\documentclass[11pt,notitlepage]{article}
\def\bare{n}
\usepackage[english,greek]{babel}
\usepackage{amsmath, graphicx, amssymb, datetime, multicol, stmaryrd, 
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\usepackage{tensor}
\usepackage{txfonts}	% for the likes of \coloneqq.
\usepackage[usenames,dvipsnames]{xcolor}
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\parindent 0in

% 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
\usepackage[framemethod=tikz]{mdframed}

\def\myurl{http://www.math.toronto.edu/~drorbn}
\def\thistalk{LesDiablerets-1608}
\def\title{The Brute and the Hidden Paradise}

\def\navigator{{
  \href{\myurl}{Dror Bar-Natan}:
  \href{\myurl/Talks}{Talks}:
  \href{\myurl/Talks/\thistalk/}{\thistalk}:
}}
\def\webdef{{{\greektext web}$\coloneqq$\url{http://drorbn.net/\thistalk/}}}
\def\web#1{{\href{\myurl/Talks/\thistalk/#1}{{\greektext web}/#1}}}
\def\titleA{{\title}}
\def\titleB{{\title}}
\def\titleC{{\title}}

\def\todo#1{\text{\Huge #1}}

\def\blue{\color{blue}}
\def\red{\color{red}}
\def\cbox#1#2{{\setlength{\fboxsep}{0pt}\colorbox{#1}{#2}}}

\def\arXiv#1{{\href{http://front.math.ucdavis.edu/#1}{arXiv:\linebreak[0]#1}}}
\def\ob#1{\overbracket[0.5pt][1pt]{#1}}
\def\ub#1{\underbracket[0.5pt][1pt]{#1}}

\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\CW{\text{\it CW}}
\def\diag{\operatorname{diag}}
\def\FL{\text{\it FL}}
\def\IHX{\mathit{IHX}}
\def\PvT{{\mathit P\!v\!T}}
\def\remove{\!\setminus\!}
\def\STU{\mathit{STU}}
\def\TC{\mathit{TC}}
\def\tr{\operatorname{tr}}
\def\vT{{\mathit v\!T}}

\def\bbE{{\mathbb E}}
\def\bbO{{\mathbb O}}
\def\bbQ{{\mathbb Q}}
\def\bbR{{\mathbb R}}
\def\bbZ{{\mathbb Z}}
\def\calA{{\mathcal A}}
\def\calD{{\mathcal D}}
\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\Abstract{{\raisebox{2mm}{\parbox[t]{3.95in}{
{\red Abstract.} There is expected to be a hidden paradise of poly-time 
computable knot polynomials lying just beyond the Alexander polynomial. I
will describe my brute attempts to gain entry.
}}}}

\def\Expected{{\raisebox{2mm}{\begin{minipage}[t]{3.95in}
{\red Why ``expected''?} Finite-type invariants include all coefficients of
all quantum knot polynomials (appropriately parametrized), and each is
computable in poly-time. Yet

\vspace{-2mm}

\definecolor{lightgreen}{RGB}{127,255,127}
\definecolor{lightergreen}{RGB}{191,255,191}
\definecolor{lightestgreen}{RGB}{223,255,223}
\begin{center}\begin{tabular}{r|ccc>{\columncolor{lightgreen}}c>{\columncolor{lightergreen}}c>{\columncolor{lightestgreen}}cccc}
  $d$ &                             2 & 3 & 4 &         5 & 6 & 7 & 8  &
$\cdots$ \\
  \hline
  \it known f.t.\ invts in $O(n^d)$ &  1 & 1 & $\infty$ &  3 & 4 & 8 & 11
& $\cdots$
\end{tabular}\end{center}
\vskip -3mm

This is an unreasonable picture! So there ought to be further poly-time
polynomial invariants.

\parshape 3 0in 3.6in 0in 3.6in 0in 3.95in
{\red Also.} $\bullet$ The line above the Alexander line in the
Melvin-Morton \cite{MM,Ro} expansion of the coloured Jones polynomial.
$\bullet$ The 2-loop contribution to the Kontsevich integral.
\end{minipage}}}}

\def\Paradise{{\raisebox{2mm}{\begin{minipage}[t]{3.95in}
{\red Why ``paradise''?} Foremost answer: {\red\sl OBVIOUSLY.}
{\footnotesize Cf.\ proving (incomputable $A$)$=$(incomputable $B$), or
categorifying (incomputable $C$).}
\par Secondary answer: may disprove $\{$ribbon$\}=\{$slice$\}$:\hfill\text{(see~\cite{K17})}
\end{minipage}}}}

\def\Ribbon{{\raisebox{2mm}{\parbox[t]{3.95in}{
{\red A bit about ribbon knots.} A ``ribbon knot'' is a knot that can be
presented as the boundary of a disk that has ``ribbon singularities'', but
no ``clasp singularities''. A ``slice knot'' is a knot in $S^3=\partial
B^4$ which is the boundary of a non-singular disk in $B^4$. Every ribbon
knots is clearly slice, yet,

{\red Conjecture.} Some slice knots are not ribbon.

