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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{Kyoto-230727}
\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 Kyoto!}}
\def\webdef{{{\greektext web}$\coloneqq$\href{http://drorbn.net/k23}{http://drorbn.net/k23}}}
\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}}

\definecolor{mgray}{HTML}{B0B0B0}
\definecolor{morange}{HTML}{FFA50A}
\definecolor{mpink}{HTML}{F35EB7}
\def\blue{\color{blue}}
\def\gray{\color{gray}}
\def\mgray{\color{mgray}}
\def\morange{\color{morange}}
\def\mpink{\color{mpink}}
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\def\cbox#1#2{{\setlength{\fboxsep}{0pt}\colorbox{#1}{#2}}}

\def\arXiv#1{{\href{https://arxiv.org/abs/#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\bbN{{\mathbb N}}
\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]{1.35in}{
Continues Rozansky \cite{Ro, Rozansky:Burau, Rozansky:U1RCC}, Ohtsuki \cite{Ohtsuki:2Loop}, Overbay
\cite{Overbay:Thesis}, joint with van der Veen \cite{BV}.
}}}}

\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 $C^{\pm 1}\in QU$}}
\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 $\exists \epsilon^{-1}$,
$sl_{2+}^\epsilon/\langle t\rangle\cong sl_2$.\hfill{\tiny \web{oa}}

$U$ is either $CU=\calU(sl_{2+}^\epsilon)\llbracket\hbar\rrbracket$ or
$QU=\calU_\hbar(sl_{2+}^\epsilon) = A\langle y,b,a,x\rangle\llbracket\hbar\rrbracket$ 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
$\mor(A\to B)$:
\[ \left\{\calF =\omega\exp(Q+P) \right\} \subset \bbQ\llbracket\zeta_A,z_B,\epsilon\rrbracket \]
Where:
$\bullet$~$\omega$ is a scalar.$^{\dagger 3}$
$\bullet$~$Q$ is a ``small'' $\epsilon$-free 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 0.5mm
{\bf\red Cool!} $(V^*)^{\otimes\Sigma}\otimes V^{\otimes S}$ explodes; the ranks of quadratics
and
bounded-degree polynomials grow slowly!$^{\dagger 7}$
\hfill\text{\red Representation theory is over-rated!}
\vskip 0.5mm
{\bf\red Cool!} How often do you see a computational toolbox so successful?
}}}}

\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,{\gray \tau_i,}\alpha_i,\xi_i\}_{i\in\Sigma}\to
    \{y_i,b_i,{\gray t_i,}a_i,x_i\}_{i\in S})$,
  where $\wt(\eta_i,\xi_i,y_i,x_i)=1$ and $\wt(\beta_i,{\gray \tau_i,}\alpha_i;b_i,{\gray
  t_i},a_i)=(2,2,0;0,0,2)$.
\par\hangindent=5mm$\dagger 9$.~For tangle invariants the wt-0 power
  series are always rational functions in the exponentials of the wt-0
  variables (for knots: just one variable), with degrees bounded linearly by the crossing
  number.
}}}}

\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]{2.75in}{
Where
$\bullet$\quad $\omega\!=\!\omega_1\omega_2\det(I-F_2G_1)^{-1}$.
\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 $E \!=\! \sum_r E_1(F_2G_1)^rE_2 \!=\! E_1(I-F_2G_1)^{-1}E_2$.
\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
\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 $\bbQ[y_i,b_i,a_i,x_i]_{i\in
S}\llbracket\hbar\rrbracket$;
so it is enough to understand $\Hom(\bbQ[z_A]\to\bbQ[z_B])$ for finite sets $A$ and $B$.

{\bf Claim.} $F\in\Hom(\bbQ[z_A]\to\bbQ[z_B])
\xrightarrow[\calD]{\raisebox{-0.75ex}[0ex][0ex]{$\sim$}}
\bbQ[z_B]\llbracket\zeta_A\rrbracket\ni \calF$ via
\[ \calD(F)
  \coloneqq \sum_{n\in\bbN^A}\frac{\zeta_A^n}{n!}F(z_A^n)
  = F \left(\bbe^{\sum_{a\in A}\zeta_a z_a}\right)
  = \calF,
\]
\[ \calD^{-1}(\calF)(p) =
\left(\left.p\right|_{z_a\to\partial_{\zeta_a}}\calF\right)_{\zeta_a=0}
  \quad\text{for $p\in\bbQ[z_A]$}.
\]

{\bf Claim.} Assuming convergence, if $F\in\Hom(\bbQ[
z_A]\to\bbQ[ z_B])$, $G\in\Hom(\bbQ[
z_B]\to\bbQ[ z_C])$, $\calF=\calD(F)$, 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!

