\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} %\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{Hefei-1811} \def\title{Computation without Representation} \def\navigator{{ \href{\myurl}{Dror Bar-Natan}: \href{\myurl/Talks}{Talks}: \href{\myurl/Talks/\thistalk/}{\thistalk}: }} \def\thanks{{Thanks for inviting me to \magenta Hefei!}} \def\webdef{{{\greektext web}$\coloneqq$\href{http://drorbn.net/hef18}{http://drorbn.net/hef18/}}} \def\web#1{{\href{\myurl/Talks/\thistalk/#1}{{\greektext web}/#1}}} \def\titleA{{\title}} \def\titleB{{\title}} \def\titleC{{\title}} \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\pink{\color{pink}} \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\barT{{\bar T}} \def\bbe{\mathbbm{e}} \def\bbE{{\mathbb E}} \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\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\credits{{\raisebox{0.75mm}{\parbox[t]{1.8in}{ Follows Rozansky \cite{Ro, Rozansky:Burau, Rozansky:U1RCC} and \text{Overbay} \cite{Overbay:Thesis}, joint with van der Veen. More at \cite{PP1} and at \web{talks}. }}}} \def\Abstract{{\raisebox{2mm}{\parbox[t]{3.95in}{ {\red\bf Abstract.} A major part of ``quantum topology'' is the definition and computation of various knot invariants by carrying out \text{computations} in quantum groups. Traditionally these computations are carried out ``in a representation'', but this is very slow: one has to use tensor powers of these representations, and the dimensions of powers grow exponentially fast. In my talk, I will describe a direct method for carrying out such computations without having to choose a representation and explain why in many ways the results are better and faster. The two key points we use are a technique for composing infinite-order ``perturbed Gaussian'' differential operators, and the little-known fact that every semi-simple Lie algebra can be approximated by solvable Lie algebras, where computations are easier. }}}} \def\KPort{{\raisebox{2mm}{\parbox[t]{2in}{ {\red\bf A Knot Theory Portfolio.} \begin{itemize}[leftmargin=*,labelindent=0pt,itemsep=-2pt,topsep=0pt] \item Has operations $\sqcup$, $m^{ij}_k$, $\Delta^i_{jk}$, $S_i$. \item All tangloids are generated by $R^{\pm 1}$ and $C^{\pm 1}$ (so ``easy'' to produce invariants). \item Makes some knot properties (``genus'', ``ribbon'') become ``definable''. \end{itemize} (more to say, but not now). }}}} \def\metaassoc{{$m^{ik}_i\circ m^{ij}_i = m^{ij}_i\circ m^{jk}_j$}} \def\UPort{{\raisebox{2mm}{\parbox[t]{3.95in}{ {\red\bf A ``Quantum Group'' Portfolio} consists of an vector space $U$ along with maps \hfill\text{\footnotesize(and some axioms\ldots)} \vskip -5mm \[ \xymatrix@C=11.5mm{ \bbQ=\hat{U}^{\otimes\emptyset} \ar[r]^<>(0.5){C_i} & \hat{U}^{\otimes \{i\}} \ar@`{p+(15,15),p+(-15,15)}^{S_i} \ar@/^/[r]^<>(0.5){\Delta^i_{jk}} & \hat{U}^{\otimes\{j,k\}} \ar@/^/[l]^<>(0.5){m^{jk}_i} & \bbQ=\hat{U}^{\otimes\emptyset} \ar[l]_<>(0.5){R_{jk}} \\ \hat\calS\left(\emptyset\right) \ar[r]^<>(0.5){C_i} \ar[u]_<>(0.5){\bbO_{()}} & \hat\calS\left(B_i\right) \ar@`{p+(15,-15),p+(-15,-15)}_{S_i} \ar@/^/[r]^<>(0.5){\Delta^i_{jk}} \ar[u]_<>(0.5){\bbO_{y_ix_i\ldots}} & \hat\calS\left(B_j,B_k\right) \ar@/^/[l]^<>(0.5){m^{jk}_i} \ar[u]_<>(0.5){\bbO_{y_jx_j\ldots\otimes y_kx_k\ldots}} & \hat\calS\left(\emptyset\right) \ar[l]_<>(0.5){R_{jk}} \ar[u]_<>(0.5){\bbO_{()}} } \] }}}} \def\PBW{{\raisebox{2mm}{\parbox[t]{3.95in}{ {\red\bf PBW Bases.