{\red Fox-Milnor.} The Alexander polynomial of a ribbon knot is always of
the form $A(t)=f(t)f(1/t)$.\hfill{(also~for~slice)}
}}}}

\def\Ta{$\calT_{2n}$}
\def\Tb{$U\in\calT_n$}
\def\Tc{ribbon $K\in\calT_1$}
\def\Aa{$\calA_{2n}$}
\def\Ab{$1\in\calA_n$}
\def\Ac{$z(K)\in\calR\subseteq\calA_1$}
\def\Ra{with $\calR\coloneqq$}
\def\Rb{$\kappa(\tau^{-1}(1))$}

\def\GST{\parbox{0.5in}{\tiny
  Gompf, Scharlemann, Thompson
\cite{GompfScharlemannThompson:Counterexample}
}}

\def\MetaAssoc{{\parbox{1.2in}{
  (meta-associativity: $m^{ab}_x\act m^{xc}_y=m^{bc}_x\act m^{ax}_y$)
}}}

\def\Gold{{\raisebox{5mm}{\begin{minipage}[t]{3.95in}
\pichskip{0mm}\parpic[l]{
  \includegraphics[height=0.4in]{../Greece-1607/IASLogo.png}
} \picskip{2}
{\red The Gold Standard} is set by the ``$\Gamma$-calculus'' Alexander formulas
\cite{Bar-NatanSelmani:MetaMonoids, KBH}. An $S$-component
tangle $T$ has $\Gamma(T) \in R_S\times M_{S\times S}(R_S) =
\left\{\begin{array}{c|c}\omega&S\\\hline S&A\end{array}\right\}$ with
$R_S\coloneqq\bbZ(\{t_a\colon a\in S\})$:

$\displaystyle
  \left(\tensor[_a]{\overcrossing}{_b},\tensor[_b]{\undercrossing}{_a}\right)
  \to
  \begin{array}{c|cc} 1 & a & b \\ \hline a & 1 & 1-t_a^{\pm 1} \\ b & 0 &
t_a^{\pm 1} \end{array}
$
\hfill$\displaystyle
  T_1\sqcup T_2
  \to
  \begin{array}{c|cc} \omega_1\omega_2 & S_1 & S_2 \\ \hline S_1 & A_1 & 0
\\ S_2 & 0 & A_2 \end{array}
$
\vskip -1mm
\[ \begin{CD}
  \begin{array}{c|ccc}
    \omega & a & b & S \\
    \hline
    a & \alpha & \beta & \theta \\
    b & \gamma & \delta & \epsilon \\
    S & \phi & \psi & \Xi
  \end{array}
%  @>{\displaystyle m^{ab}_c}>{\displaystyle t_a,t_b\to t_c\atop\displaystyle \mu\coloneqq 1-\beta}>
  @>{\displaystyle m^{ab}_c}>{\displaystyle t_a,t_b\to t_c}>
  \left(\!\begin{array}{c|cc}
%    \mu\omega & c & S \\
%    \hline
%    c & \gamma+\alpha\delta/\mu & \epsilon+\delta\theta/\mu \\
%    S & \phi+\alpha\psi/\mu & \Xi+\psi\theta/\mu
    (1-\beta)\omega & c & S \\
    \hline
    c & \gamma+\frac{\alpha\delta}{1-\beta} & \epsilon+\frac{\delta\theta}{1-\beta} \\
    S & \phi+\frac{\alpha\psi}{1-\beta} & \Xi+\frac{\psi\theta}{1-\beta}
  \end{array}\!\right)
\end{CD} \]
\end{minipage}}}}

\def\Dunfield{{\raisebox{0mm}{\parbox[t]{2.85in}{
For long knots, $\omega$ is Alexander, and that's the fastest Alexander
algorithm I know!
\newline\null\hfill\text{\footnotesize Dunfield: 1000-crossing fast.}
}}}}

