\documentclass[11pt,notitlepage]{article}
\def\bare{n}
\usepackage[all]{xy}
\usepackage[english,greek]{babel}
\usepackage{dbnsymb, amsmath, graphicx, amssymb, multicol, stmaryrd, pifont,
  amscd, colortbl, mathtools, wasysym, needspace, import, longtable, overpic,
  enumitem, bbm, pdfpages, ../picins, array, setspace, datetime, amsthm}
\usepackage[normalem]{ulem} % For \sout.
\usepackage[export]{adjustbox} % Follows https://tex.stackexchange.com/questions/6073/scale-resize-large-images-graphics-that-exceed-page-margins
\usepackage{tensor}
\usepackage{txfonts}	% for the likes of \coloneqq.
\usepackage{fontawesome} % for \faPlay
\usepackage[usenames,dvipsnames]{xcolor}
\usepackage{utfsym}	% for the likes of \car=\usym{1F697}
\usepackage{soul} % for strikeouts, \st.
\usepackage[textwidth=8in,textheight=10.5in,centering]{geometry}
%\newgeometry{textwidth=8in,textheight=10.5in}
\parindent 0in
\usepackage[makeroom]{cancel}

% Following http://tex.stackexchange.com/a/847/22475:
\usepackage[setpagesize=false]{hyperref}
\hypersetup{colorlinks,
  linkcolor={blue!50!black},
  citecolor={blue!50!black},
  urlcolor={blue!50!black}
}

% Following http://tex.stackexchange.com/questions/59340/how-to-highlight-an-entire-paragraph
\usepackage[framemethod=tikz]{mdframed}

\usepackage[T1]{fontenc}

\def\myurl{http://www.math.toronto.edu/~drorbn}
\def\thistalk{Banff-2607}
\def\title{$\Theta$: $4\smiley4\frownie$}

\def\navigator{{
  \href{\myurl}{Dror Bar-Natan}:
  \href{\myurl/Talks}{Talks}:
  \href{\myurl/Talks/\thistalk/}{\thistalk}:
}}
\def\thanks{{OMG, thanks! (Again!)}}
\def\webdef{{{\greektext web}$\coloneqq$\href{http://drorbn.net/ba26}{http://drorbn.net/ba26}}}
\def\web#1{{\href{\myurl/Talks/\thistalk/#1}{{\greektext web}/#1}}}
\def\titleA{{\title}}
\def\titleB{{\title}}
\def\titleC{{\title}}

\definecolor{mblue}{HTML}{E0E0FF}
\definecolor{mgray}{HTML}{B0B0B0}
\definecolor{mgreen}{HTML}{00B000}
\definecolor{morange}{HTML}{FFA50A}
\definecolor{mpink}{HTML}{FFE0E0}
\definecolor{mpurple}{HTML}{B000B0}
\definecolor{myellow}{HTML}{FFFF00}
\def\blue{\color{blue}}
\def\gray{\color{gray}}
\def\magenta{\color{magenta}}
\def\mgray{\color{mgray}}
\def\mgreen{\color{mgreen}}
\def\morange{\color{morange}}
\def\mpurple{\color{mpurple}}
\def\pink{\color{pink}}
\def\purple{\color{purple}}
\def\red{\color{red}}
\def\yellowm#1{{\setlength{\fboxsep}{0pt}\colorbox{yellow}{$#1$}}}
\def\myellowm#1{{\setlength{\fboxsep}{0pt}\colorbox{myellow}{$#1$}}}
\def\mpinkm#1{{\setlength{\fboxsep}{0pt}\colorbox{mpink}{$#1$}}}
\def\mbluem#1{{\setlength{\fboxsep}{0pt}\colorbox{mblue}{$#1$}}}
\def\cbox#1#2{{\setlength{\fboxsep}{0pt}\colorbox{#1}{#2}}}

\def\arXiv#1{{\href{http://arxiv.org/abs/#1}{{\tiny arXiv:}\linebreak[0]{#1}}}}

\def\ds{\displaystyle}
\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\eps{\epsilon}
\def\FL{\text{\it FL}}
\def\Hom{\operatorname{Hom}}
\def\IHX{\mathit{IHX}}
\def\Kh{\text{\it Kh}}
\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\Vol{\text{\it Vol}}
\def\vT{{\mathit v\!T}}

\def\bara{{\bar a}}
\def\barb{{\bar b}}
\def\barT{{\bar T}}
\def\bbE{{\mathbb E}}
\def\bbe{\mathbbm{e}}
\def\bbH{{\mathbb H}}
\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\calB{{\mathcal B}}
\def\calD{{\mathcal D}}
\def\calF{{\mathcal F}}
\def\calG{{\mathcal G}}
\def\calH{{\mathcal H}}
\def\calI{{\mathcal I}}
\def\calK{{\mathcal K}}
\def\calL{{\mathcal L}}
\def\calM{{\mathcal M}}
\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}}

\def\car{\reflectbox{\usym{1F697}}}
\def\rac{\usym{1F697}}

\def\ip{{i^+}} \def\jp{{j^+}} \def\kp{{k^+}}
\def\ipp{{i^{+\!+}}} \def\jpp{{j^{+\!+}}} \def\kpp{{k^{+\!+}}}

