\documentclass[11pt,notitlepage,a4paper]{article} \def\bare{n} \usepackage[all]{xy} \usepackage[english,greek]{babel} \usepackage{dbnsymb, amsmath, graphicx, amssymb, datetime, multicol, stmaryrd, pifont, amscd, colortbl, mathtools, wasysym, needspace, import, longtable, overpic, enumitem, bbm, pdfpages, ../picins} \usepackage{tensor} \usepackage{txfonts} % For \coloneqq; but harms \calA. %\usepackage{mathtools} % For \coloneqq. %\usepackage{mathbbol} % For \bbe; sometimes harmed by later packages. \usepackage[usenames,dvipsnames]{xcolor} \usepackage[textwidth=210mm,textheight=297mm,centering]{geometry} \parindent 0in % 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{Matemale-1804} \def\title{Solvable Approximations of the Quantum $sl_2$ Portfolio} \def\navigator{{ \href{\myurl}{Dror Bar-Natan}: \href{\myurl/Talks}{Talks}: \href{\myurl/Talks/\thistalk/}{\thistalk}: }} \def\webdef{{{\greektext web}$\coloneqq$\href{http://drorbn.net/mm18}{http://drorbn.net/mm18/}}} \def\web#1{{\href{\myurl/Talks/\thistalk/#1}{{\greektext web}/#1}}} \def\titleA{{\bf\title}} \def\titleB{{\title}} \def\titleC{{\title}} % Following http://tex.stackexchange.com/questions/23521/tabular-vertical-alignment-to-top: \def\imagetop#1{\vtop{\null\hbox{#1}}} \definecolor{mgray}{HTML}{B0B0B0} \definecolor{morange}{HTML}{FFA50A} \def\blue{\color{blue}} \def\gray{\color{gray}} \def\mgray{\color{mgray}} \def\morange{\color{morange}} \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\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\ced{{\linebreak[1]\null\hfill\text{$\bigcirc$}}} \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\qed{{\linebreak[1]\null\hfill\text{$\Box$}}} \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\barT{{\bar T}} \def\bbe{\mathbbm{e}} \def\bbD{{\mathbb D}} \def\bbE{{\mathbb E}} \def\bbO{{\mathbb O}} \def\bbQ{{\mathbb Q}} \def\bbR{{\mathbb R}} \def\bbT{{\mathbb T}} \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\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{\tensor[^t]{\Delta}{}} \def\tf{\tensor[^t]{f}{}} \def\tF{\tensor[^t]{F}{}} \def\tg{\tensor[^t]{g}{}} \def\tI{\tensor[^t]{I}{}} \def\tm{\tensor[^t]{m}{}} \def\tR{\tensor[^t]{R}{}} \def\tsigma{\tensor[^t]{\sigma}{}} \def\tS{\tensor[^t]{S}{}} \def\tSW{\tensor[^t]{\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\credits{With \href{http://www.rolandvdv.nl/}{Roland van der Veen}} \def\Main{{\raisebox{2mm}{\parbox[t]{3.96in}{ {\red\bf Our Main Theorem} (loosely stated). Everything that matters in the quantum $sl_2$ portfolio can be continuously expressed in terms of docile perturbed Gaussians using solvable approximations.~\ced {\red\bf Our Main Points.} \begin{itemize}[leftmargin=*,labelindent=0pt,itemsep=-2pt,topsep=0pt] \item What's the ``quantum $sl_2$ portfolio''? \item What in it ``matters'' and why?\hfill\text{\blue\small(the most important question)} \item What's ``solvable approximation''? What's ``continuously''? \item What are ``docile perturbed Gaussians''? \item Why do they matter?\hfill\text{\blue\small($2^{nd}$ most important)} \item How proven?\hfill\text{\blue\small(docile)} \item How implemented?