\def\EKTheorem{{\raisebox{2mm}{\begin{minipage}[t]{3.95in}
{\red Theorem} \cite{EtingofKazhdan:BialgebrasI,
Haviv:DiagrammaticAnalogue, Enriquez:Quantization,
Severa:BialgebrasRevisited}. There is a ``homomorphic expansion''
\[ \arraycolsep=0pt \renewcommand{\arraystretch}{1}
  \begin{array}{c}
    Z\colon\left\{\parbox{0.85in}{$S$-component ($v/b$-)tangles}\right\}
    \to \calA^v_S \coloneqq \\
  \parbox{0.45\linewidth}{\vskip 2mm
  \includegraphics[height=0.4in]{../../Projects/Gallery/Etingof.jpg}%
  \hfill\includegraphics[height=0.4in]{../../Projects/Gallery/Kazhdan.jpg}%
  \hfill\includegraphics[height=0.4in]{../../Projects/Gallery/Haviv.jpg}%
  \hfill\includegraphics[height=0.4in]{../../Projects/Gallery/Enriquez.jpg}%
  \hfill\includegraphics[height=0.4in]{../../Projects/Gallery/Severa.jpg}
  \hfill} \\
  \parbox{0.45\linewidth}{\tiny Etingof \hfill Kazhdan \hfill Haviv \hfill
Enriquez \hfill \v{S}evera \hfill}
  \end{array}
  \begin{array}{c}\resizebox{0.55\linewidth}{!}{\input{figs/Av.pdf_t}}\end{array}
\]
(it is enough to know $Z$ on $\overcrossing$ and have disjoint union and
stitching formulas)\hfill\text{\red\ldots exponential and too hard!}

{\red Idea.} Look for ``ideal'' quotients of $\calA^v_S$ that have
poly-sized descriptions; \hfill\text{\ldots specifically, limit the
co-brackets.}
\end{minipage}}}}

\def\OneCoTwoCo{{\raisebox{2mm}{\parbox[t]{1.6in}{
{\red 1-co and 2-co,} aka $\TC$ and $\TC^2$, on the right. The primitives that
remain are:
}}}}

\def\TwoD{{\raisebox{2mm}{\parbox[t]{3.1in}{
{\red The $2D$ relations} come from the relation with 2D Lie bialgebras:
}}}}

\def\TwoTwoDefs{{\raisebox{0mm}{\parbox[t]{3.95in}{
We let $\calA^{2,2}$ be $\calA^v$ modulo 2-co and $2D$, and $z^{2,2}$ be
the projection of $\log Z$ to $\calP^{2,2}\coloneqq\pi\calP^v$, where
$\calP^v$ are the primitives of $\calA^v$.

{\red Main Claim.} $z^{2,2}$ is poly-time computable.
}}}}

\def\MainPoint{{\raisebox{2mm}{\parbox[t]{3.95in}{
{\red Main Point.} $\calP^{2,2}$ is poly-size, so how hard can it be?
Indeed, as a module over $\bbQ\llbracket b_i\rrbracket$, $\calP^{2,2}$ is at
most
\vskip 23mm
{\red Claim.} $R_{jk}=e^{a_{jk}}e^{\rho_{jk}}$ is a solution of the
Yang-Baxter / R3 equation $R_{12}R_{13}R_{23}= R_{23}R_{13}R_{12}$ in
$\exp\calP^{2,2}$, with $\rho_{jk} \coloneqq$
\[
  \psi(b_j)\left(-c_k + \frac{c_ka_{jk}}{b_j} - \frac{\delta
    a_{jk}a_{jk}}{b_j^2}\right)
  + \frac{\phi(b_j)\psi(b_k)}{b_k\phi(b_k)}\left(c_ka_{kk} - \frac{\delta
    a_{jk}a_{kk}}{b_j}\right),
\]
and with $\phi(x)\coloneqq e^{-x}-1 = -x+x^2/2-\dots$, and
$\psi(x)\coloneqq\left((x+2)e^{-x}-2+x\right)/(2x) = x^2/12-x^3/24+\dots$.
}}}}

\def\SnG{{\raisebox{2mm}{\parbox[t]{3.95in}{
{\red Problem.} How do we multiply in $\exp(\calP^{2,2})$? How do we
stitch?
}}}}

\def\Smidgen{{\raisebox{2mm}{\parbox[t]{3.95in}{
\parshape 6 0in 3.45in 0in 3.45in 0in 3.45in 0in 3.45in 0in 3.25in 0in 3.95in 
{\red 1-Smidgen $sl_2$} (with van der Veen). Let {\red $\frakg_1$}
be the 4D Lie algebra $\frakg_1=\langle b,c,u,w\rangle$ over
$\bbQ[\epsilon]/(\epsilon^2=0)$, with $b$ central and \cbox{yellow}{$[w,c]=w$,
$[c,u]=u$, and $[u,w]=b-2\epsilon c$}, with CYBE $r_{ij}=(b_i-\epsilon
c_i)c_j+u_iw_j$ in $\calU(\frakg_1)^{\otimes\{i,j\}}$. Over $\bbQ$,
$\frakg_1$ is a {\red solvable approximation of $sl_2$}: $\frakg_1 \supset
\langle b,u,w,\epsilon b,\epsilon c,\epsilon u,\epsilon w\rangle \supset
\langle b,\epsilon b,\epsilon c,\epsilon u,\epsilon w\rangle \supset 0$.
In a certain sense, $\frakg_1$ is more valuable than $sl_2$.
}}}}