} The $U$'s we care about always have ``Poincar\'e-Birkhoff-Witt'' bases; there is some finite set $B=\{y,x,\dots\}$ of ``generators'' and isomorphisms $\bbO_{y,x,\ldots}\colon\hat\calS(B)\to U$ defined by ``ordering monomials'' to some fixed $y,x,\ldots$ order. The quantum group portfolio now becomes a ``symmetric algebra'' portfolio, or a ``power series'' portfolio. }}}} \def\oaoa{$f\in\Hom_\bbQ(S(B)\to S(B'))$} \def\oaob{$S(B)^\ast\otimes S(B')$} \def\oaoc{$S(B^\ast)\otimes S(B')$} \def\oaod{$S(B^\ast\sqcup B')$} \def\oaoe{$\tilde{f}\in\bbQ[\zeta_i,z'_i]$} \def\OpsAreObjects{{\raisebox{2mm}{\parbox[t]{2.5in}{ {\red\bf Operations are Objects.} \par{\red$\star$}\hfill$\displaystyle B^\ast\coloneqq\{z_i^\ast=\zeta_i\colon\,z_i\in B\}$,\hfill\null \[\langle z_i^m,\zeta_i^n\rangle=\delta_{mn}n!, \] \[ \left\langle\prod z_i^{m_i},\prod\zeta_i^{n_1i}\right\rangle = \prod\delta_{m_in_i}n_i!, \] \vskip 1mm in general, for $f\in\calS(z_i)$ and $g\in\calS(\zeta_i)$, \[ \langle f,g\rangle = \left.f(\partial_{\zeta_i})g\right|_{\zeta_i=0} = \left.g(\partial_{z_i})f\right|_{z_i=0}. \] {\red The Composition Law.} If \[ \def\neg{\hspace{-3pt}} \begin{CD} \calS(B) @>f>\neg\tf\in\bbQ\llbracket\zeta_i,z'_j\rrbracket\neg> \calS(B) @>g>\neg\tg\in\bbQ\llbracket\zeta'_j,z''_k\rrbracket\neg> \calS(B) \end{CD} \] \parshape 1 0in 3.95in then $\widetilde{(f\act g)} = \widetilde{(g\circ f)} = \left(\left.\tg\right|_{\zeta'_j\to\partial_{z'_j}}\tf\right)_{z'_j=0} = \left(\left.\tf\right|_{z'_j\to\partial_{\zeta'_j}}\tg\right)_{\zeta'_j=0}$: }}}} \def\DOExamples{{\raisebox{0mm}{\parbox[t]{3.95in}{ \hfill{\red\bf Examples} \vspace{-5mm} \begin{enumerate}[leftmargin=*,labelindent=0pt,itemsep=-2pt,topsep=0pt] \item The $1$-variable identity map $I\colon\calS(z)\to\calS(z)$ is \newline given by $\tI_1 = \yellowm{\bbe^{z\zeta}}$ and the $n$-variable one by $\tI_n = \yellowm{\bbe^{z_1\zeta_1+\dots+z_n\zeta_n}}$: \end{enumerate} \vskip 15mm {\red Proposition.} If $F\colon\calS(B)\to\calS(B')$ is linear, then $\tF=F\left(\exp\left(\sum_{z_i\in B}\zeta_iz_i\right)\right)$ (in the 1-variable case, $=\sum F(z^n)\frac{1}{n!}\zeta^n$). \vskip 2mm \hfill{\red More Examples} \vspace{-5mm} \begin{enumerate}[leftmargin=*,labelindent=0pt,itemsep=-2pt,topsep=0pt] \setcounter{enumi}{1} \item The ``$z_i\to z_j$ variable rename map \newline $\sigma^i_j\colon\calS(z_i)\to\calS(z_j)$'' 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 $\tDelta = \yellowm{\bbe^{(z_j+z_k)\zeta_i}}$. {\gray \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 $\tR = \yellowm{\bbe^{\hbar yx}}\in\calS(y,x)$. \item The ``Weyl form of the canonical commutation relations'' states that if $[y,x]=t$ is a scalar, then $\bbe^{\xi x}\bbe^{\eta y} = \bbe^{\eta y}\bbe^{\xi x}\bbe^{-\eta\xi t}$. Thus with \[ \xymatrix{ \calS(y,x) \ar@/^/[r]^{\bbO_{xy}} \ar@/_/[r]_{\bbO_{yx}} \ar@`{p+(-16,+16),p+(-16,-16)}_{\SW_{xy}} & \calU(y,x) } \] we have $\tSW_{xy} = \yellowm{\bbe^{\eta y+\xi x-\eta\xi t}}$. } \end{enumerate} }}}} \def\Zipping{{\raisebox{2mm}{\parbox[t]{3.95in}{ \parshape 4 0in 1.5in 0in 1.5in 0in 1.5in 0in 3.95in {\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 $\langle P\rangle_\zeta = \sum n!a_{nn}$). {\red\bf Implementation.