\hfill\text{\blue\small(sacred; the work of unsung heroes)} \item Some context and background. \item What's next? \end{itemize} }}}} \def\Portfolio{{\raisebox{4mm}{\parbox[t]{3.96in}{ \parpic[r]{$\xymatrix@C=12mm{ R,s\in \{QU^{\otimes S}\} \ar@`{p+(12,12),p+(-12,12)}_{\otimes,m^{ij}_k,\Delta^i_{jk},S_i,\theta} \ar[r]^<>(0.5){A\bbD,\,S\bbD} & \{CU^{\otimes S}\} \ar@`{p+(12,12),p+(-12,12)}_{\otimes,m^{ij}_k,\Delta^i_{jk},S_i,\theta} }$} {\red\bf The quantum $sl_2$ Portfolio} includes a classical universal enveloping algebra $CU$, its quantization $QU$, their tensor powers $CU^{\otimes S}$ and $QU^{\otimes S}$ with the ``tensor operations'' $\otimes$, their products $m^{ij}_k$, coproducts $\Delta^i_{jk}$ and antipodes $S_i$, their Cartan automophisms $C\theta\colon CU\to CU$ and $Q\theta\colon QU\to QU$, the ``dequantizators'' $A\bbD\colon QU\to CU$ and $S\bbD\colon QU\to CU$, and most importantly, the $R$-matrix $R$ and the Drinfel'd element $s$. All this in any PBW basis, and change of basis maps are included. }}}} \def\SolvableApproximation{{\raisebox{2mm}{\parbox[t]{3.96in}{ {\red\bf Solvable Approximation.} A quantized universal enveloping algebra (aka ``quantum group'') is an $\infty$-dimensional inverse limit. %\[ \xymatrix{ % \text{untrimmed} \ar[r] & % \text{half-trimmed} \ar[r] & % \text{fully-trimmed} \\ % \text{unmanageable} & % \text{``solvable approximation''} & % \text{``finite-type''} %} \] }}}} \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\Recomposing{{\raisebox{3mm}{\parbox[t]{3.95in}{ {\red\bf Recomposing $gl_n$.} Half is enough! $gl_n\oplus\fraka_n = \calD(\uppertriang,b,\delta)$: \vskip 14mm Now define $gl^\epsilon_n\coloneqq\calD(\uppertriang,b,\epsilon\delta)$. Schematically, this is $[\uppertriang,\uppertriang]=\uppertriang$, $[\lowertriang,\lowertriang]=\epsilon\lowertriang$, and $[\uppertriang,\lowertriang]=\lowertriang+\epsilon\uppertriang$. In detail, it is }}}} \def\glne{{\raisebox{0mm}{\parbox[t]{3.0625in}{ $[e_{ij},e_{kl}]\!=\!\delta_{jk}e_{il}-\delta_{li}e_{kj}$ \hfill$[f_{ij},f_{kl}]\!=\!\epsilon\delta_{jk}f_{il}-\epsilon\delta_{li}f_{kj}$ \newline $[e_{ij},f_{kl}] \!=\! \delta_{jk}(\epsilon\delta_{jl}f_{il})$ \newline\null\hfill $-\delta_{li}(\epsilon\delta_{kj}f_{kj})$ \newline$[g_i,e_{jk}] \!=\! (\delta_{ij}-\delta_{ik})e_{jk}$ \hfill$[h_i,e_{jk}] \!=\! \epsilon(\delta_{ij}-\delta_{ik})e_{jk}$ \newline$[g_i,f_{jk}] \!=\! (\delta_{ij}-\delta_{ik})f_{jk}$ \hfill$[h_i,f_{jk}] \!=\! \epsilon(\delta_{ij}-\delta_{ik})f_{jk}$ }}}} \def\SolvApp{{\raisebox{2mm}{\parbox[t]{3.95in}{ {\red\bf Solvable Approximation (2).} At $\epsilon=1$ and modulo $h=g$, the above is just $gl_n$. By rescaling at $\epsilon\neq 0$, $gl_n^\epsilon$ is independent of $\epsilon$. We let $gl_n^k$ be $gl_n^\epsilon$ regarded as an algebra over $\bbQ[\epsilon]/\epsilon^{k+1}=0$. It is the ``$k$-smidgen solvable approximation'' of $gl_n$! Recall that $\frakg$ is ``solvable'' if iterated commutators in it ultimately vanish: $\frakg_2\coloneqq[\frakg,\frakg],\ \frakg_3\coloneqq[\frakg_2,\frakg_2],\ \ldots,\ \frakg_d=0$. Equivalently, if it is a subalgebra of some large-size $\uppertriang$ algebra. {\red Note.