\def\ZeroSmidgen{{\raisebox{2mm}{\parbox[t]{3.95in}{
{\red 0-Smidgen $sl_2$ \smiley{}.} Let $\frakg_0$
be $\frakg_1$ at $\epsilon=0$, or $\bbQ\langle
b,c,u,w\rangle/([b,\cdot]=0,\,[c,u]=u,\,[c,w]=-w,\,[u,w]=b$ with
$r_{ij}=b_ic_j+u_iw_j$.
It is $\fraka^\ast\rtimes\fraka$ where $\fraka$ is the 2D Lie algebra
$\bbQ\langle b,u\rangle$ and $(c,w)$ is the dual basis of $(b,u)$.
It is even more valuable than $\frakg_1$, but topology already got by
other means almost everything $\frakg_0$ has to give.
}}}}

\def\HowArose{{\raisebox{2mm}{\parbox[t]{3.95in}{
{\red How did these arise?} $sl_2=\frakb^+\oplus\frakb^-/\frakh\eqqcolon
sl_2^+/\frakh$, where $\frakb^+=\langle c,w\rangle/[w,c]=w$ is a Lie
bialgebra with $\delta\colon\frakb^+\to\frakb^+\otimes\frakb^+$ by
$\delta\colon(c,w)\mapsto(0,c\wedge w)$. Going back,
$sl_2^+=\calD(\frakb^+) = (\frakb^+)^\ast\oplus\frakb^+ = \langle
b,u,c,w\rangle/\cdots$. {\red Idea.} Replace $\delta\to\epsilon\delta$ over
$\bbQ[\epsilon]/(\epsilon^{k+1}=0)$. At $k=0$, get $\frakg_0$. At $k=1$,
get $[w,c]=w$, $[w,b']=-\epsilon w$, $[c,u]=u$, $[b',u]=-\epsilon u$,
$[b',c]=0$, and $[u,w]=b'-\epsilon c$. Now note that $b'+\epsilon c$ is
central, so switch to $b\coloneqq b'+\epsilon c$. This is $\frakg_1$.
}}}}

\def\ZeroInvariants{{\raisebox{2mm}{\parbox[t]{3.95in}{
{\red 0-Smidgen Invariants.} $r=Id\in\frakb^-\otimes\frakb^+$ solves the
CYBE $[r_{12},r_{13}]+[r_{12},r_{23}]+[r_{13},r_{23}]=0$ in
$\calU(\frakg_0)^{\otimes 3}$ and, by luck, 
}}}}

\def\Rzero{$= R_{ij} = e^{r_{ij}} = 
  e^{b_ic_j+u_iw_j} \in \calU(\frakg_{0,i}\oplus\frakg_{0,j})$}

\def\ZeroInvariantsB{{\raisebox{0mm}{\parbox[t]{3.95in}{
solves YB/R3, hence we get a tangle invariant:
}}}}

\def\ZeroInvariantsC{{\raisebox{2mm}{\parbox[t]{3.95in}{
\parshape 4 0in 3.2in 0in 3.2in 0in 3.2in 0in 3.95in
{\red Goal.} Sort $Z$ to be as on the right, with 
$f_k\in\bbQ\llbracket b_i\rrbracket$. Better, with
$\zeta\in\bbQ\llbracket b_x,c_x,u_x,w_x,b_y,c_y,u_y,w_y\rrbracket$, write
\newline\null\quad$\displaystyle 
  Z=\bbO\left(\zeta|x\colon c_xu_xw_x,\,y\colon c_yu_yw_y\right)
$\hfill{\footnotesize(cuw form)}

Here $\bbO\left(\text{\it poly}\mid\text{\it specs}\right)$ plants the
variables of {\it poly} in $\calS(\oplus_i\frakg)$ on several tensor
copies of $\calU(\frakg)$ according to {\it specs}. E.g.,
\[ \bbO\left(
    c_1^3u_1c_2e^{u_3}w_3^9|x\colon\!w_3c_1,\,y\colon\!u_1u_3c_2
  \right)
  \!=\! w^9c^3\otimes ue^uc \in \calU(\frakg)_x\otimes\calU(\frakg)_y
\]
}}}}

\def\ZeroLemma{{\raisebox{2mm}{\parbox[t]{3.95in}{
{\red Lemma.} $R_{ij} \!=\! e^{b_ic_j+u_iw_j}
\!=\! \bbO\left(
  \exp\left(b_ic_j+\frac{e^{b_i}-1}{b_i}u_iw_j\right)
    |i\colon\!u_i,\,j\colon\!c_jw_j
\right)$

{\red Example.} $Z(T_0)=$ \hfill
$=\! \sum_{m,n} \frac{b_i^{m-n}(e^{b_i}-1)^n}{m!n!}u^n\otimes c^mw^m$.