\vskip -2mm\rule{\linewidth}{1pt}\vskip -1mm

{\bf Example.} $\calD(id\colon U\to U) = \bbe^{\eta y+\beta b+\alpha a+\xi x}$.

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

\vskip -2mm\rule{\linewidth}{1pt}\vskip -1mm

{\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
$\bbO_i\colon\bbQ[ y_i,b_i,a_i,x_i]\to CU_i$ be the PBW
ordering map.
\[ \xymatrix@C=5mm@R=5mm{
  CU_i\otimes CU_j \ar[r]^<>(0.5){cm^{ij}_k} &
  CU_k \\
  \bbQ[ y_i,b_i,a_i,x_i,y_j,b_j,a_j,x_j ]
    \ar[u]_<>(0.4){\bbO_{i,j}} &
  \bbQ[ y_k,b_k,a_k,x_k ]
    \ar[u]_<>(0.4){\bbO_k} &
} \]

{\bf Claim.} Let\hfill\text{\footnotesize(all brawn and no brains)}
\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 $z\mapsto\hat{z}$ of $CU$:
\newline\null\hfill\text{(\web{cm})}

\newcount\snip
\snip=0\loop
  \advance \snip 1
  \par\needspace{0mm}\includegraphics[scale=\cellscale]{../UCLA-191101/Snips/Examples/rho-\the\snip.pdf}
\ifnum \snip<6 \repeat

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

\vskip -2mm\rule{\linewidth}{1pt}\vskip -1mm

{\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

\vskip -2mm\rule{\linewidth}{1pt}\vskip -1mm

\par{\bf\red Compositions (2).} Recall that with all indices $i$ running in some set $B$,
\[ \calF\act \calG
  = \left(\calF|_{z_i\to\partial_{\zeta_i}}\calG \right)_{\zeta_i=0}
  \overset{(1)}{=}
\left.\bbe^{\sum\partial_{z_i}\partial_{\zeta_i}}(\calF\calG)\right|_{z_i=\zeta_i=0},
  \quad\parbox{0.7in}{\tiny (1) Strictly speaking,\newline true only when\newline $B\cap(A\cup
C)=\emptyset$.}
\]
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}{!}{\import{../QMUL-1908/}{Lemmas.pdftex_t}}

\vskip -2mm\rule{\linewidth}{1pt}\vskip -1mm

\par{\bf\red Complexity} to $\epsilon^k$, for an $n$-xing width $w$
knot (by \cite{LiptonTarjan:Separator}, ${w\in O(\!\sqrt{n})}$), is
$O(n^2w^{2k+2}\log n)=O(n^{k+3}\log n)$ integer operations.

%\vskip -2mm\rule{\linewidth}{1pt}\vskip -1mm
\newpage

{\red\bf A Partial To Do List.}
\begin{itemize}[leftmargin=*,labelindent=0pt,itemsep=-4pt,topsep=0pt]
\item Understand $\tr$ and links.
\item Implement $\Phi$, $J$. Determine the appropriate $\wt$-0 ground ring.
\item Implement the ``dequantizators''.
\item Understand denominators and get rid of them.
\item Clean the program and make it efficient.
\item Understand the centre and figure out how to read the output.
\item Is the ``$+$'' really necessary in $sl_{2+}^\epsilon$? Why?
\item Extend to $sl_3$ and beyond.
\item Describe a genus bound and a Seifert formula.
\item Relate with the representation theory dogma, with Melvin-Morton-Rozansky and with
Rozansky-Overbay.
\item Understand the braid group representations that arise.
\item Relate with finite-type (Vassiliev) invariants.
\item Find a topological interpretation/foundation. The \text{Garoufalidis} - Rozansky
  ``loop expansion'' \cite{GaroufalidisRozansky:LoopExpansion}?
\item Figure out the action of the Cartan automorphism.
{\blue \item Disprove the ribbon-slice conjecture!}
\item Figure out the action of the Weyl group.
\item Use to study ``\v{S}evera quantization''.
\item Do everything at the ``arrow diagram'' level of finite-type invariants of
(rotational) virtual tangles.
\item Find ``internal'' proofs of consistency.
\item What else can you do with the ``solvable approximations''?
\item And with the ``Gaussian compositions'' technology?
\end{itemize}