} This whole process makes sense for arbitrary semi-simple Lie algebras. }}}} \def\Docile{{\raisebox{2mm}{\parbox[t]{3.95in}{ \parshape 5 0in 2.25in 0in 2.25in 0in 2.25in 0in 2.25in 0in 3.95in {\red\bf Definition.} A ``docile perturbed Gaussian'' in the variables $(z_i)_{i\in S}$ \text{over} the ring $R$ is an expression of the form \[ \bbe^{q^{ij}z_iz_j}P = \bbe^{q^{ij}z_iz_j}\left(\sum_{k\geq 0}\epsilon^kP_k\right), \] where all coefficients are in $R$ and where $P$ is a ``docile series'': $\deg P_k\leq 4k$. {\red Docililty Matters!} The rank of the space of docile series to $\epsilon^k$ is polynomial in the number of variables $|S|$. }}}} \def\Faddeev{{\raisebox{2mm}{\parbox[t]{3.96in}{ \parshape 5 0in 2.5in 0in 2.5in 0in 2.5in 0in 2.5in 0in 3.96in {\red\bf Faddeev's Formula} (In as much as \text{we} can tell, first appeared w/o proof in Faddeev~\cite{Faddeev:ModularDouble}, rediscovered and proven in Quesne~\cite{Quesne:Jackson}, and again with easier proof, in Zagier~\cite{Zagier:Dilogarithm}). 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 \[ \log\bbe_q^x = \sum_{k\geq 1}\frac{(1-q)^kx^k}{k(1-q^k)} = x + \frac{(1-q)^2x^2}{2(1-q^2)} + \ldots . \] {\red Proof.} We have that $\bbe_q^x = \frac{\bbe_q^{qx}-\bbe_q^x}{qx-x}$ (``the $q$-derivative of $\bbe_q^x$ is 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 }}}} \def\MetaAssoc{{\parbox{1.1in}{ (meta-associativity: $m^{ab}_x\act m^{xc}_y=m^{bc}_x\act m^{ax}_y$) \newline (tangles are generated by $\overcrossing$ and $\undercrossing$) }}} \def\Genus{{\raisebox{2mm}{\parbox[t]{2.96in}{ {\red\bf Genus.} Every knot is the boundary of an orientable ``Seifert Surface'' (\web{SS}), and the least of their genera is the ``genus'' of the knot. {\red Claim.} The knots of genus $\leq 2$ are precisely the images of 4-component tangles via }}}} \def\Ribbon{{\raisebox{2mm}{\parbox[t]{3.96in}{ {\red\bf A Bit about Ribbon Knots.} A ``ribbon knot'' is a knot that can be presented as the boundary of a disk that has ``ribbon singularities'', but no ``clasp singularities''. A ``slice knot'' is a knot in $S^3=\partial B^4$ which is the boundary of a non-singular disk in $B^4$. Every ribbon knots is clearly slice, yet, {\red Conjecture.} Some slice knots are not ribbon. {\red Fox-Milnor.} The Alexander polynomial of a ribbon knot is always of the form $A(t)=f(t)f(1/t)$.\hfill{(also~for~slice)} }}}} \def\GST{\parbox{0.5in}{\tiny Gompf, Scharlemann, Thompson \cite{GompfScharlemannThompson:Counterexample} }} \def\Ta{$\calT_{2n}$} \def\Tb{$U\in\calT_n$} \def\Tc{ribbon $K\in\calT_1$} \def\Aa{$\calA_{2n}$} \def\Ab{$1\in\calA_n$} \def\Ac{$z(K)\in\calR\subseteq\calA_1$} \def\Ra{with $\calR\coloneqq$} \def\Rb{$\kappa(\tau^{-1}(1))$} \def\Gold{{\raisebox{5mm}{\begin{minipage}[t]{3.95in} \pichskip{0mm}\parpic[l]{ \includegraphics[height=0.4in]{../Greece-1607/IASLogo.png} } \picskip{2} {\red\bf The Gold Standard} is set by the ``$\Gamma$-calculus'' Alexander formulas \cite{Bar-NatanSelmani:MetaMonoids, KBH}. An $S$-component tangle $T$ has $\Gamma(T) \in R_S\times M_{S\times