\vskip 1mm
$\bbO\bigg(1\exp\left(
    b_5c_1 \!+\! \frac{e^{b_5}-1}{b_5}u_5w_1
    \!+\!  b_2c_4 \!+\! \frac{e^{b_2}-1}{b_2}u_2w_4
    \!-\! b_3c_6 \!+\! \frac{e^{-b_3}-1}{b_3}u_3w_6
  \right)|$
\newline\null\hfill$x\colon\!c_1w_1u_2,\,y\colon\!u_3c_4w_4u_5c_6w_6\bigg)$
\newline\null\hfill$=\bbO\left(?|x\colon c_xu_xw_x,\,y\colon c_yu_yw_y\right)$
}}}}

\def\constraints{{\raisebox{2.5mm}{\parbox[t]{1.8in}{\footnotesize
  $\bbO\big(\omega e^{L+Q}\big)$: $L$ bilinear in $b_i$ and $c_i$,
  and $Q$ a balanced quadratic in $u_i$ and $w_i$ with coefficients in
  $\bbQ(b_i,e^{b_i})\ni\omega$.
  \centerline{``Admissible''}
}}}}

\def\Big0Lemma{{\raisebox{2mm}{\parbox[t]{3.95in}{
{\red The Big $\frakg_0$ Lemma.} Under $[c,u]=u$, $[c,w]=-w$, and $[u,w]=b$:

1. $N^{c_1c_2}\coloneqq\bbO(\zeta|c_1c_2)\overset{\to}{=}\bbO(\zeta/(c_2\to c_1)|c_1)$
   \hfill{\footnotesize (trivial, also for $b$, $u$, $w$)}

2a. $N^{uc}\coloneqq\bbO(e^{\gamma c+\beta u}|uc) \overset{\to}{=} 
     \bbO(e^{\gamma c+e^{-\gamma}\beta u}|cu)$
   \hfill{\footnotesize (means $e^{\beta u}e^{\gamma c}=e^{\gamma c}
     e^{e^{-\gamma}\beta u}$}

2b. $N^{wc}\coloneqq\bbO(e^{\gamma c+\alpha w}|wc) \overset{\to}{=} 
     \bbO(e^{\gamma c+e^{\gamma}\alpha w}|cw)$
   \hfill{\footnotesize \ldots in the $\{ax+b\}$ group)}

3. $\bbO(e^{\alpha w+\beta u}|wu) =
     \bbO(e^{-b\alpha\beta+\alpha w+\beta u}|uw)$
   \hfill{\footnotesize (the Weyl relations)}

4. $\bbO(e^{\delta uw}|wu)e^{\beta u}
     = e^{\nu\beta u}\bbO(e^{\delta uw}|wu)$,
     with \cbox{yellow}{$\nu=(1+b\delta)^{-1}$}
\newline{\footnotesize
  (a. expand and crunch.\hfill b. use $w=b\hat{x}$, $u=\partial_x$.
  \hfill c. use ``scatter and glow''.)}

5. $\bbO(e^{\delta uw}|wu) =
     \bbO(\nu e^{\nu\delta uw}|uw)$
   \hfill{\footnotesize (same techniques)}

6. $N^{wu}\coloneqq\bbO(e^{\beta u+\alpha w+\delta uw}|wu)
     \overset{\to}{=} \bbO(\nu
       e^{-b\nu\alpha\beta+\nu\alpha w+\nu\beta u+\nu\delta uw}
     |uw)$

Sneaky: $\alpha$ may contain (other) $u$'s, $\beta$ may contain (other) $w$'s.

}}}}

\def\ZeroStitching{{\raisebox{2mm}{\parbox[t]{3.95in}{
{\red Strand Stitching,} $m^{ij}_k$, is defined as the composition
\begin{multline*} c_iu_i\ob{w_ic_j}u_jw_j \xrightarrow{N^{w_ic_j}_k}
  c_i\ob{u_ic_k}\ob{w_ku_j}w_j \xrightarrow{N^{u_ic_k}_k\act N^{w_ku_j}_k}
  \ob{c_ic_k}\ob{u_ku_k}\ob{w_kw_j} \\
  \xrightarrow{N^{c_ic_k}_k\act --\act N^{w_kw_j}_k} c_ku_kw_k
\end{multline*}
}}}}

\def\OneInvariants{{\raisebox{2mm}{\parbox[t]{4in}{
{\red 1-Smidgen Invariants.} Much is the same:

