\documentclass[11pt,notitlepage]{article}
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\usepackage{tensor}
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\parindent 0in

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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{Toronto-1609}
\def\title{A Poly-Time Knot Polynomial Via Solvable Approximation}

\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}{
\parshape 3 0in 2.45in 0in 2.45in 0in 3.95in
{\red Abstract.} I will construct the first poly-time-computable knot
polynomial since Alexander's \cite[1928]{Alexander:TopologicalInvariants}
by using some new commutator-calculus techniques and a Lie algebra
$\frakg_1$ which is at the same time solvable and an approximation of
the simple Lie algebra $sl_2$.
}}}}

\def\Expected{{\raisebox{2mm}{\begin{minipage}[t]{3.95in}
{\red 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 Paradise!} Foremost reason: {\red\sl OBVIOUSLY.}
{\footnotesize Cf.\ proving (incomputable $A$)$=$(incomputable $B$), or
categorifying (incomputable $C$).}
\par Secondary reason: 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''
\[ \arraycolsep=0pt \renewcommand{\arraystretch}{1}
  \begin{array}{c}
    Z\colon\left\{\parbox{0.85in}{$S$-component ($v/b$-)tangles}\right\}
    \to \calA^v_S \coloneqq \\
  \parbox{0.45\linewidth}{\vskip 2mm
  \includegraphics[height=0.4in]{../../Projects/Gallery/Etingof.jpg}%
  \hfill\includegraphics[height=0.4in]{../../Projects/Gallery/Kazhdan.jpg}%
  \hfill\includegraphics[height=0.4in]{../../Projects/Gallery/Haviv.jpg}%
  \hfill\includegraphics[height=0.4in]{../../Projects/Gallery/Enriquez.jpg}%
  \hfill\includegraphics[height=0.4in]{../../Projects/Gallery/Severa.jpg}
  \hfill} \\
  \parbox{0.45\linewidth}{\tiny Etingof \hfill Kazhdan \hfill Haviv \hfill
Enriquez \hfill \v{S}evera \hfill}
  \end{array}
  \begin{array}{c}\resizebox{0.55\linewidth}{!}{\subimport{../LesDiablerets-1608/}{figs/Av.pdf_t}}\end{array}
\]
(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
$\psi(x)\coloneqq\left((x+2)e^{-x}-2+x\right)/(2x) = x^2/12-x^3/24+\dots$.

{\red But} 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 $sl_2$.
\hfill\text{\footnotesize(note: $\deg(b,c,u,w,\epsilon)=(1,0,1,0,1)$)}
}}}}

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

\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{3mm}{\begin{minipage}[t]{3.95in}
{\red The Big $\frakg_0$ Lemma.} Under $[c,u]=u$, $[c,w]=-w$, and $[u,w]=b$:

\vskip -1.2mm
\parpic[r]{$\xymatrix@R=0.2in@C=0.4in{
  \calS \ar[r]^{N^{c_ic_j}_k} \ar[rd]_{\bbO(-|c_ic_j)} & \calS \ar[d]^{\bbO(-|c_k)} \\
  & \calU
}$}
\picskip{1}
1. $N^{c_ic_j}_k\coloneqq\bbO(\zeta|c_ic_j)\overset{\to}{=}\bbO(\zeta/(c_i,c_j\to c_k)|c_k)$

\parshape 1 0.18in 2.5in
{\footnotesize (Meaning, $N^{c_ic_j}_k\colon\zeta\mapsto(\zeta/(c_i,c_j\to c_k))$ and
the diagram commutes. 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. $N^{wu}\coloneqq\bbO(e^{\beta u+\alpha w+\delta uw}|wu)
     \overset{\to}{=} \bbO(\nu
       e^{-b\nu\alpha\beta+\nu\alpha w+\nu\beta u+\nu\delta uw}
     |uw)$
\end{minipage}}}}

\def\Sneaky{{\raisebox{2mm}{\parbox[t]{3.95in}{
{\red 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 = &
  - b\nu\left(\nu^2\alpha^2\beta^2+4\delta\nu\alpha\beta+2\delta^2\right)/2
  - \delta\nu^3(3b\delta+2)\beta^2u^2/2 \\
& - b\delta^4\nu^3u^2w^2/2
  - \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 \\
& + \delta\nu^3(b\delta+2)\alpha^2w^2/2
  + 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\PPSeminar{{\raisebox{2mm}{\parbox[t]{2.2in}{\small
{\red Videos} of a 4-hour version of this talk are at \web{LD}. {\red
Videos} of private seminar meetings are at \web{PP}.
}}}}

\pagestyle{empty}

\begin{document} \latintext
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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}{\subimport{../LesDiablerets-1608/}{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}{!}{\subimport{../LesDiablerets-1608/}{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}

\vfill\rule{\linewidth}{1pt}%\vspace{-1mm}

\includegraphics[height=0.6in]{../../Projects/Gallery/Kronecker.jpg}
\hfill
\parbox[b]{2.7in}{
  ``God created the knots; all else in
  \newline topology is the work of mortals.''
  \vskip 1.5mm
  {\footnotesize Leopold Kronecker (modified)\hfill\href{http://katlas.org}{katlas.org}}
}
\hfill
\includegraphics[height=0.6in]{../../Projects/Gallery/The_Knot_Atlas.png}

\columnbreak

{\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?
$\bullet$~Find the 2-variable version (for knots). How complex is it?
$\bullet$~What about links / closed components?
$\bullet$~Fully digest the ``expansion'' theorem; include cuaps.
$\bullet$~Explore the \text{(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
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\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.
\hfill\text{\red Help Needed!}

\end{multicols}

%\vfill\null\hfill.
\end{document}

\endinput