% \Gint from https://tex.stackexchange.com/questions/171415/superimpose-letter-on-integral-symbol
\makeatletter
\let\DOTSI\relax % amsmath support for \dots
\newcommand*{\Gint}{%
  \DOTSI
  \mathop{%
    \mathpalette\@LetterOnInt{G}%
  }%
  \mkern-\thinmuskip % thin space is inserted between two \mathop
  \int
}
\newcommand*{\@LetterOnInt}[2]{%
  \sbox0{$#1\int\m@th$}%
  \sbox2{$%
    \ifx#1\displaystyle
      \textstyle
    \else
      \scriptscriptstyle
    \fi
    #2%
  \m@th$}%
  \dimen@=.4\dimexpr\ht0+\dp0\relax
  \ifdim\dimexpr\ht2+\dp2\relax>\dimen@
    \sbox2{\resizebox*{!}{\dimen@}{\unhcopy2}}%
  \fi
  \dimen@=\wd0 %
  \ifdim\wd2>\dimen@
    \dimen@=\wd2 %
  \fi
  \rlap{\hbox to \dimen@{\hfil
    $#1\vcenter{\copy2}\m@th$%
  \hfil}}%
  \ifdim\dimen@>\wd0 %
    \kern.5\dimexpr\dimen@-\wd0\relax
  \fi
}
\makeatother

% 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\ThickSeparator{{\vspace{-3mm}\rule{\linewidth}{1.5pt}\vspace{-1mm}}}
\def\ThinSeparator{{\vspace{-3mm}\rule{\linewidth}{0.75pt}\vspace{-1mm}}}

%%%

\def\Abstract{{\raisebox{1.6mm}{\parbox[t]{3.95in}{
\parshape 7 0in 3.3in 0in 3.3in 0in 3.3in 0in 3.3in 0in 3.3in 0in 3.3in 0in 3.95in
{\red\bf Abstract.} Recently, in September 2025, Roland van der Veen and
myself released a paper titled ``A Fast~${\smiley}$, Strong~${\smiley}$,
Topologically Meaningful~${\smiley}$, and Fun~${\smiley}$ Knot Invariant''
\cite{Theta}. More recently, in January 2025 in Les Diablerets
\cite{ld26}, I gave a talk in which I suggested 27 homework tasks related
to that same invariant $\Theta$ (plus a bonus task).  \qquad Today,
after a brief review, I'll talk a bit more about just 4 of those tasks:

$\frownie$~Figure out the relationship of $\Theta$ with Chern
Simons theory.

$\frownie$~Complete the discussion of the relationship of $\Theta$
with $sl_3$.

\hangindent=5mm\hangafter=1
$\frownie$~Find formulas for $\Theta$ corresponding to other
presentations of the Alexander module.

\hangindent=5mm\hangafter=1
$\frownie$~Fully understand $\Theta$-like formulas and their
non-uniqueness (i.e., clean the mess).

\footnotesize {\bf\red Acknowledgement.} This work was supported by NSERC
grant RGPIN-2025-06718 and by the Chu Family Foundation (NYC).
}}}}

\def\Preparation{{\raisebox{2mm}{\parbox[t]{3in}{
{\red\bf Preparation.} Draw an $n$-crossing knot $K$ as a diagram $D$
as on the right: all crossings face up, and the edges are marked with
a running index ${k\in\{1,\ldots,2n+1\}}$ and with rotation numbers
$\varphi_k$.
}}}}

\def\A{{\raisebox{1.7mm}{\parbox[t]{3.95in}{
\parshape 1 0in 3in
{\red $A$.} With $T$ an indeterminate, start from a presentation
matrix $A$ for the Alexander module of $K$, coming from the Wirtinger
presentation of $\pi_1(K)$: $A\coloneqq I_{2n+1}+\sum_c A_c$, where

\vskip -4mm
\parshape 1 0in 3.125in
\[
  \begin{array}{c}\import{../LesDiablerets-2601}{figs/Xings.pdf_t}\end{array}
  \ \rightarrow\
  \begin{array}{c|cccc}
    A_c &   i+1  &  j+1 \\
    \hline
    i & -T^s  & T^s-1 \\
    j & 0  & -1
  \end{array}
\]
\[
A=\left(
\begin{array}{ccccccc}
 1 & \mbluem{-T} & 0 & 0 & \mbluem{T-1} & 0 & 0 \\
 0 & 1 & \mpinkm{-1} & 0 & 0 & \mpinkm{\ 0\ } & 0 \\
 0 & 0 & 1 & \myellowm{-T} & 0 & 0 & \myellowm{T-1} \\
 0 & \mbluem{\ 0\ } & 0 & 1 & \mbluem{-1} & 0 & 0 \\
 0 & 0 & \mpinkm{T-1} & 0 & 1 & \mpinkm{-T} & 0 \\
 0 & 0 & 0 & \myellowm{\ 0\ } & 0 & 1 & \myellowm{-1} \\
 0 & 0 & 0 & 0 & 0 & 0 & 1 \\
\end{array}
\right)
\]
}}}}