S}(R_S) = \left\{\begin{array}{c|c}\omega&S\\\hline S&A\end{array}\right\}$ with $R_S\coloneqq\bbZ(\{t_a\colon a\in S\})$: $\displaystyle \left(\tensor[_a]{\overcrossing}{_b},\tensor[_b]{\undercrossing}{_a}\right) \to \begin{array}{c|cc} 1 & a & b \\ \hline a & 1 & 1-t_a^{\pm 1} \\ b & 0 & t_a^{\pm 1} \end{array} $ \hfill$\displaystyle T_1\sqcup T_2 \to \begin{array}{c|cc} \omega_1\omega_2 & S_1 & S_2 \\ \hline S_1 & A_1 & 0 \\ S_2 & 0 & A_2 \end{array} $ \vskip -1mm \[ \begin{CD} \begin{array}{c|ccc} \omega & a & b & S \\ \hline a & \alpha & \beta & \theta \\ b & \gamma & \delta & \epsilon \\ S & \phi & \psi & \Xi \end{array} @>{\displaystyle m^{ab}_c}>{\displaystyle t_a,t_b\to t_c}> \left(\!\begin{array}{c|cc} (1-\beta)\omega & c & S \\ \hline c & \gamma+\frac{\alpha\delta}{1-\beta} & \epsilon+\frac{\delta\theta}{1-\beta} \\ S & \phi+\frac{\alpha\psi}{1-\beta} & \Xi+\frac{\psi\theta}{1-\beta} \end{array}\!\right) \end{CD} \] {\footnotesize (Roland: ``add to $A$ the product of column $b$ and row $a$, divide by $(1-A_{ab})$, delete column $b$ and row $a$''.)} \vskip 1mm \parshape 1 0in 3.45in For long knots, $\omega$ is Alexander, and that's the fastest \text{Alexander} algorithm I know! \hfill\text{\footnotesize Dunfield: 1000-crossing fast.} \vskip 1.5mm \resizebox{\linewidth}{!}{$\displaystyle \begin{CD} \begin{array}{c|cc} \omega & a & S \\ \hline a & \alpha & \theta \\ S & \phi & \Xi \end{array} @>q\Delta^a_{bc}>{\mu\coloneqq T_a-1\atop{\nu\coloneqq \alpha-\sigma_a\atop T_a\mapsto T_bT_c}}> \left(\begin{array}{c|ccc} \omega & b & c & S \\ \hline b & (\sigma_a-\alpha T_a-\nu T_c)/\mu & (T_b-1)T_c\nu/\mu & (T_b-1)T_c\theta/\mu \\ c & (T_c-1)\nu/\mu & (\alpha-\sigma_a T_a-\nu T_c)/\mu & (T_c-1)\theta/\mu \\ S & \phi & \phi & \Xi \end{array}\right) \\ @VdS^aVT_a\to T_a^{-1}V \\ \hspace{0mm}\left(\begin{array}{c|cc} \alpha\omega/\sigma_a & a & S \\ \hline a & 1/\alpha & \theta/\alpha \\ S & -\phi/\alpha & (\alpha\Xi-\phi\theta)/\alpha \end{array}\right)\hspace{-33mm} @. \raisebox{-2mm}{\hspace{16mm}\parbox{81mm}{ Where $\sigma$ assigns to every $a\in S$ a Laurent monomial $\sigma_a$ in $\{t_b\}_{b\in S}$ subject to $\sigma\left( \tensor[_a]{\overcrossing}{_b},\tensor[_b]{\undercrossing}{_a} \right) = (a\to1,\,b\to t_a^{\pm 1})$, $\sigma(T_1\sqcup T_2)=\sigma(T_1)\sqcup\sigma(T_2)$, and $\sigma\act m^{ab}_c = \left. (\sigma\setminus\{a,b\})\cup(c\to\sigma_a\sigma_b) \right|_{t_a,t_b\to t_c}$. }} \end{CD} $} \end{minipage}}}} \def\Vo{{\raisebox{2mm}{\parbox[t]{3.5in}{ {\red\bf Vo's Thesis \cite{Vo:Thesis}.} A proof of the Fox-Milnor theorem for ribbon knots using this technology (and more). }}}} \def\Implementation{{\raisebox{2.5mm}{\parbox[t]{2in}{ \parshape 2 0in 2in 0.1in 1.5in {\red\bf Implementation} key idea: \newline $\left(\omega,A=(\alpha_{ab})\right)\leftrightarrow$ \newline $\left(\omega,\lambda=\sum\alpha_{ab}t_ah_b\right)$ }}}} \def\technique{{\raisebox{2mm}{\parbox[t]{2.75in}{ {\red\bf The Yang-Baxter Technique.} Given an algebra $U$ (typically $\hat\calU(\frakg)$ or $\hat\calU_q(\frakg)$) and elements \[ R=\sum a_i\otimes b_i\in U\otimes U \quad\text{and}\quad C\in U, \] form \hfill $\displaystyle Z=\sum_{i,j,k}Ca_ib_ja_kC^2b_ia_jb_kC.