\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, manfnt} %\usepackage{tensor} \usepackage{txfonts} % for the likes of \coloneqq. \usepackage[usenames,dvipsnames]{xcolor} \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{CRM-1907} \def\title{Everything around $sl_{2+}^\epsilon$ is {\bf DoPeGDO}. So what?} \def\navigator{{ \href{\myurl}{Dror Bar-Natan}: \href{\myurl/Talks}{Talks}: \href{\myurl/Talks/\thistalk/}{\thistalk}: }} \def\thanks{{Thanks for coming to the \magenta CRM!}} \def\webdef{{{\greektext web}$\coloneqq$\href{http://drorbn.net/m19}{http://drorbn.net/m19/}}} \def\web#1{{\href{\myurl/Talks/\thistalk/#1}{{\greektext web}/#1}}} \def\titleA{{\title}} \def\titleB{{\title}} \def\titleC{{\title}} \def\more{More at \web{talks}} \definecolor{mgray}{HTML}{B0B0B0} \definecolor{morange}{HTML}{FFA50A} \definecolor{mpink}{HTML}{F35EB7} \def\blue{\color{blue}} \def\gray{\color{gray}} \def\mgray{\color{mgray}} \def\morange{\color{morange}} \def\mpink{\color{mpink}} \def\pink{\color{pink}} \def\purple{\color{purple}} \def\magenta{\color{magenta}} \def\red{\color{red}} \def\yellowm#1{{\setlength{\fboxsep}{0pt}\colorbox{yellow}{$#1$}}} \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\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\FL{\text{\it FL}} \def\Hom{\operatorname{Hom}} \def\IHX{\mathit{IHX}} \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\vT{{\mathit v\!T}} \def\wt{\operatorname{wt}} \def\barT{{\bar T}} \def\barz{{\bar z}} \def\bbe{\mathbbm{e}} \def\bbD{{\mathbb D}} \def\bbE{{\mathbb E}} \def\bbF{{\mathbb F}} \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\calE{{\mathcal E}} \def\calF{{\mathcal F}} \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\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}} % 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\cellscale{0.645} %%% \def\Plan{{\raisebox{2mm}{\parbox[t]{3.95in}{ {\red\bf Talk Plan.} \begin{enumerate}[leftmargin=*,labelindent=0pt,itemsep=-4pt,topsep=0pt] \item Headers and abstract. \item Ops grid. \item $sl_{2+}^\epsilon$ and 4D Lie algebras, then the details of $CU$ and $QU$. \item The {\bf DoPeGDO} box, ``one abstraction level up''. \item A glance through the {\bf DoPeGDO} footnotes. \item Naive {\bf DoPeGDO} compositions. \item The ``debts'' box, and then go through the debts as follows. \item A quantum algebra example. \item A knot theory example, followed by the knot table noting affinities with topology. \end{enumerate} }}}} \def\credits{{\raisebox{0.75mm}{\parbox[t]{1.3in}{ Continues Rozansky \cite{Ro, Rozansky:Burau, Rozansky:U1RCC} and \text{Overbay} \cite{Overbay:Thesis}, joint with van der Veen \cite{PP1}. }}}} \def\AbstractA{{\raisebox{2mm}{\parbox[t]{3.95in}{\begin{hyphenrules}{nohyphenation} {\red\bf Abstract.