\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