}\hfill\web{Zip} \par\includegraphics[scale=\cellscale]{../StonyBrook-1805/Snips/ZipDemo-1.pdf} \par\includegraphics[scale=\cellscale]{../StonyBrook-1805/Snips/ZipDemo-2.pdf} \hfill\includegraphics[scale=\cellscale]{../StonyBrook-1805/Snips/ZipDemo-3.pdf} }}}} \def\ZippingThm{{\raisebox{2mm}{\parbox[t]{3.95in}{ {\red\bf The Zipping / Contraction Theorem.} If $P$ has a finite $\zeta$-degree and the $y$'s and the $q$'s are ``small'' then \[ \left\langle P(z_i,\zeta^j)\bbe^{c+\eta^iz_i+y_j\zeta^j+q^i_jz_i\zeta^j}\right\rangle_{(\zeta^j)} \!\!\!\! = \det(\tilq)\left\langle \left.P(z_i,\zeta^j)\bbe^{c+\eta^iz_i}\right|_{z_i\to\tilq_i^k(z_k+y_k)} \right\rangle_{(\zeta^j)} \] where $\tilq$ is the inverse matrix of $1-q$: $(\delta^i_j-q^i_j)\tilq^j_k=\delta^i_k$. \vskip 2mm {\red\bf Exponential Reservoirs..} The true Hilbert hotel is $\exp$! Remove one $x$ from an ``exponential reservoir'' of $x$'s and you are left with the same exponential reservoir: \[ \bbe^x = \left[\ldots+\frac{xxxxx}{120}+\ldots\right] \overset{\partial_x}{\longrightarrow} \left[\ldots+\frac{x\bcancel{x}xxx}{120}+\ldots\right] = (\bbe^x)' = \bbe^x, \] and if you let each element choose left or right, you get twice the same reservoir: \vskip -1mm \parshape 1 0in 1.75in \[ \bbe^x \xrightarrow{x\to x_l+x_r} \bbe^{x_l+x_r} = \bbe^{x_l}\bbe^{x_r}. \] \vskip 2mm \parshape 1 0in 1.75in {\red\bf A Graphical Proof.} Glue top to bottom on the right, in all possible ways. Several scenarios occur: \begin{enumerate}[leftmargin=*,labelindent=4pt,itemsep=-2pt,topsep=0pt] \item Start at $A$, go through the $q$-machine $k\geq 0$ times, stop at $B$. Get $\left\langle P\left(\sum_{k\geq 0}q^kz,\zeta\right)\right\rangle = \left\langle P\left(\tilq z,\zeta\right)\right\rangle$. \item Loop through the $q$-machine and swallow your own tail. Get $\exp\left(\sum q^k/k\right) = \exp(-\log(1-q)) = \tilq$. \item \ldots \end{enumerate} By the reservoir splitting principle, these scenarios contribute multiplicatively. \qed {\red\bf Implementation.}\hfill\web{Zip} \[\includegraphics[scale=\cellscale]{../StonyBrook-1805/Snips/GZip-1.pdf}\] }}}} \def\RealThing{{\raisebox{2mm}{\parbox[t]{3.95in}{ {\red\bf The Real Thing.} In the algebra $QU_\epsilon$ (explained later), over $\bbQ\llbracket\hbar\rrbracket$ using the $yaxt$ order, $T=\bbe^{\hbar t}$, $\barT=T^{-1}$, $\calA=\bbe^{\alpha}$, and $\bcA=\calA^{-1}$, we have \[ \tR_{ij} = \yellowm{\bbe^{\hbar(y_ix_j-t_ia_j)} \left(1+\epsilon\hbar\left( a_ia_j-\hbar^2y_i^2x_j^2/4 \right)+O(\epsilon^2)\right)} \] in $\calS(B_i,B_j)$, and in $\calS(B^\ast_1,B^\ast_2,B)$ we have \[ \tm = \yellowm{ \bbe^{(\alpha _1+\alpha _2)a+\eta _2 \xi _1(1-T)/\hbar + (\xi_1\bcA_2 + \xi _2)x + (\eta_1+\eta_2\bcA_1) y} \left(1 + \epsilon\lambda+O(\epsilon^2)\right) }, \] where {\footnotesize$\lambda = \yellowm{2a\eta_2\xi_1T + \eta_2^2\xi_1^2\left(3T^2-4T+1\right)/4\hbar - \eta_2\xi_1^2(3T-1)x\bcA_2/2}$ \newline\null\hfill$\yellowm{- \eta_2^2\xi_1(3 T-1)y\bcA_1/2} \yellowm{+ \eta_2\xi_1xy\hbar\bcA_1\bcA_2}$}. Finally, \[ \tDelta = \yellowm{\bbe^{\tau(t_1+t_1)+\eta(y_1+T_1y_2)+\alpha(a_1+a_2)+\xi(x_1+x_2)}\left(1 + O(\epsilon)\right)} \in \calS(B^\ast,B_1,B_2), \] and $\displaystyle \tS = \yellowm{\bbe^{-\tau t -\alpha a -\eta \xi (1-\barT)\calA/\hbar - \barT\eta y \calA -\xi x \calA} \left(1+O(\epsilon)\right)} \in\calS(B^\ast,B)$. }}}} \def\RealZipping{{\raisebox{2mm}{\parbox[t]{3.95in}{ {\red\bf Real Zipping} is a minor mess, and is done in two phases: \[ \begin{array}{c|cc|cc} & \multicolumn{2}{c|}{\text{$\tau a$-phase}} & \multicolumn{2}{c}{\text{$\xi y$-phase}} \\ \hline \text{$\zeta$-like variables} & \tau & a & \xi & y \\ \text{$z$-like variables} & t & \alpha & x & \eta \end{array} \] Already at $\epsilon=0$ we get the best known formulas for the Alexander polynomial! }}}} \def\Docility{{\raisebox{2mm}{\parbox[t]{3.95in}{ {\red\bf Generic Docility.