$\hfill\null {\red Problem.} Extract information from $Z$. {\red The Dogma.} Use representation theory. In principle finite, but {\em slow}. }}}} \def\Moduli{{\raisebox{2mm}{\parbox[t]{1.95in}{ {\red\bf The (fake) moduli} of Lie algebras on $V$, a quadratic variety in $(V^\ast)^{\otimes 2}\otimes V$ is on the right. We care about $sl_{17}^k \coloneqq sl_{17}^\epsilon/(\epsilon^{k+1}=0)$. }}}} \def\mmr{{\raisebox{2mm}{\parbox[t]{3.95in}{ \parshape 3 0in 2.45in 0in 2.45in 0in 3.95in {\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 \[ \left. \frac{(q^{1/2}-q^{-1/2})J_d(K)}{q^{d/2}-q^{-d/2}} \right|_{q=e^\hbar} = \sum_{j,m\geq 0} a_{jm}(K)d^j\hbar^m, \] \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}: \[ J_d(K)(q) = \frac{q^d-q^{-d}}{(q-q^{-1})\omega(K)(q^d)} \left(1+ \sum_{k=1}^\infty \frac{(q-1)^k\rho_k(K)(q^d)}{\omega^{2k}(K)(q^d)} \right). \] \parshape 1 0in 2.5in {\red Prior art.} Some amazing computations by Rozansky and Overbay in \cite{Rozansky:Burau, Rozansky:U1RCC} and in \cite{Overbay:Thesis}. }}}} \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\scalebox{0.98}{\input{Page1.pdftex_t}}\vfill\null\end{center} \eject \newgeometry{textwidth=7.84in,textheight=11.025in} \begin{center}\scalebox{0.98}{\input{Page2.pdftex_t}}\end{center} \vspace{-7mm} \begin{multicols}{2} \raggedcolumns {\red\bf $\calG\calD\calO$-Categories.} Given $\frakg$ with basis $B=\{x,y,\ldots\}$, consider the following diagram: \[ \xymatrix@C=11.5mm{ \bbQ=\hat\calU_{(q)}\left(\bigoplus_0\frakg\right) \ar[r]^<>(0.5)Z & \hat\calU_{(q)}\left(\frakg\right) \ar@/^/[r]^<>(0.5)\Delta & \hat\calU_{(q)}\left(\bigoplus_2\frakg\right) \ar@/^/[l]^<>(0.5)m \\ \hat\calS\left(\emptyset\right) \ar[r]^<>(0.5)Z \ar[u] & \hat\calS\left(B\right) \ar@/^/[r]^<>(0.5)\Delta \ar@/^/[u]^{\bbO(xy\ldots\colon\cdot)} \ar@/_/[u]_{\bbO(yx\ldots\colon\cdot)} \ar@`{p+(-16,-16),p+(+16,-16)}^{\SW_{xy}} & \hat\calS\left(B_1,B_2\right) \ar@/^/[l]^<>(0.5)m \ar[u]_{\bbO(y_1x_1\ldots\otimes y_2x_2\ldots\colon\cdot)} } \] Hence $Z$, $\SW_{xy}$, $m$, $\Delta$, (and likewise $S$ and $\theta$) are morphisms in the {\em completion} of the monoidal category $\calF$ whose objects are finite sets $B$ and whose morphisms are $\mor_\calF(B,B') \coloneqq \Hom_\bbQ\left(\calS(B)\to\calS(B')\right) = \calS\left(B^\ast,B'\right)$ (by convention, $x^\ast=\xi$, $y^\ast=\eta$, etc.). Ergo we need to {\em consolidate} (at least parts of) said completion. {\red\bf Aside.} ``Consolidate'' means ``give a finite name to an infinite object, and figure out how to sufficiently manipulate such finite names''. E.g., solving $f''=-f$ we encounter and set $\sum\frac{(-1)^kx^{2k}}{(2k)!}\leadsto\cos x$, $\sum\frac{(-1)^kx^{2k+1}}{(2k+1)!}\leadsto\sin x$, and then $\cos^2x+\sin^2x=1$ and $\sin(x+y)=\sin x\cos y+\cos x\sin y$. {\red\bf The Composition Law.} If \[ \begin{CD} \calS(B_0) @>f>\tf\in\bbQ\llbracket\zeta_{0i},z_{1j}\rrbracket> \calS(B_1) @>g>\tg\in\bbQ\llbracket\zeta_{1j},z_{2k}\rrbracket> \calS(B_2) \end{CD} \] then $\tensor[^t]{(f\act g)}{} = \tensor[^t]{(g\circ f)}{} = \left(\left.g\right|_{\zeta_{1j}\to\partial_{z_{1j}}}f\right)_{z_{1j}=0}$. {\red\bf Examples.