} I'll explain what ``everything around'' means: classical and quantum $m$, $\Delta$, $S$, $tr$, $R$, $C$, and $\theta$, as well as $P$, $\Phi$, $J$, $\bbD$, and more, and all of their compositions. What {\bf DoPeGDO} means: the category of \textcolor{ForestGreen}{Docile Perturbed Gaussian Differential Operators}. And what $sl_{2+}^\epsilon$ means: a solvable approximation of the semi-simple Lie algebra $sl_2$. \end{hyphenrules}}}}} \def\AbstractB{{\raisebox{-1mm}{\parbox[t]{3.9in}{ Knot theorists should rejoice \text{because} all this leads to very powerful and well-behaved poly-time-computable knot \text{invariants}. Quantum algebraists should rejoice because it's a realistic playground for testing complicated \text{equations} and theories. }}}} \def\maximm{{\thickmuskip=0mu $m\colon U \otimes U \to U$}} \def\maximDelta{{\thickmuskip=0mu $\Delta\colon U \to U \otimes U$}} \def\maximS{{\thickmuskip=0mu $S\colon U \to U$}} \def\maximtr{{\thickmuskip=-2mu $\tr\colon U \to U/wx=xw$}} \def\maximR{{\thickmuskip=0mu $R\in QU\otimes QU$}} \def\maximC{{\thickmuskip=0mu $C^{\pm 1}\in QU$}} \def\maximPhi{{\thickmuskip=0mu $\Phi\in CU^{\otimes 3}$}} \def\maximJ{{\thickmuskip=0mu $J\in CU\otimes CU$}} \def\Udef{{\raisebox{2mm}{\parbox[t]{3.95in}{ {\bf\red Our Algebras.} Let $sl_{2+}^\epsilon \coloneqq L\langle y,b,a,x\rangle$ subject to ${[a,x]=x}$, $[b,y]=-\epsilon y$, $[a,b]=0$, $[a,y]=-y$, $[b,x]=\epsilon x$, and $[x,y]=\epsilon a+b$. So $t\coloneqq\epsilon a-b$ is central and if $\exists \epsilon^{-1}$, $sl_{2+}^\epsilon/\langle t\rangle\cong sl_2$. $U$ is either $CU=\hat\calU(sl_{2+}^\epsilon)$ or $QU=\calU_\hbar(sl_{2+}^\epsilon) = A\langle y,b,a,x\rangle$ with $[a,x]=x$, $[b,y]=-\epsilon y$, $[a,b]=0$, $[a,y]=-y$, $[b,x]=\epsilon x$, and $xy-qyx=(1-AB)/\hbar$, where $q=\bbe^{\hbar\epsilon}$, $A=\bbe^{-\hbar\epsilon a}$, and $B=\bbe^{-\hbar b}$. Set also $T=A^{-1}B=\bbe^{\hbar t}$. {\red The Quantum Leap.} Also decree that in $QU$, \[ \Delta(y,b,a,x) = (y_1+B_1y_2, b_1+b_2, a_1+a_2, x_1+A_1x_2), \] \[ S(y,b,a,x) = (-B^{-1}y, -b, -a, -A^{-1}x),\] \vskip 3pt and $R=\sum\hbar^{j+k}y^kb^j\otimes a^jx^k/j![k]_q!$. }}}} \def\Conventions{{\raisebox{0mm}{\parbox[t]{3.9in}{ {\bf Conventions.} 1.~For a set $A$, let $z_A\coloneqq\{z_i\}_{i\in A}$ and let ${\zeta_A\coloneqq\{z^*_i=\zeta_i\}_{i\in A}}$.$^{\dagger 1}$ 2.~Everything converges! }}}} \def\DoPeGDO{{\raisebox{1mm}{\parbox[t]{2.96in}{ $\text{\bf DoPeGDO} \coloneqq$ The category with objects finite sets$^{\dagger 2}$ and $\mor(A\to B)$: \[ \left\{\calF =\omega\exp(Q+P) \right\} \subset \bbQ\llbracket\zeta_A,z_B\rrbracket \] Where: $\bullet$~$\omega$ is a scalar.$^{\dagger 3}$ $\bullet$~$Q$ is a ``small'' quadratic in $\zeta_A\cup z_B$.$^{\dagger 4}$ $\bullet$~$P$ is a ``docile perturbation'': $P=\sum_{k\geq 1}\epsilon^kP^{(k)}$, where $\deg P^{(k)}\leq 2k+2$.$^{\dagger 5}$ $\bullet$~Compositions:$^{\dagger 6}$ \[ \calF\act \calG = \calG\circ \calF \coloneqq \left( \calG|_{\zeta_i\to\partial_{z_i}}\calF \right)_{z_i=0} \!\! = \left( \calF|_{z_i\to\partial_{\zeta_i}}\calG \right)_{\zeta_i=0}. \] \vskip 0.5mm {\bf\red Cool!} $(V^*)^{\otimes\Sigma}\otimes V^{\otimes S}$ explodes; the ranks of quadratics and bounded-degree polynomials grow slowly!$^{\dagger 7}$ \hfill\text{\red Representation theory is over-rated!