\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, multicol} \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=8.5in,textheight=11in,centering]{geometry} \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{MonteVerita-2604} \def\title{Chern-Simons-Witten Theory Near The Co-Commutative Limit} \def\navigator{{ \href{\myurl}{Dror Bar-Natan}: \href{\myurl/Talks}{Talks}: \href{\myurl/Talks/\thistalk/}{\thistalk}: }} \def\thanks{{Thanks for inviting me to Monte Verit\`a!}} \def\webdef{{{\greektext web}$\coloneqq$\href{http://drorbn.net/mv26}{http://drorbn.net/mv26}}} \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} 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\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\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}} % From http://tex.stackexchange.com/questions/154672/how-to-get-a-medium-sized-otimes \DeclareMathOperator*{\midotimes}{\text{\raisebox{0.25ex}{\scalebox{0.8}{$\bigotimes$}}}} % \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 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1mm\par\noindent\includegraphics[valign=t]{#1}} \def\nbpdfgraphOutput#1{\vskip 1mm\par\noindent\Out\includegraphics[width=1.5in valign=t]{#1}} %%% \def\Abstract{{\raisebox{2mm}{\parbox[t]{3.95in}{ \parshape 7 0in 3.35in 0in 3.35in 0in 3.35in 0in 3.35in 0in 3.35in 0in 3.35in 0in 3.95in {\red\bf Abstract.} Perhaps every algebra meeting should have one analysis talk (and vice versa), lest we forget that the other exists. In my role as the outsider, I will tell you today about the other -- perturbative -- evaluation of path integrals, where instead of hoping that nature will help us compute faster, we approximate nature by things we already can compute quickly. Specifically I will tell you how in the Chern-Simons-Witten theory you can perturb the base Lie algebra from where it's easy towards where it's strong, leading to the strongest genuinely computable knot invariant we presently have. I wish I could give my talk in the language of the Kabbalah, but I ain't smart enough for that. So I'll highlight the Kabbalistic points that we're still missing, and then stick to the Talmud. \vskip 3.5pt {\bf\red Acknowledgement.} Work supported by NSERC grant RGPIN-2025-06718 and by the Chu Family Foundation (NYC). }}}} \def\Acknowledgement{{\raisebox{2.25mm}{\parbox[t]{3.25in}{\footnotesize {\bf\red Acknowledgement.} Work supported by NSERC grant RGPIN-2025-06718 and by the Chu Family Foundation (NYC). }}}} %\def\Speculation{{\raisebox{2.25mm}{\parbox[t]{3.95in}{ %{\bf\red Non-Apology.} %This talk touches the disreputable subject of {\em speculation}. But it's speculation in %which the results are already there; we merely speculate on where they may be coming from. %}}}} \def\Speculation{{\raisebox{2.25mm}{\parbox[t]{3.95in}{ {\bf\red Compliance Statement.} This talk touches {\em speculative} Kabbalah. I'm over 40, and it's speculation with Talmudic results. I merely speculate on where they may be coming from. }}}} \def\CSW{{\raisebox{1mm}{\parbox[t]{2.85in}{ Chern-Simons-Witten with metrized Lie algebra $\frakg$ and representation $V$: \[ \int_{\mathrlap{A\in\Omega^1(\bbR^3;\frakg)}}\calD A\, \bbe^{ \frac{i}{4\pi}\int_{\bbR^3}\tr\left( A\wedge dA+\frac{2\sqrt{\hbar}}{3}A\wedge A\wedge A \right) } \tr_V\calP\!