\def\Alexander{{\raisebox{2mm}{\parbox[t]{1.5in}{
{\bf\red Note.} Alexander's $\Delta$ is
\[ \Delta = T^{(-\varphi-w)/2}\det(A), \]
with $\varphi = \sum_k \varphi_k$, $w = \sum_c s$.
}}}}

\def\G{{\raisebox{2mm}{\parbox[t]{3.95in}{
{\red $G$.} Let $G = (g_{\alpha\beta}) \coloneqq A^{-1}$, the ``two point function'':

$G=\left( \arraycolsep=3.5pt
\begin{array}{ccccccc}
 1 & T & 1 & T & 1 & T & 1 \\
 0 & 1 & \frac{1}{T^2-T+1} & \frac{T}{T^2-T+1} & \frac{T}{T^2-T+1} &
\frac{T^2}{T^2-T+1} &
   1 \\
 0 & 0 & \frac{1}{T^2-T+1} & \frac{T}{T^2-T+1} & \frac{T}{T^2-T+1} &
\frac{T^2}{T^2-T+1} &
   1 \\
 0 & 0 & \frac{1-T}{T^2-T+1} & \frac{1}{T^2-T+1} & \frac{1}{T^2-T+1} &
\frac{T}{T^2-T+1} &
   1 \\
 0 & 0 & \frac{1-T}{T^2-T+1} & -\frac{(T-1) T}{T^2-T+1} & \frac{1}{T^2-T+1} &
   \frac{T}{T^2-T+1} & 1 \\
 0 & 0 & 0 & 0 & 0 & 1 & 1 \\
 0 & 0 & 0 & 0 & 0 & 0 & 1 \\
\end{array}
\right)$

Let $T_1$ and $T_2$ be new indeterminates, let $T_3=T_1T_2$, and
let $G_\nu=(g_{\nu\alpha\beta})$ be $G$ with $T\to T_\nu$, for $\nu=1,2,3$. Likewise for
$\Delta_\nu$.
}}}}

\def\thet{{\raisebox{3mm}{\parbox[t]{2.25in}{
\[
  {\red\theta} \sim \Delta_1\Delta_2\Delta_3\sum_{c_0,c_1}g_{1i_0i_1}g_{2i_0i_1}g_{3i_1i_0}
  + \text{l.o.}
\]
\[ {\red\Theta} = (\Delta,\theta)\in\bbZ[T^{\pm1}]\times\bbZ[T_1^{\pm1},T_2^{\pm1}] \]
}}}}

\def\Theorem{{\raisebox{2mm}{\parbox[t]{3.95in}{
\parshape 4 0in 2.875in 0in 2.875in 0in 2.875in 0in 3.95in
{\bf\red Theorem} \cite{Theta}. With $c=(s,i,j)$, $c_0=(s_0,i_0,j_0)$,
and $c_1=(s_1,i_1,j_1)$ denoting crossings, there is a quadratic
$F_1(c)\in\bbQ(T_\nu)[g_{\nu\alpha\beta}:\alpha,\beta\in\{i,j\}]$,
a cubic $F_2(c_0,c_1) \in
\bbQ(T_\nu)[g_{\nu\alpha\beta}:\alpha,\beta\in\{i_0,j_0,i_1,j_1\}]$, and a
linear $F_3(\varphi,k)$ such that $\theta$ is a knot invariant:
\[
  \theta(D) \coloneqq \Delta_1\Delta_2\Delta_3
%  \underbrace{\Delta_1\Delta_2\Delta_3}_{\parbox{0.66in}{\scriptsize\centering
%    normalization, see later
%  }}
  \left(\sum_c F_1(c) + \sum_{c_0,c_1} F_2(c_0,c_1) + \sum_kF_3(\varphi_k,k)\right),
\]
\vskip 27mm
%\vskip 22mm
If these pictures remind you of Feynman diagrams, it's because they are Feynman
diagrams~\cite{bo25}.
}}}}

\def\egA{{e.g. $g_{2ii} g_{3jj}$}}
\def\egB{{e.g. $g_{3j_0i_1}g_{1j_1i_0}g_{2i_1i_0}$}}
\def\egC{{e.g. $g_{3kk}$}}

