}{\ding{52}}}

\newlength{\outerboxwidth} \newlength{\innerboxwidth}

\def\headcell{\parbox[t]{\outerboxwidth}{
  \hspace{-4pt}\parbox[b]{0.4in}{knot diag}
  \parbox[b]{\innerboxwidth}{
    {\purple $n_k^t$}\quad Alexander's $\omega^+$\hfill genus / \textcolor{ForestGreen}{ribbon} \\
    {\red $(\rho'_1)^+$} \hfill unknotting \# / \textcolor{Orange}{amphi?}
  }\\
  \centering{\tiny\blue $(\rho'_2)^+$}
  \vskip 1pt
}}
\def\rolcell#1#2#3#4#5#6#7#8#9{\parbox[t]{\outerboxwidth}{
  \hspace{-4pt}\parbox[b]{0.4in}{\raisebox{-3pt}{\includegraphics[height=21pt]{../UNC-1610/KnotFigs/#1.pdf}}}
  \parbox[b]{\innerboxwidth}{
    {\purple $#2$}\quad $#3$\hfill $#6$ / #8 \\
    {\red $#4$} \hfill $#7$ / #9
  }\\
  \centering{\tiny\blue $#5$}
  \vskip 1pt
}}

\setlength{\outerboxwidth}{2.5in} \setlength{\innerboxwidth}{2.1in}
{\footnotesize \begin{longtable}{|c|c|c|}
\hline \headcell & \headcell & \headcell \\ \endhead
\hline
\input table1.tex
\end{longtable}}

\setlength{\outerboxwidth}{3.83in} \setlength{\innerboxwidth}{3.43in}
{\footnotesize \begin{longtable}{|c|c|c|}
\hline \headcell & \headcell \\ \endhead
\hline
\input table2.tex
\end{longtable}}

\begin{center}
\input{DoPeGDO3.pdftex_t}
\end{center}

\end{document}
\endinput