\exp_\gamma(\sqrt{\hbar}A) \] }}}} \def\RT{{\raisebox{1mm}{\parbox[t]{1.85in}{ Reshetikhin-Turaev for $(\frakg,V)$ at $q=\bbe^{\hbar}$ }}}} \def\OldInvariants{{\raisebox{1mm}{\parbox[t]{2.6in}{ Finite type invariants and the Kontsevitch Integral: Very powerful, but except for low cases, impossible to compute. }}}} \def\OldKabbalah{{\raisebox{0mm}{\parbox[t]{2.4in}{ \parshape 3 0in 2.5in 0.2in 2.2in 0.4in 2.0in Old Kabbalah: Perturbed Gaussian Integration, Feynman diagrams, configuration space integrals. }}}} \def\hbarexp{{\raisebox{2mm}{\parbox[t]{1.75in}{ {\centering $\hbar$-expansion (B-N, Lin, \ldots)} }}}} \def\SolvApp{{\raisebox{0mm}{\parbox[t]{1.65in}{\centering The co-commutative limit: replace $\frakg$ by $\frakg_\eps$. ``Solvable Approximation''. }}}} \def\CSWeps{{\raisebox{1mm}{\parbox[t]{2.75in}{ Chern-Simons-Witten with $\frakg_\eps$: \[ \int_{\mathrlap{A\in\Omega^1(\bbR^3;\frakg_\eps)}}\calD A\, \bbe^{ \frac{i}{4\pi}\int_{\bbR^3}\tr\left( A\wedge dA+\frac{2\sqrt{\hbar}}{3}A\wedge A\wedge A \right) } \calP\!\exp_\gamma(\sqrt{\hbar}A) \] }}}} \def\URTeps{{\raisebox{1mm}{\parbox[t]{1.85in}{ Universal Reshetikhin-Turaev for $\frakg_\eps$ with $q=\bbe^{\hbar\eps}$ }}}} \def\Invariants{{\raisebox{1mm}{\parbox[t]{2.6in}{ Today's invariants: \[ \left(\begin{array}{c|c|c|c} \eps=0 & \eps^2=0 &\eps^3=0 & \ldots \\ \text{Alexander's }\Delta & \rho_1,\theta & \rho_2,\ldots & \ldots \end{array}\right) \] }}}} \def\epsexp{{\raisebox{2mm}{\parbox[t]{1.65in}{ {\centering $\eps$-expansion: perturbation theory, Feynman diagrams \vskip 3mm \includegraphics[width=\linewidth]{../Kyoto-230727/DoPeGDO-thumb.pdf} } \vskip -10mm \rightline{\footnotesize In Kyoto, \web{k23}} }}}} \def\NewKabbalah{{\raisebox{0mm}{\parbox[t]{2.4in}{ New Kabbalah (missing): \par \quad 1.~Exact evaluation at $\eps=0$ giving $\Delta$. \par \qquad 2.~Then perturbation theory. }}}} \def\Gaussian{{\raisebox{2mm}{\parbox[t]{3.9in}{ {\bf\red Perturbed Gaussian Integration (PGI) / Feynman Diagrams.} With Einstein sumintegration and with $G=A^{-1}=(a_{ij})^{-1}$, \[ \int_{\mathrlap{x\in\bbR^M}}dx\,\bbe^{-a^{ij}x_ix_j/2+\eps V(x)} = \frac{1}{\det^{1/2}(A)}\left. \bbe^{g_{ij}\partial_{x_i}\partial_{x_j}}\bbe^{\eps V(x)}\right|_{x=0}. \] \parshape 5 0in \linewidth 0in \linewidth 0in \linewidth 1.4in 2.25in 2.25in 1.4in $\bullet$~Kabbalah: If $M$ is continuous, a reduction of path integrals to finite dimensional integration. $\bullet$~Talmud: If $M$ is discrete, approximates an exponential-time computation with a polynomial-time one! $\bullet$~Cements my status as an outsider here\ldots }}}} \def\bracket{$b({\red\uppertriang})=b\colon{\red\uppertriang}\otimes\!{\red\uppertriang} \to{\red\uppertriang}$} \def\cobracket{$b({\blue\lowertriang})\leadsto\delta\colon{\red\uppertriang} \to{\red\uppertriang}\otimes\!{\red\uppertriang}$} \def\CCL{{\raisebox{2mm}{\parbox[t]{3.75in}{ {\red\bf Co-Commutative Limit / Solvable Approximation.} In $gl_n$, half is enough! Indeed $gl_n\oplus\fraka_n = \calD(\uppertriang,b,\delta)$: \vskip 12mm Now define $gl^\epsilon_n\coloneqq\calD(\uppertriang,b,\epsilon\delta)$. 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, and at $\epsilon^{k+1}=0$ always yields a solvable Lie algebra. }}}} \def\Conventions{{\raisebox{2mm}{\parbox[t]{3.95in}{ {\bf\red Conventions.} $T$, $T_1$, and $T_2$ are indeterminates and $T_3\coloneqq T_1T_2$. }}}} \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\TrafficRules{{\raisebox{2mm}{\parbox[t]{3in}{ \parshape 4 0in 3in 0in 3in 0in 3in 0.72in 2.28in {\bf\red Model $T$ Traffic Rules.