\def\Ouch{{\raisebox{4.5mm}{\scalebox{0.8}{\parbox[t]{5in}{
\begin{multline*}
  F_1(c) \!=\! s
    \left[ 1/2 - g_{3ii} +  T_2^s g_{1ii} g_{2ji} - T_2^s g_{3jj} g_{2ji}
      - (T_2^s\!-\!1) g_{3ii} g_{2ji} \right. \\
    \left. + (T_3^s\!-\!1) g_{2ji} g_{3ji} - g_{1ii} g_{2jj} + 2 g_{3ii} g_{2jj}
      + g_{1ii} g_{3jj} - g_{2ii} g_{3jj} \right] \\
  + \frac{s}{T_2^s\!-\!1}
    \left[
      (T_1^s\!-\!1)T_2^s \left( g_{3jj} g_{1ji} - g_{2jj} g_{1ji} + T_2^s g_{1ji} g_{2ji}
\right) \right. \\
      + (T_3^s\!-\!1) \left( g_{3ji} - T_2^s g_{1ii} g_{3ji} + g_{2ij} g_{3ji}
      + (T_2^s\!-\!2) g_{2jj} g_{3ji} \right) \\
    \left. - (T_1^s\!-\!1) (T_2^s\!+\!1) (T_3^s\!-\!1) g_{1ji} g_{3ji} \right]
\end{multline*}
\[
  F_2(c_0,c_1) \!=\!
    \frac{s_1 (T_1^{s_0}\!-\!1) (T_3^{s_1}\!-\!1) g_{1j_1i_0} g_{3j_0i_1}}{T_2^{s_1}\!-\!1}
    \left(T_2^{s_0} g_{2i_1i_0}+g_{2j_1j_0} - T_2^{s_0} g_{2j_1i_0}-g_{2i_1j_0} \right)
\]
\hspace{4mm}$\ds F_3(\varphi,k) = \varphi(-1/2+g_{3kk})$
}}}}}

\def\random{{\raisebox{1mm}{\parbox[t]{1.75in}{
A random 300 xing knot from \cite{DHOEBL:Random}. For most invariants,
300 is science fiction.
}}}}

\def\Strong{{\raisebox{2mm}{\parbox[t]{4in}{
\parshape 1 0in 1.8in
{\bf\red $\smiley$~Strong.} $\Theta$ vs.\ a slew of other reasonably-computable invariants
(deficits shown):

\!\resizebox{4.02in}{!}{\def\s{$\sim$}
\begin{tabular}{c|c|c|c|c|c|c}
\hline
$n$&                            $\leq 10$&      $\leq 11$&      $\leq 12$&      $\leq 13$&      $\leq 14$&      $\leq 15$ \\ \hline
knots&                          249&            801&            2,977&          12,965&         59,937&         313,230 \\ \hline
$\Delta$&                       (38)&           (250)&          (1,204)&        (7,326)&        (39,741)&       (236,326) \\ \hline
$\sigma_{LT}$&                  (108)&          (356)&          (1,525)&        (7,736)&        (40,101)&       (230,592) \\ \hline
$J$&                            (7)&            (70)&           (482)&          (3,434)&        (21,250)&       (138,591) \\ \hline
$\Kh$&                          (6)&            (65)&           (452)&          (3,226)&        (19,754)&       (127,261) \\ \hline
$H$&                            (2)&            (31)&           (222)&          (1,839)&        (11,251)&       (73,892) \\ \hline
$\Vol$&                         (\s6)&          (\s25)&         (\s113)&        (\s1,012)&      (\s6,353)&      (\s43,607) \\ \hline
$(\Kh,H,\Vol)$&                 (\s0)&          (\s14)&         (\s84)&         (\s911)&        (\s5,917)&      (\s41,434) \\ \hline
$(\Delta,\rho_1)$&              (0)&            (14)&           (95)&           (959)&          (6,253)&        (42,914) \\ \hline
$(\Delta,\rho_1,\rho_2)$&       (0)&            (14)&           (84)&           (911)&          (5,926)&        (41,469) \\ \hline
$(\rho_1,\rho_2,\Kh,H,\Vol)$&   (0)&            (\s14)&         (\s84)&         (\s911)&        (\s5,916)&      (\s41,432) \\ \hline
\rowcolor{yellow}
$\Theta$&                       (0)&            (3)&            (19)&           (194)&          (1,118)&        (6,758) \\ \hline
$(\Theta,\rho_2)$&              (0)&            (3)&            (10)&           (169)&          (982)&          (6,341) \\ \hline
$(\Theta,\sigma_{LT})$&         (0)&            (3)&            (19)&           (194)&          (1,118)&        (6,758) \\ \hline
$(\Theta,\Kh)$&                 (0)&            (3)&            (18)&           (185)&          (1,062)&        (6,555) \\ \hline
$(\Theta,H)$&                   (0)&            (3)&            (18)&           (185)&          (1,064)&        (6,563) \\ \hline
$(\Theta,\Vol)$&                (0)&            (\s3)&          (\s10)&         (\s169)&        (\s973)&        (\s6,308) \\ \hline
$(\Theta,\rho_2,\Kh,H,\Vol)$&   (0)&            (\s3)&          (\s10)&         (\s169)&        (\s972)&        (\s6,304) \\ \hline
\end{tabular}}
}}}}

\def\TopMean{{\raisebox{2mm}{\parbox[t]{4in}{
{\bf\red $\smiley$~Topologically Meaningful.} $\theta$ is near $\Delta$ and
we dream that anything $\Delta$ can do, $\theta$ does too (sometimes
better).
\par{\bf\red Theorem.} $\deg_{T_1}\theta(K)\leq 2g(K)$.
\par{\bf\red Conjecture} (verified to 13 xings). If $K$ is a fibered
knot and $d$ is the degree of $\Delta(K)$ (the highest power of $T$),
then the coefficient of $T_2^{2d}$ in $\theta(K)$, which is a polynomial
in $T_1$, is an integer multiple of $T_1^d\Delta(K)|_{T\to T_1}$.
\par{\bf\red Dream.} $\theta$ has something to say about ribbon
\newline knots.
}}}}