{\red The Big $\frakg_1$ Lemma.} Parts 1 and 2 are the same, yet

6. \cbox{yellow}{$\bbO\left(e^{\alpha w+\beta u+\delta uw}|wu\right) = \bbO\left(\nu
(1+\epsilon\nu\Lambda) e^{\nu(-b\alpha\beta+\alpha w+\beta u+\delta uw)}|cuw\right)$}

Here $\Lambda$ is for {\greektext L'ogos}, ``a principle of order and knowledge'', a balanced
quartic in $\alpha$, $\beta$, $c$, $u$, and $w$: 
\begin{align*} \Lambda = &
  -\frac12 b\nu\left(\nu^2\alpha^2\beta^2+4\delta\nu\alpha\beta+2\delta^2\right)
  - \frac12 \delta\nu^3(3b\delta+2)\beta^2u^2 \\
& - \frac12 b\delta^4\nu^3u^2w^2
  - \delta^2\nu^3(2b\delta+1)\beta u^2w \\
& - \nu^2(2b\delta+1)(\nu\alpha\beta+2\delta)\beta u
  - 2b\delta^2\nu^2(\nu\alpha\beta+\delta)uw \\
& + \frac12 \delta\nu^3(b\delta+2)\alpha^2w^2
  + 2(\nu\alpha\beta+\delta)c
  + 2\delta\nu\beta cu
  + 2\delta^2\nu cuw \\
& + 2\delta\nu \alpha cw
  + \delta^2\nu^3\alpha uw^2
  + \nu^2(\nu\alpha\beta+2\delta)\alpha w.
\end{align*}

{\red Proof.} A brutal hell.

{\red Problem.} We now need to normal-order perturbed Gaussians!

{\red Solution.} Borrow some tactics from QFT:
\begin{multline*} \bbO(\epsilon P(c,u)e^{\gamma c+\beta u}|uc) =
  \bbO(\epsilon P(\partial_\gamma,\partial_\beta)e^{\gamma c+\beta u}|uc) = \\
  \bbO(\epsilon P(\partial_\gamma,\partial_\beta)e^{\gamma c+e^{-\gamma}\beta u}|cu),
\end{multline*}
\vskip -3mm
and likewise
\newline\null\hfill\cbox{yellow}{$\displaystyle
  \bbO\left(\epsilon P(u,w)e^{\alpha w+\beta u+\delta uw}|wu\right) 
    \!=\! \bbO\left(\epsilon P(\partial_\beta,\partial_\alpha)\nu
     e^{\nu(-b\alpha\beta+\alpha w+\beta u+\delta uw)}|cuw\right)
$}\hfill\null

\vskip 3pt
{\red Note.} Strand stitching requires a tiny extra step.

{\red Finally,} the values of the generators $\overcrossing$,
$\undercrossing$, $\overrightarrow{n}$, $\overleftarrow{n}$,
$\underrightarrow{u}$, and $\underleftarrow{u}$, are set by brutally
solving many equations, non-uniquely.
}}}}

\def\Pragmatic{{\raisebox{2mm}{\parbox[t]{3.95in}{
{\red Pragmatic Simplifications.} Get rid of $\zeta=(e^b-1)/b$ factors by rescaling
$u\to\bar{u}=\zeta u$. Complement this with $\beta\to\bar{\beta}=\zeta^{-1}\beta$,
$\delta\to\bar{\delta}=\zeta^{-1}\delta$, $\epsilon\to\bar{\epsilon}=\zeta^{-1}\epsilon$.
Simplify further by naming $e^b\to t$; e.g.,
$\nu\to\bar{\nu}=(1+(t-1)\delta)^{-1}$. Get confused by renaming
$(\bar{u},\bar\beta,\bar\delta,\bar\nu)\to(u,\beta,\delta,\nu)$, and more confused by working
with $\mu=\nu^{-1}$ and $\bbE(\omega,L,Q,P) \coloneqq
\omega^{-1}(1+\epsilon\omega^{-4}P)e^{L+\omega^{-1}Q}$, where $\omega\in R\coloneqq \bbQ(t_k)$,
$L=\sum l_{ij}b_ic_j$ with $l_{ij}\in\bbZ$, $Q=\sum q_{ij}u_iw_j$ with
$q_{ij}\in R$, and $P$ is a balanced quartic polynomial in $c_i$, $u_i$, and $w_i$ with
coefficients in $R$. Magically, all coefficients are now Laurent polynomials in the $t_k$'s.
}}}}

\def\Complexity{{\raisebox{15pt}{\begin{minipage}[t]{3.95in}{%\footnotesize
\parpic[r]{$
  \ub{n}_A
  \ob{
    \sum\nolimits_{\scriptstyle d=0}^4
      \ub{w^{4-d}}_E\ub{w^d}_F
  }^B
  \ub{n^2}_G
  = n^3w^4\in[n^5,n^7]
$}
{\red Rough complexity estimate,} after $t_k\to t$. $n$: xing number; $w$: width, maybe
${\sim\sqrt{n}}$. $A$: go over stitchings in order. $B$: multiplication
ops per $N^{u_iw_j}$. $d$: deg of $u_i,w_j$ in $P$. $E$: $\#$terms of
deg $d$ in $P$. $F$: ops per term. $G$: cost per polynomial multiplication op.