} Cars always drive forward. When a car crosses over a sign-$s$ bridge it goes through with (algebraic) probability $T^s\sim 1$, but falls off with probability $1-T^s\sim 0$. At the very end, cars fall off and disappear. On various edges {\em traffic counters} are placed. See also~\cite{Jones:Hecke, LinTianWang:RandomWalk}. }}}} \def\dtA{{\tiny image credits:}} \def\dtB{{\tiny \href{https://diamondtraffic.com/productcategory/Portable-Counters}{diamondtraffic.com}}} \def\DallEA{{\tiny image credits:}} \def\DallEB{{\tiny \href{https://labs.openai.com/}{Dall-E}}} \def\gab{{\raisebox{2mm}{\parbox[t]{3.95in}{ \parshape 4 0in 2.9in 0in 2.9in 0in 2.9in 0in 3.95in {\bf\red Definition.} The {\em traffic function} $G=(g_{\alpha\beta})$ (also, the {\em Green function} or the {\em two-point function}) is the reading of a traffic counter at $\beta$, if car traffic is injected at $\alpha$ (if $\alpha=\beta$, the counter is {\em after} the injection point). There are also model-$T_\nu$ traffic functions $(g_{\nu\alpha\beta})$ for $\nu=1,2,3$. }}}} \def\kinkA{{$\sum_{p\geq 0}(1\!-\!T)^p=T^{-1}$}} \def\kinkG{{$G=\begin{pmatrix}1&T^{-1}&1\\0&T^{-1}&1\\0&0&1\end{pmatrix}$}} \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: \[ \theta(D) \coloneqq \underbrace{\Delta_1\Delta_2\Delta_3}_{\parbox{0.66in}{\scriptsize\centering normalization, see below }} \left(\sum_c F_1(c) + \sum_{c_0,c_1} F_2(c_0,c_1) + \sum_kF_3(\varphi_k,k)\right), \] \vskip 23mm These pictures should remind you of Feynman diagrams! \vskip 2.5mm \par\rightline{$\Delta_\nu$ is the normalized Alexander polynomial at $T_\nu$} \par\rightline{$F_1$, $F_2$, and $F_3$ are below} }}}} \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\LemmaA{{\raisebox{2mm}{\parbox[t]{3.95in}{ {\bf\red Lemma 1.} The traffic function $g_{\alpha\beta}$ is a ``relative invariant'': }}}} \def\messA{{$(1\!-\!T)^2\!+\!T(1\!-\!T)$}} \def\messB{{$(1\!-\!T)T$}} \def\messC{{$T(1\!-\!T)$}} \def\messD{{$1\!-\!T$}} \def\i{{$i$}} \def\j{{$j$}} \def\k{{$k$}} \def\m{{$m$}} \def\n{{$n$}} \def\s{{$s$}} \def\ip{{$i^+$}} \def\jp{{$j^+$}} \def\kp{{$k^+$}} \def\ipp{{$i^{+\!+}$}} \def\jpp{{$j^{+\!+}$}} \def\kpp{{$k^{+\!+}$}} \def\LemmaB{{\raisebox{2mm}{\parbox[t]{3.95in}{ \parshape 3 0in 3in 0in 3in 0in 3.95in {\bf\red Lemma 2.} With $k^+\coloneqq k+1$, the ``$g$-rules'' hold near a crossing $c=(s,i,j)$: \[ g_{j\beta} = g_{j^+\beta} + \delta_{j\beta} \quad g_{i\beta} = T^sg_{i^+\beta} + (1-T^s)g_{j^+\beta} + \delta_{i\beta} \quad g_{2n^+,\beta} = \delta_{2n^+,\beta} \] \[ g_{\alpha i^+} = T^sg_{\alpha i} + \delta_{\alpha i^+} \quad g_{\alpha j^+} = g_{\alpha j} + (1-T^s)g_{\alpha i} + \delta_{\alpha j^+} \quad g_{\alpha,1} = \delta_{\alpha,1} %\quad g_{\alpha,2n^+} = 1 \] {\bf\red Corollary 1.} $G$ is easily computable, for $AG=I$ ($=GA$), with $A$ the $(2n+1)\times(2n+1)$ identity matrix with additional contributions: \[ c=(s,i,j) \mapsto \begin{array}{c|cccc} A & \text{col }i^+ & \text{col }j^+ \\ \hline \text{row }i & -T^s & T^s-1 \\ \text{row }j & 0 & -1 \end{array} \] For the trefoil example, we have that $A,G=$ \resizebox{\linewidth}{!}{$ \left(\setlength{\arraycolsep}{2pt} \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), \left(\setlength{\arraycolsep}{2pt} \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) $} }}}} \def\Alexander{{\raisebox{2mm}{\parbox[t]{3.95in}{ {\bf\red Note.} The Alexander polynomial $\Delta$ is given by \newline\null \hfill$\Delta = T^{(-\varphi-w)/2}\det(A)$, \hfill with $\varphi = \sum_k \varphi_k$, $w = \sum_c s$. \hfill\null }}}} \def\Invariance{{\raisebox{2mm}{\parbox[t]{4in}{ {\bf\red Corollary 2.} Proving invariance is easy:\hfill(Theta.nb at \web{ap}) \vskip 31mm \scalebox{0.85}{\parbox{5in}{\input{Invariance.tex}}} %\input{Invariance.tex} }}}} \def\Strong{{\raisebox{3mm}{\parbox[t]{3.95in}{ {\bf\red Strong.