\def\sltwoexample{{\bf\red The $sl_2^{/\eps^2}$ Example.} With $T$ an indeterminate and with
$\eps^2=0$:}

\def\vs#1{$\bbR^2_{p_#1x_#1}$}
\def\rp#1#2{$\calL(X^+_{#1#2})$}
\def\gp#1#2{$\calL(C^{#1}_#2)$}
\def\ta#1{$\tau(p_#1,x_#1)$}
\def\fintexample{$\ds Z =
  \underset{\bbR^{14}_{p_ix_i}\mathrlap{\ \text{
    measure on $\bbR$ is $(2\pi)^{-1/2}\cdot${\it standard}
  }}}{\Gint}
  {\red\calL(X^+_{15})} {\mgreen\calL(X^+_{62})} {\blue\calL(X^+_{37})}
{\mpurple\calL(C^{-1}_4)}
$}

\def\sltwodefs{{\raisebox{0mm}{\parbox[t]{2.6in}{\setstretch{1.25}
where $\calL(X^s_{ij}) = T^{s/2}\bbe^{L(X^s_{ij})}$ and
$\calL(C^\varphi_i) = T^{\varphi/2}\bbe^{L(C^\varphi_i)}$, and
\par $\ds L(X^s_{ij}) = x_i(p_{i+1}-p_i) + x_j(p_{j+1}-p_j)$
\par\hfill $\ds + (T^s-1)x_i(p_{i+1}-p_{j+1})$
\vskip 3pt\par\hfill $\ds +  \frac{\eps s}{2} \left(
    x_i (p_i-p_j) \left({(T^s-1)x_ip_j\quad}\atop{\quad+2(1-x_jp_j)}\right)-1
  \right)$
\vskip 5pt\par$\ds L(C^\varphi_i) = x_i(p_{i+1}-p_i) + \eps\varphi(1/2-x_ip_i)$
}}}}

\def\sltwoZ{{\raisebox{0mm}{\parbox[t]{3.96in}{
So $Z = T \Gint \bbe^{L(\righttrefoil)}dp_1\dots dp_7dx_1\dots dx_7$, where
$L(\righttrefoil)=$
$\ds
  \sum\nolimits_{i=1}^7 \hspace{-2mm} x_i(p_{i+1}-p_i)
  \ \ +\ \ (T-1)( {\red x_1(p_2-p_6)} + {\mgreen x_6(p_7-p_3)} + {\blue x_3(p_4-p_8)})
$
\null\hfill$\ds
  + \frac{\eps}{2}\left(\begin{array}{c}
    {\red x_1 (p_1-p_5) \left((T-1)x_1p_5+2(1-x_5p_5)\right)-1} \\
    + {\mgreen x_6 (p_6-p_2) \left((T-1)x_6p_2+2(1-x_2p_2)\right)-1} \\
    + {\blue x_3 (p_3-p_7) \left((T-1)x_3p_7+2(1-x_7p_7)\right)-1} \\
    + {\mpurple 2x_4p_4-1}
  \end{array}\right),
$
\newline and so $Z =
(T-1+T^{-1})^{-1}\exp\left(\eps\cdot\frac{(T-2+T^{-1})(T+T^{-1})}{(T-1+T^{-1})^2}\right)
=
\Delta^{-1}\exp\left(\eps\cdot\frac{(T-2+T^{-1})\rho_1}{\Delta^2}\right)$.
Here $\Delta$ is the Alexander polynomial and $\rho_1$ is the Rozansky-Overbay polynomial
\cite{Rozansky:Contribution, Rozansky:Burau, Rozansky:U1RCC, Overbay:Thesis, PP1, APAI}. It
is a
reduction of $\theta$.
}}}}

\def\slthree{{\raisebox{2mm}{\parbox[t]{3.95in}{
{\bf\red The $sl_3^{/\eps^2}$ Example} \cite{Theta}. Here we have two formal variables
$T_1$ and $T_2$, we set $T_3\coloneqq T_1T_2$, we integrate over
6 variables for each edge: $p_{1i}$, $p_{2i}$, $p_{3i}$, $x_{1i}$,
$x_{2i}$, and $x_{3i}$, with the Lagrangian given by:
\vskip 1mm \par \includegraphics[width=4in]{../MonteVerita-2604/Snips/LXInput.pdf}
\vskip 1mm \par \includegraphics[width=4in]{../MonteVerita-2604/Snips/LCInput.pdf}
\vskip 1mm \par {\bf\red Theorem.} Here,
$\ds Z=\frac{1}{\Delta_1\Delta_2\Delta_3}\exp
  \left(
    \eps\frac{\theta}{\Delta_1\Delta_2\Delta_3}
  \right)
$.
}}}}

\def\Combinatorics{{\raisebox{2mm}{\parbox[t]{3.95in}{
{\bf\red$\frownie$~Combinatorics: We still don't really understand
the $\theta$ formulas $F_{1,2,3}$.} How unique are they? What others
like them there are? What special games they play when the knot is
non-virtual? Perhaps with Alexander numbering? Is there an indigenous theory
here, or only exogenous results?
}}}}