} \vskip 0.5mm {\bf\red Cool!} How often do you see a computational toolbox so successful? }}}} \def\Footnotes{{\raisebox{2mm}{\parbox[t]{3.96in}{\small\hangindent=5mm {\bf\red DoPeGDO Footnotes.} $\dagger 1$.~Each variable has a ``weight''$\in\{0,1,2\}$, and always $\wt z_i+\wt\zeta_i=2$. \par$\dagger 2$.~Really, ``weight-graded finite sets'' $A=A_0\sqcup A_1\sqcup A_2$. \par$\dagger 3$.~Really, a power series in the weight-0 variables$^{\dagger 9}$. \par\hangindent=5mm$\dagger 4$.~The weight of $Q$ must be 2, so it decomposes as $Q=Q_{20}+Q_{11}$. The coefficients of $Q_{20}$ are rational numbers while the coefficients of $Q_{11}$ may be weight-0 power series$^{\dagger 9}$. \par\hangindent=5mm$\dagger 5$.~Setting $\wt\epsilon=-2$, the weight of $P$ is $\leq 2$ (so the powers of the weight-0 variables are not constrained$^{\dagger 9}$). \par\hangindent=5mm$\dagger 6$.~There's also an obvious product \newline\null\hfill$ \mor(A_1\to B_1)\times\mor(A_2\to B_2)\to\mor(A_1\sqcup A_2\to B_1\sqcup B_2). $\hfill\null \par\hangindent=5mm$\dagger 7$.~That is, if the weight-0 variables are ignored. Otherwise more care is needed yet the conclusion remains. \par\hangindent=5mm$\dagger 8$.~$\Hom(U^{\otimes\Sigma}\to U^{\otimes S}) \leadsto\mor(\{\eta_i,\beta_i,\tau_i,\alpha_i,\xi_i\}_{i\in\Sigma}\to \{y_i,b_i,t_i,a_i,x_i\}_{i\in S})$, where $\wt(\eta_i,\xi_i,y_i,x_i)=1$ and $\wt(\beta_i,\tau_i,\alpha_i;b_i,t_i,a_i)=(2,2,0;0,0,2)$. \par\hangindent=5mm$\dagger 9$.~For tangle invariants the weight-0 power series are always rational functions in the exponentials of the weight-0 variables (for knots: just one variable). }}}} \def\QEFG{{In $\mor(A\!\to\!B)$, $Q\!= \hspace{-2mm}\sum\limits_{i\in A,j\in B}\hspace{-2mm}E_{ij}\zeta_iz_j \!+\! \frac12\hspace{-1mm}\sum\limits_{i,j\in A}\hspace{-1mm}F_{ij}\zeta_i\zeta_j \!+\! \frac12\hspace{-1mm}\sum\limits_{i,j\in B}\hspace{-1mm}G_{ij}z_iz_j $}} \def\Compositions{{\raisebox{1mm}{\parbox[t]{2.65in}{ Where $\bullet$\quad $E=E_1(I-F_2G_1)^{-1}E_2$. \newline$\bullet$\quad $F=F_1+E_1F_2(I-G_1F_2)^{-1}E_1^T$. \newline$\bullet$\quad $G=G_2+E_2^TG_1(I-F_2G_1)^{-1}E_2$. \newline$\bullet$\quad $\omega=\omega_1\omega_2\det(I-F_2G_1)^{-1}$. \newline$\bullet$\quad $P$ is computed using ``connected Feynman diagrams'' or as the solution of a messy PDE (yet we're still in algebra!). %\newline\null\hfill\text{\mpink \ldots a presentation / implementation dead end.} }}}} \def\oneup{{\raisebox{1mm}{\parbox[t]{1.08in}{ One abstraction level up from tangles! \par $\{\text{tangles}\}\to\left\{\rule{0pt}{10pt}\hspace{11pt}\right\}$ \par with compositions: }}}} \def\OverratedOne{{\raisebox{2mm}{\parbox[t]{3.96in}{ {\bf\red Categories are overrated (1)!} $\bullet$~Tangles are artificially made to have a ``top'' and a ``bottom''. $\bullet$~Tangles are accessed by their ends and not by their strands; crossings are named by their position and not by the strands involved: }}}} \def\TwoReidemeister{{\raisebox{1mm}{\parbox[t]{2.15in}{ Is this $\sigma_1\sigma_2\sigma_1=\sigma_2\sigma_1\sigma_2$ \newline or $\sigma_{12}\sigma_{13}\sigma_{23}=\sigma_{23}\sigma_{13}\sigma_{12}$? \vskip 3mm $\bullet$~Easier to talk about ``skein theory''. \newline $\bullet$~Harder to talk about ``universal quantum invariants''. }}}} \def\Debts{{\raisebox{2mm}{\parbox[t]{3.95in}{ {\bf\red Mid-Talk Debts.