}\end{minipage}}}}

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

\def\mmr{{\raisebox{2mm}{\parbox[t]{3.95in}{
\parshape 1 0in 2.45in
{\red Expectation.} Our invariant is the ``1-higher diagonal'' in the MMR expansion of the coloured Jones polynomial $J_\lambda$.

{\red 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
\[ \left.
  \frac{(q^{1/2}-q^{-1/2})J_d(K)}{q^{d/2}-q^{-d/2}}
  \right|_{q=e^\hbar} =
  \sum_{j,m\geq 0} a_{jm}(K)d^j\hbar^m,
\]
\parshape 1 0in 3in
``below diagonal'' coefficients vanish, $a_{jm}(K)=0$ if $j>m$, and
``on diagonal'' coefficients give the inverse of the
Alexander polynomial:
$\left(\sum_{m=0}^\infty a_{mm}(K)\hbar^m\right)\cdot A(K)(e^\hbar)=1$.
}}}}

\def\Help{{\raisebox{2mm}{\parbox[t]{2.4in}{\small
{\red Help Needed!} Disorganized videos of talks in a private
seminar are at \web{PP}.
}}}}

\pagestyle{empty}

\begin{document} \latintext
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\begin{center}
\null\vfill\input{BHP1.pdftex_t}\vfill\null\eject
\null\vfill\input{BHP2.pdftex_t}\vfill\null\eject
\end{center}
\eject
\newgeometry{textwidth=8in,textheight=10.5in}
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\begin{multicols}{2} \raggedcolumns

{\large\red Demo Programs for 0-Co.}\hfill\web{Demo}

\vskip -2mm
\begin{mdframed}[hidealllines=true,backgroundcolor=yellow!20,innerleftmargin=3pt,innerrightmargin=3pt,leftmargin=-3pt,rightmargin=-3pt]
\hfill{\red The $R$-matrices}
\vskip -3mm\includegraphics[scale=\cellscale]{Snips/0R.pdf}
\end{mdframed}
\vskip -2mm

\vskip -2mm
\begin{mdframed}[hidealllines=true,backgroundcolor=green!10,innerleftmargin=3pt,innerrightmargin=3pt,leftmargin=-3pt,rightmargin=-3pt]
\hfill{\red Utilities}
\vskip -3mm\includegraphics[scale=\cellscale]{Snips/0Util.pdf}

\hfill{\red Normal Ordering Operators}
\vskip -3mm\includegraphics[scale=\cellscale]{Snips/0NO.pdf}

\hfill{\red Stitching}
\vskip -3mm\includegraphics[scale=\cellscale]{Snips/0m.pdf}
\end{mdframed}
\vskip -2mm

\hfill{\red Some calculations for $T_0$}
\vskip -3mm\includegraphics[scale=\cellscale]{Snips/T00.pdf}

\includegraphics[scale=\cellscale]{Snips/T01.pdf}

\includegraphics[scale=\cellscale]{Snips/T02.pdf}

\includegraphics[scale=\cellscale]{Snips/T03.pdf}

\includegraphics[scale=\cellscale]{Snips/T04.pdf}

\hfill{\red Verifying meta-associativity}
\includegraphics[scale=\cellscale]{Snips/0Q0.pdf}

%\includegraphics[scale=\cellscale]{Snips/0NODemo.pdf}

%\includegraphics[scale=\cellscale]{Snips/0mDemo.pdf}

\includegraphics[scale=\cellscale]{Snips/0MetaAssoc.pdf}

\Needspace{2cm}
\hfill{\red Testing R3}
\vskip -3mm\includegraphics[scale=\cellscale]{Snips/0R3Left.pdf}

\includegraphics[scale=\cellscale]{Snips/0R3.pdf}

\hfill{\red $8_{17}$}
\vskip -4mm
\hfill\resizebox{!}{38mm}{\input{figs/817.pdf_t}}\hfill\null

\includegraphics[scale=\cellscale]{Snips/0817.pdf}

{\large\red Demo Programs for 1-Co.}\hfill\web{Demo}

\vskip -2mm
\begin{mdframed}[hidealllines=true,backgroundcolor=yellow!20,innerleftmargin=3pt,innerrightmargin=3pt,leftmargin=-3pt,rightmargin=-3pt]
\includegraphics[scale=\cellscale]{Snips/Logos.pdf}
\vskip -6mm
\hfill{\red The {\greektext L'ogos}}
\end{mdframed}
\vskip -2mm