\def\Topology{{\raisebox{2mm}{\parbox[t]{3.95in}{
{\bf\red$\frownie$~Topology: We don't have formulas coming from other
presentations of the Alexander module.} Such may give the
definitive proofs of the genus theorem and fibered conjecture, explain
the hexagonal symmetry, and clarify the relation with the two-loop
invariant~\cite{GaroufalidisRozansky:LoopExpansion, Kricker:Lines,
Ohtsuki:2Loop}. Why should Wirtinger be special? Are there efficient 3D formulas for
$\theta$ \cite{YarnBallKnots}?
}}}}

\pagestyle{empty}

\begin{document} \latintext
%\setlength{\jot}{0ex}
\setlength{\abovedisplayskip}{0.5ex}
\setlength{\belowdisplayskip}{0.5ex}
\setlength{\abovedisplayshortskip}{0ex}
\setlength{\belowdisplayshortskip}{0ex}

%\begin{center} \null\vfill\input{4S4FT1.pdftex_t}\vfill\null \end{center}

%\newpage\begin{center} \input{4S4FT2.pdftex_t} \end{center}
\enlargethispage{\baselineskip}\input{4S4FT1.pdftex_t}
\newpage\input{4S4FT2.pdftex_t}

\newpage

\begin{multicols}{2}

{\bf\red$\frownie$~Quantum Algebra: We still haven't implemented the relationship with $sl_3$ and with other
quantum groups.}

\ThinSeparator

\parpic[r]{\scalebox{0.9}{\import{.}{Puddle.pdftex_t}}}
{\red\bf Co-Commutative Limit / Solvable Approximation.} In $sl_n$, half is enough! Indeed
${sl_n\oplus\fraka_{n-1}} = \calD(\uppertriang,b,\delta)$.  Now define
$sl^\epsilon_{n+}\coloneqq\calD(\uppertriang,b,\epsilon\delta)$.
Schematically, this is ${[\uppertriang,\uppertriang]=\uppertriang}$,
$[\lowertriang,\lowertriang]=\epsilon\lowertriang$, and
$[\uppertriang,\lowertriang]=\lowertriang+\epsilon\uppertriang$. The same
process works for all semi-simple Lie algebras $\frakg$, and at $\epsilon^{k+1}=0$
always yields a solvable Lie algebra.

\vskip -3mm
\def\bracket{$b({\red\uppertriang})=b\colon{\red\uppertriang}\otimes\!{\red\uppertriang}
  \to{\red\uppertriang}$}
\def\cobracket{$b({\blue\lowertriang})\leadsto\delta\colon{\red\uppertriang}
  \to{\red\uppertriang}\otimes\!{\red\uppertriang}$}
\[ \hspace{-35mm}\import{../UCLA-191101}{Double.pdftex_t} \]

{\red Note.} $\epsilon=0$ is w-knots \cite{WKO, KBH}.

{\red By PBW,} $\calU(\frakg^\epsilon_+)$ and $\calU_q(\frakg^\epsilon_+)$
are isomorphic as vector spaces to polynomial rings. So we care for maps
between polynomial rings.

\ThinSeparator

{\bf\red Docile Perturbed Gaussians.} For a finite
set $A$, let $z_A\coloneqq\{z_i\}_{i\in A}$ and let
${\zeta_A\coloneqq\{z^*_i=\zeta_i\}_{i\in A}}$ with $\langle
z_i,\zeta_j\rangle=\delta_{ij}$. We have
\begin{multline*}
  \Hom(\bbQ[z_A]\to\bbQ[z_B])
  \simeq \bbQ[z_A]^*\otimes\bbQ[z_B] \\
  \simeq \bbQ\llbracket \zeta_A\rrbracket\otimes\bbQ[z_B]
  \simeq \bbQ[z_B]\llbracket \zeta_A\rrbracket
  \subset \bbQ\llbracket \zeta_A,z_B\rrbracket,
\end{multline*}
so every morphism has a {\em generating function}.

{\bf Future Theorem.} The generating functions of all the relevant
morphisms for $\calU(\frakg^\epsilon_+)$ and $\calU_q(\frakg^\epsilon_+)$
(meaning, $m$, $\Delta$, $S$, $R$, etc.) are docile perturbed
(\raisebox{-0.3ex}{\shortstack{\tiny two\\\tiny step}}) Gaussians.

\ThinSeparator

{\bf Theorem.} The composition of two
docile perturbed Gaussians is again a docile perturbed Gaussian, computable efficiently.

{\bf Proof Sketch.}  In $\mor(A\!\to\!B)$,
\[ Q = \sum_{i\in A,j\in B}E_{ij}\zeta_iz_j
  + \frac12\sum_{i,j\in A}F_{ij}\zeta_i\zeta_j
  + \frac12\sum_{i,j\in B}G_{ij}z_iz_j,
\]
\vskip -3mm and so
\[ \import{../Sydney-191002/}{Compositions.pdftex_t} \]
\parpic[r]{\scalebox{0.9}{\import{../UCLA-191101}{FeynmanDiagrams.pdftex_t}}}
where $\bullet$\ $E=E_1(I-F_2G_1)^{-1}E_2$.
\newline$\bullet$\ $F=F_1+E_1F_2(I-G_1F_2)^{-1}E_1^T$.
\newline$\bullet$\ $G=G_2+E_2^TG_1(I-F_2G_1)^{-1}E_2$.
\newline$\bullet$\ $\omega=\omega_1\omega_2\det(I-F_2G_1)^{-1}$.
\newline$\bullet$\ $P$ is computed as the solution of a
  messy PDE or using ``connected Feynman diagrams'' (yet we're still in pure algebra!).
Docility is preserved.