} $\bullet$~What is this good for in quantum algebra? \par$\bullet$~In knot theory? \par\hangindent=5mm$\bullet$~How does the ``inclusion'' $\calD\colon\Hom(U^{\otimes\Sigma}\to U^{\otimes S}) \leadsto$ \text{\bf DoPeGDO} work? \par$\bullet$~Proofs that everything around $sl_{2+}^\epsilon$ really is {\bf DoPeGDO}. \par$\bullet$~Relations with prior art. \par$\bullet$~The rest of the ``compositions'' story. }}}} \def\mmr{{\raisebox{2mm}{\parbox[t]{3.95in}{ \parshape 3 0in 2.45in 0in 2.45in 0in 3.95in \begin{hyphenrules}{nohyphenation}{\red\bf 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 \end{hyphenrules} \[ \left. \frac{(q^{1/2}-q^{-1/2})J_d(K)}{q^{d/2}-q^{-d/2}} \right|_{q=e^\hbar} = \sum_{j,m\geq 0} a_{jm}(K)d^j\hbar^m, \] \parshape 5 0in 3in 0in 3in 0in 3in 0in 3in 0in 3.95in ``below diagonal'' coefficients vanish, $a_{jm}(K)=0$ if $j>m$, and ``on diagonal'' coefficients give the inverse of the Alexander polynomial: $\left(\sum_{m=0}^\infty a_{mm}(K)\hbar^m\right)\cdot \omega(K)(e^\hbar)=1$. ``Above diagonal'' we have {\red Rozansky's Theorem} \cite[(1.2)]{Rozansky:U1RCC}: \[ J_d(K)(q) = \frac{q^d-q^{-d}}{(q-q^{-1})\omega(K)(q^d)} \left(1+ \sum_{k=1}^\infty \frac{(q-1)^k\rho_k(K)(q^d)}{\omega^{2k}(K)(q^d)} \right). \] }}}} \def\MMG{\parbox{0.6in}{\scriptsize\raggedright Melvin, Morton, Garoufalidis }} \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{DoPeGDO1.pdftex_t}\vfill\null \end{center} \eject \newgeometry{textwidth=8in,textheight=10.5in} \begin{multicols}{2} {\bf\red $\calD\colon\Hom(U^{\otimes\Sigma}\to U^{\otimes S}) \to \bbQ\llbracket \eta_\Sigma,\beta_\Sigma,\alpha_\Sigma,\xi_\Sigma,y_S,b_S,a_S,x_S \rrbracket$.} %$ \text{\bf DoPeGDO}.} The PBW theorem for $CU$ (always in the $ybax$ order), or its quantum analog for $QU$, say that if $U=CU\text{ or }QU$ then $U^{\otimes S}$ is isomorphic as a vector space to $\bbQ\llbracket y_i,b_i,a_i,x_i\rrbracket_{i\in S}$; so it is enough to understand $\Hom(\bbQ\llbracket z_A\rrbracket\to\bbQ\llbracket z_B\rrbracket)$ for finite sets $A$ and $B$. Using the pairing \[ \langle z_i^m,\zeta_j^n \rangle = \left.\partial_{\zeta_i}^m\zeta_j^n\right|_{\zeta_A\to 0} = \delta_{ij}\delta_{mn}n!, \] we get a map \begin{multline*} \calD\colon\Hom(\bbQ\llbracket z_A\rrbracket\to\bbQ\llbracket z_B\rrbracket) \cong\bbQ\llbracket z_A\rrbracket^*\otimes\bbQ\llbracket z_B\rrbracket \\ \cong\bbQ\llbracket \zeta_A\rrbracket\otimes\bbQ\llbracket z_B\rrbracket \cong\bbQ\llbracket \zeta_A,z_B \rrbracket \end{multline*} {\bf Example.} $\calD(id\colon\bbQ\llbracket z\rrbracket\to\bbQ\llbracket z\rrbracket)=\bbe^{\zeta z}$. Indeed, \[ \langle z^n, \bbe^{\zeta z}\rangle = \left\langle z^n, \sum_m\frac{(\zeta z)^m}{m!}\right\rangle = \sum_m\frac{z^m}{m!