} \begin{enumerate}[leftmargin=*,labelindent=0pt,itemsep=-2pt,topsep=0pt] \item The $1$-variable identity map $I\colon\calS(z)\to\calS(z)$ is given by $\tI_1 = \yellowm{\bbe^{z\zeta}}$ and the $n$-variable one by $\tI_n = \yellowm{\bbe^{z_1\zeta_1+\dots+z_n\zeta_n}}$. \needspace{9mm} \item The ``$z_i\to z_j$ variable rename map $\sigma^i_j\colon\calS(z_i)\to\calS(z_j)$ becomes $\tsigma^i_j = \yellowm{\bbe^{z_j\zeta_i}}$, and it's easy to rename several variables simultaneously. \item The ``archetypal multiplication map $m^{ij}_k\colon\calS(z_i,z_j)\to\calS(z_k)$'' has $\tm = \yellowm{\bbe^{z_k(\zeta_i+\zeta_j)}}$. \item The ``archetypal coproduct $\Delta^i_{jk}\colon\calS(z_i)\to\calS(z_j,z_k)$'', given by $z_i\to z_j+z_k$ or $\Delta z = z\otimes 1+1\otimes z$, has $\tDelta = \yellowm{\bbe^{(z_j+z_k)\zeta_i}}$. \item $R$-matrices tend to have terms of the form $\bbe_q^{\hbar y_1x_2} \in \calU_q\otimes\calU_q$. The ``baby $R$-matrix'' is $\tR = \yellowm{\bbe^{\hbar yx}\in\calS(y,x)}$. \end{enumerate} {\red\bf Proposition.} If $F\colon\calS(B)\to\calS(B')$ is linear and ``continuous'', then $\tF=\exp\left(\sum_{z_i\in B}\zeta_iz_i\right)\act F$. {\red\bf The Heisenberg Example.} The ``Weyl form of the canonical commutation relations'' states that if $[y,x]=t$ and $t$ is central, then $\bbe^{\xi x}\bbe^{\eta y} = \bbe^{\eta y}\bbe^{\xi x}\bbe^{-\eta\xi t}$. Thus with \[ \xymatrix{ \calS(t,y,x) \ar@/^/[r]^{\bbO_{xy}} \ar@/_/[r]_{\bbO_{yx}} \ar@`{p+(-16,+16),p+(-16,-16)}_{\SW_{xy}} & \calU(t,y,x) } \] we have $\tSW_{xy} = \yellowm{\bbe^{\tau t+\eta y+\xi x-\eta\xi t}}$. \parpic[r]{\input{figs/Zipping.pdf_t}} {\red\bf The Zipping Issue} (between unbound and bound lies half-zipped). {\red\bf Zipping.} If $P(\zeta^j,z_i)$ is a polynomial, or whenever otherwise convergent, set \[ \left\langle P(\zeta^j,z_i)\right\rangle_{(\zeta^j)} = \left.P\left(\partial_{\!z_j},z_i\right)\right|_{z_i=0}. \] (E.g., if $P=\sum a_{nm}\zeta^nz^m$ then $\langle P\rangle_\zeta = \sum n!a_{nn}$). \needspace{15mm} {\red\bf The Zipping / Contraction Theorem.} If $P$ has a finite $\zeta$-degree and the $y$'s and the $q$'s are ``small'' then \[ \left\langle P(z_i,\zeta^j)\bbe^{\eta^iz_i+y_j\zeta^j}\right\rangle_{(\zeta^j)} = \left\langle P(z_i+y_i,\zeta^j)\bbe^{\eta^i(z_i+y_i)}\right\rangle_{(\zeta^j)}, \] (proof: replace $y_j\to\hbar y_j$ and test at $\hbar=0$ and at $\partial_\hbar$), and \begin{multline*} \left\langle P(z_i,\zeta^j)\bbe^{c+\eta^iz_i+y_j\zeta^j+q^i_jz_i\zeta^j}\right\rangle_{(\zeta^j)} \\ = \det(\tilq)\left\langle P(\tilq_i^k(z_k+y_k),\zeta^j)\bbe^{c+\eta^i\tilq^k_i(z_k+y_k)} \right\rangle_{(\zeta^j)} \end{multline*} where $\tilq$ is the inverse matrix of $1-q$: $(\delta^i_j-q^i_j)\tilq^j_k=\delta^i_k$ (proof: replace $q^i_j\to\hbar q^i_j$ and test at $\hbar=0$ and at $\partial_\hbar$). {\red\bf Implementation.}\hfill\web{ZipBindDemo} \def\cellscale{0.645} \newcount\snip \snip=1\loop \par\includegraphics[scale=\cellscale]{Snips/ZipBindDemo-\the\snip.pdf} \advance \snip 1 \ifnum \snip<23 \repeat {\red\bf The 2D Lie Algebra.