\ThinSeparator

All this is fully implemented for $sl_2$ and in the works for $sl_n$. Hopefully Roland and I
will have news by my (generalized) 61th birthday.

\ThickSeparator

{\bf\red$\frownie$~Quantum Field Theory: Chern-Simons-Witten theory is
screaming at us and we aren't listening.} There is an alternative formula
for $\theta$, and it just can't be that it isn't directly related to
Chern-Simons-Witten with gauge group $\frakg_\eps$,
\[
  \int_{\mathrlap{A\in\Omega^1(\bbR^3;\frakg_\eps)}}\calD A\, \bbe^{
    \frac{i}{4\pi}\int_{\bbR^3}\tr\left(
      A\wedge dA+\frac{2\sqrt{\hbar}}{3}A\wedge A\wedge A
    \right)
  }
  \calP\!\exp_\gamma(\sqrt{\hbar}A)
  \in \calU(\frakg_\eps)\llbracket\hbar\rrbracket.
\]

\vskip 4mm
\ThickSeparator

\newcommand{\refssize}{\fontsize{7pt}{8pt}\selectfont}
\def\bysame{{---}}

\hfill{\normalsize\red\bf References.}

\par\vspace{-10mm}
\renewcommand{\section}[2]{}%
\begin{thebibliography}{}
\setlength{\parskip}{0pt}
\setlength{\itemsep}{0pt plus 0.3ex}
{\refssize \input{refs.tex}}
\end{thebibliography}

\end{multicols}

\vfill
\begin{center}
\includegraphics[width=0.715\linewidth]{../../Projects/Theta/Theta_16up_1.pdf}
\parbox[b]{0.23\linewidth}{\begin{center}
  \includegraphics[width=0.92\linewidth]{../LesDiablerets-2601/figs/P41.pdf}
  \newline A $(2,41,-41)$ pretzel for
  \newline dessert
\end{center}}%
\end{center}%

\newpage
\begin{center}
\includegraphics[height=\textheight, page=1]{../../Projects/Theta/Theta_16up_2.pdf}
\newpage\includegraphics[height=\textheight, page=2]{../../Projects/Theta/Theta_16up_2.pdf}
\end{center}

%\newpage
%{\bf\red $\smiley$~Fun.} The Rolfsen Table:
%
%\vfill
%
%\includegraphics[width=\linewidth]{../UBC-241004/g250@.png}
%
%\vfill
%
%\parbox[t]{5.45in}{
%  The 132-crossing torus knot $T_{22/7}$:\hfill(many more at \web{TK})
%  \newline\includegraphics[width=\linewidth]{../Toronto-241030/T227Plot.pdf}
%}
%\hfill
%\parbox[t]{2.45in}{
%  Random knots from \cite{DHOEBL:Random} with 51 -- 75
%  crossings: (many more at \web{DK})
%  \newline
%  \vskip -7mm\includegraphics[height=\linewidth,angle=-90]{../LesDiablerets-2601/figs/Beehive.pdf}
%}