}\delta_{mn}n! = z^n. \] {\bf Example.} $\calD(id\colon U\to U) = \bbe^{\eta y+\beta b+\alpha a+\xi x}$. {\bf Claim.} Assuming convergence, if $F\in\Hom(\bbQ\llbracket z_A\rrbracket\to\bbQ\llbracket z_B\rrbracket)$, $G\in\Hom(\bbQ\llbracket z_B\rrbracket\to\bbQ\llbracket z_C\rrbracket)$, $\calF=\calD(F)$, and $\calG=\calD(G)$, then \[ \calD(F\act G) = \left( \calF|_{z_i\to\partial_{\zeta_i}}\calG \right)_{\zeta_i=0}. \] And so the title of the talk finally makes sense! \rule{\linewidth}{1pt} {\bf\red Other GDOs.} {\bf Claim.} If $L\colon\bbQ\llbracket z_A\rrbracket\to\bbQ\llbracket z_B\rrbracket$ is linear, then $\calD(L)=L\left(\bbe^{\sum_{i\in A}\zeta_i z_i}\right)$.\hfill\text{{\bf Proof.} Exercise.} {\bf Example.} Let $c\Delta^i_{jk}\colon CU^{\otimes\{i\}}\to CU^{\otimes\{j,k\}}$ be the standard co-product, given by $c\Delta^i_{jk}(y_i,b_i,a_i,x_i) = (y_j+y_k, b_j+b_k, a_j+a_k, x_j+x_k)$. Then \begin{multline*} \calD(c\Delta^i_{jk}) = c\Delta^i_{jk}(\bbe^{\eta_iy_i+\beta_ib_i+\alpha_ia_i+\xi_ix_i}) \\ = \bbe^{\eta_i(y_j+y_k)+\beta_i(b_j+b_k)+\alpha_i(a_j+a_k)+\xi_i(x_j+x_k)}. \end{multline*} \needspace{1.25cm} \parpic[r]{$\xymatrix@C=5mm@R=5mm{ \bbQ\llbracket z\rrbracket_i\otimes\bbQ\llbracket z\rrbracket_j \ar[r]^<>(0.5){m^{ij}_k} \ar@{=}[d] & \bbQ\llbracket z\rrbracket_k \ar@{=}[d] \\ \bbQ\llbracket z_i,z_j\rrbracket \ar[r]^<>(0.5){m^{ij}_k} & \bbQ\llbracket z_k\rrbracket }$} {\bf Example.} The standard commutative product $m^{ij}_k$ of polynomials is given by $z_i,z_j\to z_k$. Hence $\calD(m^{ij}_k) = m^{ij}_k(\bbe^{\zeta_i z_i+\zeta_j z_j}) = \bbe^{(\zeta_i+\zeta_j)z_k}$. \rule{\linewidth}{1pt} {\red\bf A real DoPeGDO Example.} Let $cm^{ij}_k\colon CU_i\otimes CU_j\to CU_k$ be ``classical multiplication'' for $sl_{2+}^\epsilon$, and let $\bbO_i\colon\bbQ\llbracket y_i,b_i,a_i,x_i\rrbracket\to CU_i$ be the PBW ordering map. \[ \xymatrix@C=5mm@R=5mm{ CU_i\otimes CU_j \ar[r]^<>(0.5){cm^{ij}_k} & CU_k \\ \bbQ\llbracket y_i,b_i,a_i,x_i,y_j,b_j,a_j,x_j \rrbracket \ar[u]_<>(0.4){\bbO_{i,j}} & \bbQ\llbracket y_k,b_k,a_k,x_k \rrbracket \ar[u]_<>(0.4){\bbO_k} & } \] {\bf Claim.} Let \begin{multline*} \Lambda = \left(\eta _i+\frac{e^{-\alpha _i-\epsilon \beta _i} \eta _j}{1+\epsilon \eta _j \xi _i}\right) y_k+\left(\beta _i+\beta _j+\frac{\log \left(1+\epsilon \eta _j \xi _i\right)}{\epsilon }\right) b_k+ \\ \left(\alpha _i+\alpha _j+\log \left(1+\epsilon \eta _j \xi _i\right)\right) a_k+\left(\frac{e^{-\alpha _j-\epsilon \beta _j} \xi _i}{1+\epsilon \eta _j \xi _i}+\xi _j\right) x_k \end{multline*} Then $\bbe^{\eta_iy_i+\beta_ib_i+\alpha_ia_i+\xi_ix_i + \eta_jy_j+\beta_jb_j+\alpha_ja_j+\xi_jx_j} \act \bbO_{i,j} \act cm^{ij}_k = \bbe^\Lambda \act \bbO_k$, and hence $\calD(cm^{ij}_k)=\bbe^\Lambda$ and $cm^{ij}_k$ is DoPeGDO. \needspace{3\baselineskip} {\bf Proof.} We compute in a faithful 2D representation $\rho$ of $CU$: \newline\null\hfill\text{(\web{cm})} \newcount\snip \snip=0\loop \advance \snip 1 \par\needspace{0mm}\includegraphics[scale=\cellscale]{../DaNang-1905/Snips/Examples/rho-\the\snip.pdf} \ifnum \snip<6 \repeat (Shame, but this technique fails for $QU$). \rule{\linewidth}{1pt} {\red\bf Claim. In $QU$, $R$ is {\bf DoPeGDO}.