} Clever people know$^\ast$ that if $[a,x] = \gamma x$ then $\bbe^{\xi x}\bbe^{\alpha a} = \bbe^{\alpha a}\bbe^{\bbe^{-\gamma\alpha}\xi x}$. Ergo with \[ \xymatrix{ \calS(a,x) \ar@/^/[r]^{\bbO_{ax}} \ar@/_/[r]_{\bbO_{xa}} \ar@`{p+(-16,+16),p+(-16,-16)}_{\SW_{ax}} & \calU(a,x) } \] we have $\tSW_{ax} = \yellowm{\bbe^{\alpha a + \bbe^{-\gamma\alpha}\xi x}}$. {\footnotesize $\ast$ Indeed $xa = (a-\gamma)x$ thus $xa^n = (a-\gamma)^nx$ thus $x\bbe^{\alpha a} = \bbe^{\alpha(a-\gamma)}x = \bbe^{-\gamma\alpha}\bbe^{\alpha a}x$ thus $x^n\bbe^{\alpha a} = \bbe^{\alpha a}(\bbe^{-\gamma\alpha})^nx^n$ thus $\bbe^{\xi x}\bbe^{\alpha a} = \bbe^{\alpha a}\bbe^{\bbe^{-\gamma\alpha}\xi x}$.} {\red\bf The Real Thing.} In $QU/(\epsilon^2=0)$ over $\bbQ\llbracket\hbar\rrbracket$ using the $yax$ order, $T=\bbe^{\hbar t}$, $\barT=T^{-1}$, $\calA=\bbe^{\gamma\alpha}$, and $\bcA=\calA^{-1}$, we have \[ \tR_{ij} = \yellowm{\bbe^{\hbar(y_ix_j-t_ia_j/\gamma)} \left(1+\epsilon\hbar\left( a_ia_j/\gamma-\gamma\hbar^2y_i^2x_j^2/4 \right)\right)} \] in $\calS(B_i,B_j)$, and in $\calS(B^\ast_1,B^\ast_2,B)$ we have \[ \tm \!=\! \yellowm{ \bbe^{(\alpha _1+\alpha _2)a+\eta _2 \xi _1(1-T)/\hbar + (\xi_1\bcA_2 + \xi _2)x + (\eta_1+\eta_2\bcA_1) y} \left(1 \!+\! \epsilon\lambda_m\right) }, \] where {\footnotesize$\lambda_m \!=\! \yellowm{2a\eta_2\xi_1T \!+\! \frac{1}{4}\gamma\eta_2^2\xi_1^2\left(3T^2\!-\!4T\!+\!1\right)/\hbar \!-\! \frac{1}{2}\gamma\eta_2\xi_1^2(3T\!-\!1)x\bcA_2}$ $\yellowm{- \frac{1}{2}\gamma\eta_2^2\xi_1(3 T\!-\!1)y\bcA_1} \yellowm{+ \gamma\eta_2\xi_1xy\hbar\bcA_1\bcA_2}$}. Similar formulas delight us for $\tDelta$ and $\tS$. {\red\bf A generic morphism.} \[ \input{figs/GDOMorphism.pdf_t} \] \end{multicols} \vskip -3mm {\red\bf Implementation.}\hfill\web{SL2Portfolio} \vskip -5mm \def\cellscale{0.45} \begin{tabular}{ccc} \includegraphics[scale=\cellscale]{figs/QZip.pdf} & \includegraphics[scale=\cellscale]{figs/LZip.pdf} & \includegraphics[scale=\cellscale]{figs/Bind.pdf} \end{tabular} \begin{multicols}{2} {\red\bf A Partial To Do List.} \begin{itemize}[leftmargin=*,labelindent=0pt,itemsep=-4pt,topsep=0pt] \item Complete all ``docility'' arguments by identifying a ``contained'' docile substructure. \item Understand denominators and get rid of them. \item See if much can be gained by including $P$ in the exponential: $\bbe^{L+Q}P\leadsto\bbe^{L+Q+P}$? \item Clean the program and make it efficient. \item Run it for all small knots and links, at $k=2,3$. \item Understand the centre and figure out how to read the output. \item Execute the Drinfel'd double procedule at $\bbE$-level (and thus get rid of \verb$DeclareAlgebra$ and all that is around it!). \item Extend to $sl_3$ and beyond. \item Do everything with \verb$Zip$ and \verb$Bind$ as the fundamentals, without ever referring back to (quantized) Lie algebras. \item Prove a genus bound and a Seifert formula. \item Obtain ``Gauss-Gassner formulas'' (\web{NCSU}). \item Relate with Melvin-Morton-Rozansky and with Rozansky-Overbay. \item Understand the braid group representations that arise. \item Find a topological interpretation. The