\vskip -2mm
\begin{mdframed}[hidealllines=true,backgroundcolor=green!10,innerleftmargin=3pt,innerrightmargin=3pt,leftmargin=-3pt,rightmargin=-3pt]
\hfill{\red Differential Polynomials}
\includegraphics[scale=\cellscale]{Snips/1DP.pdf}

\hfill{\red Utilities}
\vskip -3mm\includegraphics[scale=\cellscale]{Snips/1Util.pdf}

\hfill{\red Normal Ordering Operators}
\includegraphics[scale=\cellscale]{Snips/1NOc.pdf}

\includegraphics[scale=\cellscale]{Snips/1NOuw.pdf}

\hfill{\red Stitching}
\vskip -3mm\includegraphics[scale=\cellscale]{Snips/1m.pdf}
\end{mdframed}
\vskip -2mm

\vskip -2mm
\begin{mdframed}[hidealllines=true,backgroundcolor=yellow!20,innerleftmargin=3pt,innerrightmargin=3pt,leftmargin=-3pt,rightmargin=-3pt]
\hfill{\red The Generators}
\vskip -3mm\includegraphics[scale=\cellscale]{Snips/1Gens.pdf}
\end{mdframed}

\resizebox{\linewidth}{!}{\input{figs/SwirlAndTrefoil.pdf_t}}

\includegraphics[scale=\cellscale]{Snips/1SwirlLeft.pdf}

\includegraphics[scale=\cellscale]{Snips/1Swirl.pdf}

\includegraphics[scale=\cellscale]{Snips/131.pdf}

\includegraphics[scale=\cellscale]{Snips/131a.pdf}

{\red Questions and To Do List.}
$\bullet$~Clean up and write up.
$\bullet$~Implement well, compute for everything in sight.
$\bullet$~Why are our quantities polynomials rather than just rational functions?
$\bullet$~Bounds on their degrees?
$\bullet$~What about links / closed components?
$\bullet$~Fully digest the ``expansion'' theorem.
$\bullet$~Explore the (non-)dependence on $R$.
$\bullet$~Is there a canonical $R$?
$\bullet$~What does ``group like'' mean?
$\bullet$~Strand removal? Strand doubling? Strand reversal?
$\bullet$~Say something about knot genus.
$\bullet$~Find the EK/AT/KV ``vertex''.
$\bullet$~Use as a playground to study associators/braidors.
$\bullet$~Restate in topological language.
$\bullet$~Study the associated (v-)braid representations.
$\bullet$~Study mirror images and the $\frakb^+\leftrightarrow\frakb^-$ involution.
$\bullet$~Study ribbon knots.
$\bullet$~Make precise the relationship with $\Gamma$-calculus and Alexander.
$\bullet$~Relate to the coloured Jones polynomial.
$\bullet$~Relate with ``ordinary'' $q$-algebra.
$\bullet$~$k$-smidgen $sl_n$, etc.
$\bullet$~Are there ``solvable'' CYBE algebras not arising from semi-simple algebras?
$\bullet$~Categorify and appease the Gods.

\vskip 2mm
{\red References.}{\footnotesize
%\par\vspace{-3mm}
\renewcommand{\section}[2]{}%
\begin{thebibliography}{}
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\setlength{\itemsep}{0pt plus 0.3ex}
\input refs.tex
\end{thebibliography}}

{\red Disclaimer.} This is all quite new. The overall picture is correct, yet some details
might be somewhat off. Many pieces are certainly not in their final form yet.

\end{multicols}

\parpic[r]{\parbox{0.65in}{
  \includegraphics[width=0.65in]{Evil.jpg}
  \newline\footnotesize\web{Joker}
}}
\picskip{5}
{\footnotesize {\red Dire Warning.} On Tuesday night agents of the
Evil Galactic Empire will lock all participants of this workshop in
separate sound proof, electromagnetically sealed, neutrino hardened,
and gravitational wave resistant secret rooms in Hotel Les Sources. In the
rooms they will place identical countable sequences of numbered boxes,
each one containing a real number (the same sequence of real numbers
in each room). By morning, each participant must open all but one of
their boxes in the order of their liking, and guess the number in the
remaining one. If more than one participant guesses wrong, breakfast will
be poisoned. {\red Do Something!} We must devise a strategy during Tuesday's hike
or else we will miss Thomas' talks!

--- ``Saw Omega'' from Alfonso Gracia-Saz from Mira Bernstein from
\web{SO} (spoilers). Deadly serious.
}

\end{document}

\endinput