\newpage

\begin{center}
\adjustbox{valign=m}{\resizebox*{!}{10.5in}{
\def\k#1#2{{\parbox{0.5in}{\centering
  \vskip 2pt
  \includegraphics[width=\linewidth,height=\linewidth]{../../Projects/Theta/KnotFigs/#1_#2.pdf}
  \newline\href{https://katlas.org/wiki/#1_#2}{#1\_#2}
  \vskip 2pt
}}}
\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|}
  \hline \k{0}{1} & \k{3}{1} & \k{4}{1} & \k{5}{1} & \k{5}{2} & \k{6}{1} & \k{6}{2} & \k{6}{3} & \k{7}{1} & \k{7}{2} \\
  \hline \k{7}{3} & \k{7}{4} & \k{7}{5} & \k{7}{6} & \k{7}{7} & \k{8}{1} & \k{8}{2} & \k{8}{3} & \k{8}{4} & \k{8}{5} \\
  \hline \k{8}{6} & \k{8}{7} & \k{8}{8} & \k{8}{9} & \k{8}{10} & \k{8}{11} & \k{8}{12} & \k{8}{13} & \k{8}{14} & \k{8}{15} \\
  \hline \k{8}{16} & \k{8}{17} & \k{8}{18} & \k{8}{19} & \k{8}{20} & \k{8}{21} & \k{9}{1} & \k{9}{2} & \k{9}{3} & \k{9}{4} \\
  \hline \k{9}{5} & \k{9}{6} & \k{9}{7} & \k{9}{8} & \k{9}{9} & \k{9}{10} & \k{9}{11} & \k{9}{12} & \k{9}{13} & \k{9}{14} \\
  \hline \k{9}{15} & \k{9}{16} & \k{9}{17} & \k{9}{18} & \k{9}{19} & \k{9}{20} & \k{9}{21} & \k{9}{22} & \k{9}{23} & \k{9}{24} \\
  \hline \k{9}{25} & \k{9}{26} & \k{9}{27} & \k{9}{28} & \k{9}{29} & \k{9}{30} & \k{9}{31} & \k{9}{32} & \k{9}{33} & \k{9}{34} \\
  \hline \k{9}{35} & \k{9}{36} & \k{9}{37} & \k{9}{38} & \k{9}{39} & \k{9}{40} & \k{9}{41} & \k{9}{42} & \k{9}{43} & \k{9}{44} \\
  \hline \k{9}{45} & \k{9}{46} & \k{9}{47} & \k{9}{48} & \k{9}{49} & \k{10}{1} & \k{10}{2} & \k{10}{3} & \k{10}{4} & \k{10}{5} \\
  \hline \k{10}{6} & \k{10}{7} & \k{10}{8} & \k{10}{9} & \k{10}{10} & \k{10}{11} & \k{10}{12} & \k{10}{13} & \k{10}{14} & \k{10}{15} \\
  \hline \k{10}{16} & \k{10}{17} & \k{10}{18} & \k{10}{19} & \k{10}{20} & \k{10}{21} & \k{10}{22} & \k{10}{23} & \k{10}{24} & \k{10}{25} \\
  \hline \k{10}{26} & \k{10}{27} & \k{10}{28} & \k{10}{29} & \k{10}{30} & \k{10}{31} & \k{10}{32} & \k{10}{33} & \k{10}{34} & \k{10}{35} \\
  \hline \k{10}{36} & \k{10}{37} & \k{10}{38} & \k{10}{39} & \k{10}{40} & \k{10}{41} & \k{10}{42} & \k{10}{43} & \k{10}{44} & \k{10}{45} \\
  \hline \k{10}{46} & \k{10}{47} & \k{10}{48} & \k{10}{49} & \k{10}{50} & \k{10}{51} & \k{10}{52} & \k{10}{53} & \k{10}{54} & \k{10}{55} \\
  \hline \k{10}{56} & \k{10}{57} & \k{10}{58} & \k{10}{59} & \k{10}{60} & \k{10}{61} & \k{10}{62} & \k{10}{63} & \k{10}{64} & \k{10}{65} \\
  \hline \k{10}{66} & \k{10}{67} & \k{10}{68} & \k{10}{69} & \k{10}{70} & \k{10}{71} & \k{10}{72} & \k{10}{73} & \k{10}{74} & \k{10}{75} \\
  \hline \k{10}{76} & \k{10}{77} & \k{10}{78} & \k{10}{79} & \k{10}{80} & \k{10}{81} & \k{10}{82} & \k{10}{83} & \k{10}{84} & \k{10}{85} \\
  \hline \k{10}{86} & \k{10}{87} & \k{10}{88} & \k{10}{89} & \k{10}{90} & \k{10}{91} & \k{10}{92} & \k{10}{93} & \k{10}{94} & \k{10}{95} \\
  \hline \k{10}{96} & \k{10}{97} & \k{10}{98} & \k{10}{99} & \k{10}{100} & \k{10}{101} & \k{10}{102} & \k{10}{103} & \k{10}{104} & \k{10}{105} \\
  \hline \k{10}{106} & \k{10}{107} & \k{10}{108} & \k{10}{109} & \k{10}{110} & \k{10}{111} & \k{10}{112} & \k{10}{113} & \k{10}{114} & \k{10}{115} \\
  \hline \k{10}{116} & \k{10}{117} & \k{10}{118} & \k{10}{119} & \k{10}{120} & \k{10}{121} & \k{10}{122} & \k{10}{123} & \k{10}{124} & \k{10}{125} \\
  \hline \k{10}{126} & \k{10}{127} & \k{10}{128} & \k{10}{129} & \k{10}{130} & \k{10}{131} & \k{10}{132} & \k{10}{133} & \k{10}{134} & \k{10}{135} \\
  \hline \k{10}{136} & \k{10}{137} & \k{10}{138} & \k{10}{139} & \k{10}{140} & \k{10}{141} & \k{10}{142} & \k{10}{143} & \k{10}{144} & \k{10}{145} \\
  \hline \k{10}{146} & \k{10}{147} & \k{10}{148} & \k{10}{149} & \k{10}{150} & \k{10}{151} & \k{10}{152} & \k{10}{153} & \k{10}{154} & \k{10}{155} \\
  \hline \k{10}{156} & \k{10}{157} & \k{10}{158} & \k{10}{159} & \k{10}{160} & \k{10}{161} & \k{10}{162} & \k{10}{163} & \k{10}{164} & \k{10}{165} \\
  \hline
\end{tabular}
}}
$\underset{\Theta}{\rightarrow}$
\includegraphics[height=10.5in,valign=m]{../../Projects/Theta/figs/Theta4Rolfsen.pdf}
\end{center}

\end{document}

\endinput