} {\bf Proof.} Recall that with $q=\bbe^{\hbar\epsilon}$, \[ R = \sum\hbar^{j+k}y^kb^j\otimes a^jx^k/j![k]_q! = \bbO\left(\bbe^{\hbar b_1a_2}\bbe_q^{\hbar y_1x_2}\right). \] Now expand $\bbe_q^{\hbar y_1x_2}$ in powers of $\epsilon$ using: {\red Faddeev's Formula} (In as much as we can tell, first appeared without proof in Faddeev~\cite{Faddeev:ModularDouble}, rediscovered and proven in Quesne~\cite{Quesne:Jackson}, and again with easier proof, in Zagier~\cite{Zagier:Dilogarithm}). \text{With} ${[n]_q\coloneqq\frac{q^n-1}{q-1}}$, with $[n]_q!\coloneqq[1]_q[2]_q\cdots[n]_q$ and with $\bbe_q^x\coloneqq\sum_{n\geq 0}\frac{x^n}{[n]_q!}$, we have \[ \log\bbe_q^x = \sum_{k\geq 1}\frac{(1-q)^kx^k}{k(1-q^k)} = x + \frac{(1-q)^2x^2}{2(1-q^2)} + \ldots . \] {\bf Proof.} We have that $\bbe_q^x = \frac{\bbe_q^{qx}-\bbe_q^x}{qx-x}$ (``the $q$-derivative of $\bbe_q^x$ is \text{itself}''), and hence $\bbe_q^{qx} = (1+(1-q)x)\bbe_q^x$, and \[ \log\bbe_q^{qx} = \log(1+(1-q)x) + \log\bbe_q^x. \] Writing $\log\bbe_q^x=\sum_{k\geq 1}a_kx^k$ and comparing powers of $x$, we get $q^ka_k=-(1-q)^k/k+a_k$, or $a_k=\frac{(1-q)^k}{k(1-q^k)}$. \qed \rule{\linewidth}{1pt} \par{\bf\red Compositions (2).} Recall that with all indices $i$ running in some set $B$, \[ \calF\act \calG = \left(\calF|_{z_i\to\partial_{\zeta_i}}\calG \right)_{\zeta_i=0} = \left.\bbe^{\sum\partial_{z_i}\partial_{\zeta_i}}(\calF\calG)\right|_{z_i=\zeta_i=0}, \] so in general we wish to understand \[ [F\colon\calE]_B\coloneqq \bbe^{\frac12\sum_{i,j\in B} F_{ij}\partial_{z_i}\partial_{z_j}}\calE \quad\text{and}\quad \langle F\colon\calE\rangle_B\coloneqq \left.[F\colon\calE]_B\right|_{z_B\to 0}, \] where $\calE$ is a docile perturbed Gaussian. The following lemma allows us to restrict to the case where $\calE$ has no $B$-$B$ quadratic part: {\bf Lemma 1.} With convergences left to the reader, \[ \left\langle F\colon\calE\,\bbe^{\frac12\sum_{i,j\in B}G_{ij}z_iz_j}\right\rangle_B = \det(1-GF)^{-1/2}\left\langle F(1-GF)^{-1} \colon \calE \right\rangle_B. \] The next lemma dispatches the case where $\calE$ has a $B$-linear part: {\bf Lemma 2.} $\displaystyle \left\langle F\colon\calE\,\bbe^{\sum_{i\in B}y_iz_i}\right\rangle_B = \bbe^{\frac12\sum_{i,j\in B}F_{ij}y_iy_j} \left\langle F\colon \left.\calE\right|_{z_B\to z_B+Fy_B}\right\rangle_B$. \needspace{4\baselineskip} Finally, we deal with the docile perturbation case: {\bf Lemma 3.} With an extra variable $\lambda$, $Z_\lambda\coloneqq\log[\lambda F\colon \bbe^P]_B$ satisfies and is determined by the following PDE / IVP: \[ Z_0=P \quad\text{and}\quad \partial_\lambda Z_\lambda = \frac12\sum_{i,j\in B}F_{ij}\left( \partial_{z_i}\partial_{z_j}Z_\lambda + (\partial_{z_i}Z_\lambda)(\partial_{z_j}Z_\lambda) \right). \] \resizebox{\linewidth}{!}{\input{Lemmas.pdftex_t}} \rule{\linewidth}{1pt} {\bf Warning.