Garoufalidis-Rozansky ``loop expansion'' \cite{GaroufalidisRozansky:LoopExpansion}? \item Figure out the action of the Cartan automorphism. \item Disprove the ribbon-slice conjecture! \item Figure out the action of the Weyl group. \item Do everything at the ``arrow diagram'' level of finite-type invariants of (rotational) virtual tangles. \item What else can you do with the ``solvable approximations''? \item And with the ``Gaussian zip and bind'' technology? \end{itemize} \end{multicols}\vspace{-6mm}\begin{multicols}{2} {\red\bf References.}{\footnotesize %\par\vspace{-3mm} \renewcommand{\section}[2]{}% \begin{thebibliography}{} \setlength{\parskip}{0pt} \setlength{\itemsep}{0pt plus 0.3ex} \bibitem[BN1]{KBH} D.~Bar-Natan, {\em Balloons and Hoops and their Universal Finite Type Invariant, BF Theory, and an Ultimate Alexander Invariant,} \web{KBH}, \arXiv{1308.1721}. \bibitem[BN2]{K17} D.~Bar-Natan, {\em Polynomial Time Knot Polynomial,} research proposal for the 2017 Killam Fellowship, \web{K17}. \bibitem[BNG]{Bar-NatanGaroufalidis:MMR} D.~Bar-Natan and S.~Garoufalidis, {\em On the Melvin-Morton-Rozansky conjecture,} Invent.\ Math.\ {\bf 125} (1996) 103--133. \bibitem[BNS]{Bar-NatanSelmani:MetaMonoids} D.~Bar-Natan and S.~Selmani, {\em Meta-Monoids, Meta-Bicrossed Products, and the Alexander Polynomial,} J.\ of Knot Theory and its Ramifications {\bf 22-10} (2013), \arXiv{1302.5689}. \bibitem[BV]{PP1} D.~Bar-Natan and R.~van der Veen, {\em A Polynomial Time Knot Polynomial,} Proc.\ Amer.\ Math.\ Soc., to appear, \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[GST]{GompfScharlemannThompson:Counterexample} R.~E.~Gompf, M.~Scharlemann, and A.~Thompson, {\em Fibered Knots and Potential Counterexamples to the Property 2R and Slice-Ribbon Conjectures,} Geom.\ and Top.\ {\bf 14} (2010) 2305--2347, \arXiv{1103.1601}. \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[Vo]{Vo:Thesis} H.~Vo, {\em Alexander Invariants of Tangles via Expansions,} University of Toronto Ph.D.\ thesis, in preparation. \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}\vspace{-6mm}\begin{multicols}{2} {\red\bf The Complete Implementation.}\hfill\web{SL2Portfolio} An even fuller implementation is at \web{FullImp}. \def\cellscale{0.645} \newcount\snip \snip=0\loop \advance \snip 1 \par\needspace{20mm}\includegraphics[scale=\cellscale]{Snips/SL2Portfolio-\the\snip.pdf} \ifnum \snip<58 \repeat \end{multicols} \def\N{\ding{56}} \def\gY{\textcolor{ForestGreen}{\ding{52}}} \def\oY{\textcolor{Orange}{\ding{52}}} \def\headcell{ diagram & \parbox{3in}{{\blue $n^t_k$}\quad Alexander's $\omega^+$ \hfill genus / \textcolor{ForestGreen}{ribbon} \newline {\red Today's / Rozansky's $\rho_1^+$} \hfill unknotting number / \textcolor{Orange}{amphicheiral}} } \def\rolcell#1#2#3#4#5#6#7#8{ \raisebox{-3pt}{\includegraphics[height=23pt]{../UNC-1610/KnotFigs/#1.pdf}} & \parbox[b]{3.12in}{ {\blue $#2$}\quad $#3$\hfill $#5$ / #7 \\ {\red $#4$} \hfill $#6$ / #8 }} {\footnotesize \begin{longtable}{|cl|cl|cl|} \hline \headcell & \headcell \\ \endhead \hline \input ../GWU-1612/table.tex \end{longtable}} \end{document} \endinput