\end{multicols}

\rule{\linewidth}{1pt}

\begin{multicols}{2}

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

\parpic[r]{\scalebox{0.85}{\import{../QMUL-1908/}{ZippingTheorem.pdf_t}}\quad}
{\red\bf The Zipping Theorem.} If $P$ has a finite
$\zeta$-degree and $\tilde{q}$ is the inverse matrix of $1-q$:
$(\delta^i_j-q^i_j)\tilde{q}^j_k=\delta^i_k$, then
\begin{multline*} \left\langle
    P(z_i,\zeta^j)\bbe^{c+\eta^iz_i+y_j\zeta^j+q^i_jz_i\zeta^j}
  \right\rangle
  \\
  = |\tilde{q}|\bbe^{c+\eta^i\tilde{q}_i^ky_k}
    \left\langle P\left( \tilde{q}_i^k(z_k+y_k),\zeta^j+\eta^i\tilde{q}_i^j \right)
\right\rangle.
\end{multline*}
\picskip{0}

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

\newcount\snip
\snip=0\loop
  \advance \snip 1
  \par\needspace{0mm}\includegraphics[scale=\cellscale]{../../Projects/SL2Portfolio2/Snips/Engine-Speedy/Program-\the\snip.pdf}
\ifnum \snip<33 \repeat

\rule{\linewidth}{1pt}

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

\newcount\snip
\snip=0\loop
  \advance \snip 1
  \par\needspace{0mm}\includegraphics[scale=\cellscale]{../../Projects/SL2Portfolio2/Snips/Objects/Program-\the\snip.pdf}
\ifnum \snip<21 \repeat

\rule{\linewidth}{1pt}
\par{\red\bf Some of the Atoms.}\hfill\text{\web{atoms}}

With $\calA_i\coloneqq\bbe^{\alpha_i}$ and $B_i=\bbe^{-b_i}$,

\newcount\snip
\snip=0\loop
  \advance \snip 1
  \par\needspace{0mm}\includegraphics[scale=\cellscale]{../UCLA-191101/Snips/Examples/Atoms-\the\snip.pdf}
\ifnum \snip<2 \repeat

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

{\bf Proto-Proposition}$^{\dagger 0}$ (with Jesse Frohlich and Roland van der Veen,
near~\cite[Proposition~1.7.3]{Majid:Foundations}).
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-proposition'' is something that will become a
  proposition 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]{../UCLA-191101/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]{../UCLA-191101/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]{BV} D.~Bar-Natan and R.~van~der~Veen,
  {\em A Polynomial Time Knot Polynomial,}
  Proc.\ Amer.\ Math.\ Soc.\ {\bf 147} (2019) 377--397, \arXiv{1708.04853};
  {\em Perturbed Gaussian Generating Functions for Universal Knot Invariants,}
  \arXiv{2109.02057};
  {\em A Perturbed-Alexander Invariant,} \arXiv{2206.12298}.

\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 Expansion of the Kontsevich Integral, the Null-Move, and $S$-Equivalence,}
  \arXiv{math.GT/0003187}.

\bibitem[LT]{LiptonTarjan:Separator} R.~J.~Lipton and R.~E.~Tarjan,
  {\em A Separator Theorem for Planar Graphs,}
  SIAM J.\ Appl.\ Math.\ {\bf 36-2} (1979) 177--189.

\bibitem[Ma]{Majid:Foundations} S.~Majid,
  {\em Foundations of Quantum Group Theory,}
  Cambridge University Press, 1995.

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

\bibitem[Oh]{Ohtsuki:2Loop} T.~Ohtsuki,
  {\em On the 2–Loop Polynomial of Knots,}
  Geom.\ Top.\ {\bf 11-3} (2007) 1357--1475.

\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}

\rule{\linewidth}{1pt}

{\red\bf Acknowledgement.} This work was partially supported by NSERC
grant RGPIN-2018-04350 and by the Chu Family Foundation (NYC).

\end{multicols}

\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
\import{../QMUL-1908/}{table1.tex}
\end{longtable}}

\setlength{\outerboxwidth}{3.83in} \setlength{\innerboxwidth}{3.43in}
{\footnotesize \begin{longtable}{|c|c|c|}
\hline \headcell & \headcell \\ \endhead
\hline
\import{../QMUL-1908/}{table2.tex}
\end{longtable}}

\end{document}
\endinput