} Some implementation details match earlier versions of the theory. \null\hfill\web{engine} \vskip -9mm \newcount\snip \snip=0\loop \advance \snip 1 \par\needspace{0mm}\includegraphics[scale=\cellscale]{../../Projects/SL2Portfolio2/Snips/Engine-Speedy/Program-\the\snip.pdf} \ifnum \snip<33 \repeat \rule{\linewidth}{1pt} \null\hfill\web{objects} \vskip -6mm \newcount\snip \snip=0\loop \advance \snip 1 \par\needspace{0mm}\includegraphics[scale=\cellscale]{../../Projects/SL2Portfolio2/Snips/Objects/Program-\the\snip.pdf} \ifnum \snip<21 \repeat \rule{\linewidth}{1pt} \par{\red\bf A Quantum Algebra Example.}\hfill\text{\web{qa}} {\bf Proto-Proposition}$^{\dagger 0}$ (with Jesse Frohlich and Roland van der Veen, near~\cite[Proposition~1.7.3]{Majid:Foundations}). Let $H$ be a finite dimensional Hopf algebra and let $U=H^{\ast\text{\it cop}}\otimes H$ be its Drinfel'd double, with $R$-matrix $R\in H^\ast\otimes H\subset U\otimes U$. Write $R^{\dagger 1}=\sum\rho_a\otimes r_a$, and let $\langle\cdot\mid\cdot\rangle\colon H^\ast\otimes H\to\bbF$ be the duality pairing. Then the functional $\int\in U^\ast$ defined by \[ \int\phi\otimes x\coloneqq\sum\langle\phi\rho_a^{\dagger 2}\mid xr_a^{\dagger 3}\rangle \] is a right$^{\dagger 4}$ integral in $U^\ast$. (Meaning $\Delta^i_{jk}\act\int_j = \int_i\act\epsilon_k$ in $\Hom(U^{\otimes\{i\}}\to U^{\otimes\{k\}})$). {\footnotesize $\dagger 0$~A ``proto-proposition'' is something that will become a proposition once you figure out the correct statement. $\dagger 1$~Or did we want it to be $R\act S_1^2$? Or $R\act S_2^2$? $\dagger 2$~Or is it $\rho_a\phi$? $\dagger 3$~Or is it $r_ax$? $\dagger 4$~Or maybe ``left''? } \newcount\snip \snip=0\loop \advance \snip 1 \par\needspace{0mm}\includegraphics[scale=\cellscale]{../DaNang-1905/Snips/Examples/Integrals-\the\snip.pdf} \ifnum \snip<2 \repeat \rule{\linewidth}{1pt} \par{\red\bf A Knot Theory Example.}\hfill\text{\web{kt}} \newcount\snip \snip=0\loop \advance \snip 1 \par\needspace{0mm}\includegraphics[scale=\cellscale]{../DaNang-1905/Snips/Examples/Trefoil-\the\snip.pdf} \ifnum \snip<3 \repeat \rule{\linewidth}{1pt} \par{\red\bf References.} %\par\vspace{-3mm} \renewcommand{\section}[2]{}% \begin{thebibliography}{} \setlength{\parskip}{0pt} \setlength{\itemsep}{0pt plus 0.3ex} \bibitem[BG]{Bar-NatanGaroufalidis:MMR} D.~Bar-Natan and S.~Garoufalidis, {\em On the Melvin-Morton-Rozansky conjecture,} Invent.\ Math.\ {\bf 125} (1996) 103--133. \bibitem[BV]{PP1} D.~Bar-Natan and R.~van~der~Veen, {\em A Polynomial Time Knot Polynomial,} \arXiv{1708.04853}. \bibitem[Fa]{Faddeev:ModularDouble} L.~Faddeev, {\em Modular Double of a Quantum Group,} \arXiv{math/9912078}. %\bibitem[GR]{GaroufalidisRozansky:LoopExpansion} S.~Garoufalidis and L.~Rozansky, % {\em The Loop Exapnsion of the Kontsevich Integral, the Null-Move, and $S$-Equivalence,} % \arXiv{math.GT/0003187}. \bibitem[Ma]{Majid:Foundations} S.~Majid, {\em Foundations of Quantum Group Theory,} Cambridge University Press, 1995. \bibitem[MM]{MM} P.~M.~Melvin and H.~R.~Morton, {\em The coloured Jones function,} Commun.\ Math.