} Testing $\Theta=(\Delta,\theta) \in \bbZ[T^{\pm 1}]\times\bbZ[T_1^{\pm 1},T_2^{\pm 1}]$ vs. a slew of other reasonably-computable invariants on prime knots up to mirrors and reversals, counting the number of distinct values (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\GC{{\raisebox{3mm}{\parbox[t]{3.95in}{ \parshape 1 0in 1.75in {\red\bf Fast.} Here's $\Theta$ on a random 300 crossing knot (from \cite{DHOEBL:Random}). This proves that I don't belong. %For almost every other invariant, that's science fiction. \parshape 8 0in 1.75in 0in 1.75in 0in 1.75in 0in 1.75in 0in 1.75in 0in 1.75in 0in 1.75in 0in \linewidth {\red\bf Fun.} There's so much more to see in 2D pictures than in 1D ones! Yet almost nothing of the patterns you see \text{we} know how to prove. {\text We'll} have fun with that over the next few years. Would you join? \parshape 3 0in 2in 0in 2.25in 0in \linewidth {\bf\red Meaningful.} $\theta$ gives a genus bound (with yet-unwritten proof). Also, $\theta$ seems to give a criterion for a knot to be fibered (conjectured with a large scale verification). There are ``safe'' conjectured characterizations of $\theta$ as ``the two loop invariant'' and as ``the one cobracket invariant''. We hope (with reason) $\theta$ will say something about ribbon 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]{Snips/LXInput.pdf} \vskip 1mm \par \includegraphics[width=4in]{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\Believe{{\raisebox{2mm}{\parbox[t]{3.55in}{ {\bf\red Why Believe in the New Kabbalah?} $\bullet$~There is no doubt that the $\eps$-expansion of $RT(\frakg_\eps)$ gives $\theta$ etc. That's how \text{we} got to $\theta$. $\bullet$~There is no doubt that the $\eps$-expansion of $\text{CSW}(\frakg_\eps)$ / the Kontsevich integral is the loop expansion and it leads to $\theta$. It works on the level of weight systems, and the match with Ohtsuki~\cite{Ohtsuki:2Loop} is perfect. $\bullet$~In a similar context, Rozansky has a non-perturbative derivation of $\Delta$ from CSW \cite{Rozansky:Contribution}. $\bullet$~There's perturbed Gaussian integration at the tail of the missing arrow and at the head of that arrow. What else the arrow may be, other than integrating out {\em most} of the degrees of freedom (but not all)? $\bullet$~I had ideas, but not the time to implement them\ldots }}}} \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{CSWCCLT1.pdftex_t}\vfill\null \end{center} \def\In{\smiley} \def\Out{\footnotesize\faLaptop} \begin{center} \null\vfill\import{}{CSWCCLT2.pdftex_t}\vfill\null \end{center} \begin{center} \null\vfill\import{}{CSWCCLT3.pdftex_t}\vfill\null \end{center} \newgeometry{textwidth=8in,textheight=10.5in} \begin{multicols*}{2} \def\In{\rlap{\protect\makebox[-4mm]{\smiley}}} \def\Out{\rlap{\protect\makebox[-4mm]{\footnotesize\faLaptop}}} \input{Program.tex} \newcommand{\refssize}{\fontsize{10pt}{11pt}\selectfont} \def\bysame{{---}} \needspace{2cm} {\normalsize\red\bf References.} \par\vspace{-3mm} \renewcommand{\section}[2]{}% \begin{thebibliography}{} \setlength{\parskip}{0pt} \setlength{\itemsep}{0pt plus 0.3ex} %{\input{refs.tex}} {\refssize \input{refs.tex}} \end{thebibliography} \parbox[t]{\linewidth}{ Random knots from \cite{DHOEBL:Random} with 51 -- 77 crossings: (many more at \web{DK}) \newline % \vskip -7mm\includegraphics[height=\linewidth]{Beehive.pdf} \vskip -7mm\includegraphics[height=\linewidth,angle=-90]{Beehive.pdf} } \end{multicols*} \end{document} \endinput