\ Phys.\ {\bf 169} (1995) 501--520. \bibitem[Ov]{Overbay:Thesis} A.~Overbay, {\em Perturbative Expansion of the Colored Jones Polynomial,} University of North Carolina PhD thesis, \web{Ov}. \bibitem[Qu]{Quesne:Jackson} C.~Quesne, {\em Jackson's $q$-Exponential as the Exponential of a Series,} \arXiv{math-ph/0305003}. \bibitem[Ro1]{Ro} L.~Rozansky, {\em A contribution of the trivial flat connection to the Jones polynomial and Witten's invariant of 3d manifolds, I,} Comm.\ Math.\ Phys.\ {\bf 175-2} (1996) 275--296, \arXiv{hep-th/9401061}. \bibitem[Ro2]{Rozansky:Burau} L.~Rozansky, {\em The Universal $R$-Matrix, Burau Representation and the Melvin-Morton Expansion of the Colored Jones Polynomial,} Adv.\ Math.\ {\bf 134-1} (1998) 1--31, \arXiv{q-alg/9604005}. \bibitem[Ro3]{Rozansky:U1RCC} L.~Rozansky, {\em A Universal $U(1)$-RCC Invariant of Links and Rationality Conjecture,} \arXiv{math/0201139}. \bibitem[Za]{Zagier:Dilogarithm} D.~Zagier, {\em The Dilogarithm Function,} in Cartier, Moussa, Julia, and Vanhove (eds) {\em Frontiers in Number Theory, Physics, and Geometry II.} Springer, Berlin, Heidelberg, and \web{Za}. \end{thebibliography} \end{multicols} \rule{\linewidth}{1pt} \begin{multicols}{2} {\bf\red KiW 43 Abstract} (\web{kiw}). Whether or not you like the formulas on this page, they describe the strongest truly computable knot invariant we know.%\hfill\text{\footnotesize(experimental analysis @\web{kiw})} {\bf\red Observations.} $\bullet$~Separates the Rolfsen table; does better than Khovanov plus HOMFLY-PT on knots with up to 12 crossings (not tested beyond). $\bullet$~The degrees are bounded by the genus! $\bullet$~$\rho_1$ vanishes for amphichiral knots. $\bullet$~Has a chance of detecting non-ribbonness (\web{ind})! \end{multicols} %\vskip -7mm \def\N{\ding{56}} \def\gY{\textcolor{ForestGreen}{\ding{52}}} \def\oY{\textcolor{Orange}{\ding{52}}} \newlength{\outerboxwidth} \newlength{\innerboxwidth} \def\headcell{\parbox[t]{\outerboxwidth}{ \hspace{-4pt}\parbox[b]{0.4in}{knot diag} \parbox[b]{\innerboxwidth}{ {\purple $n_k^t$}\quad Alexander's $\omega^+$\hfill genus / \textcolor{ForestGreen}{ribbon} \\ {\red $(\rho'_1)^+$} \hfill unknotting \# / \textcolor{Orange}{amphi?} }\\ \centering{\tiny\blue $(\rho'_2)^+$} \vskip 1pt }} \def\rolcell#1#2#3#4#5#6#7#8#9{\parbox[t]{\outerboxwidth}{ \hspace{-4pt}\parbox[b]{0.4in}{\raisebox{-3pt}{\includegraphics[height=21pt]{../UNC-1610/KnotFigs/#1.pdf}}} \parbox[b]{\innerboxwidth}{ {\purple $#2$}\quad $#3$\hfill $#6$ / #8 \\ {\red $#4$} \hfill $#7$ / #9 }\\ \centering{\tiny\blue $#5$} \vskip 1pt }} \setlength{\outerboxwidth}{2.5in} \setlength{\innerboxwidth}{2.1in} {\footnotesize \begin{longtable}{|c|c|c|} \hline \headcell & \headcell & \headcell \\ \endhead \hline \input ../DaNang-1905/table1.tex \end{longtable}} \setlength{\outerboxwidth}{3.83in} \setlength{\innerboxwidth}{3.43in} {\footnotesize \begin{longtable}{|c|c|c|} \hline \headcell & \headcell \\ \endhead \hline \input ../DaNang-1905/table2.tex \end{longtable}} \end{document} \endinput