} 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 Our Docility.} In the case of $QU_\epsilon$, all invariants and operations are of the form $\bbe^{L+Q}P$, where \begin{itemize}[leftmargin=*,labelindent=0pt,itemsep=-2pt,topsep=0pt] \item $L$ is a quadratic of the form $\sum l_{z\zeta}z\zeta$, where $z$ runs over $\{t_i,\alpha_i\}_{i\in S}$ and $\zeta$ over $\{\tau_i,a_i\}_{i\in S}$, with integer coefficients $l_{z\zeta}$. \item $Q$ is a quadratic for the form $\sum q_{z\zeta}z\zeta$, where $z$ runs over $\{x_i,\eta_i\}_{i\in S}$ and $\zeta$ over $\{\xi_i,y_i\}_{i\in S}$, with coefficients $q_{z\zeta}$ in the ring $R_S$ of rational functions in $\{T_i,\calA_i\}_{i\in S}$. \item $P$ is a docile power series in $\{y_i,a_i,x_i,\eta_i,\xi_i\}_{i\in S}$ with coefficients in $R_S$, and where $\deg(y_i,a_i,x_i,\eta_i,\xi_i)=(1,2,1,1,1)$. \end{itemize} {\red Docililty Matters!} The rank of the space of docile series to $\epsilon^k$ is polynomial in the number of variables $|S|$.\hfill{\red !!!!!} \begin{itemize}[leftmargin=*,labelindent=0pt,itemsep=-2pt,topsep=0pt] \item At $\epsilon^2=0$ we get the Rozansky-Overbay \cite{Ro, Rozansky:Burau, Rozansky:U1RCC, Overbay:Thesis} invariant, which is stronger than HOMFLY-PT polynomial and Khovanov homology taken together! \item In general, get ``higher diagonals in the Melvin-Morton-Rozansky expansion of the coloured Jones polynomial'' \cite{MM, Bar-NatanGaroufalidis:MMR}, but why spoil something good? \end{itemize} }}}} \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\SolvApprox{{\raisebox{2mm}{\parbox[t]{3.95in}{ {\red\bf 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. {\red $CU$ and $QU$.} Starting from $sl_2$, get $CU_\epsilon = \langle y,a,x,t\rangle/([t,-]=0,\, [a,y]=-y,\, [a,x]=x,\, [x,y]=2\epsilon a-t)$. Quantize using standard tools (I'm sorry) and get $QU_\epsilon = \langle y,a,x,t\rangle/([t,-]=0,\, [a,y]=-y,\, [a,x]=x,\, xy-\bbe^{\hbar\epsilon}yx=(1-T\bbe^{-2\hbar\epsilon a})/\hbar)$. }}}} \def\refs{{\raisebox{3mm}{\parbox[t]{3.95in}{ %{\red\bf References.} {\footnotesize %\par\vspace{-3mm} \renewcommand{\section}[2]{}% \begin{thebibliography}{} \setlength{\parskip}{0pt} \setlength{\itemsep}{0pt plus 0.3ex} \bibitem[BNG]{Bar-NatanGaroufalidis:MMR} D.~Bar-Natan and S.~Garoufalidis, {\em On the Melvin-\newline 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[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[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}. \end{thebibliography}} }}}} \def\Sketch{{\raisebox{2mm}{\parbox[t]{3.95in}{\footnotesize {\red Sketch.} (Total 57m) \begin{enumerate}[leftmargin=*,labelindent=4pt,itemsep=-2pt,topsep=0pt] \item (5m) The knot theory portfolio. \item (5m) The (quantum) group portfolio. (Write in general-algebra language, not in universal enveloping algebra language). \item (3m) Quantum groups have ``PBW'' basis. Hence they are symmetric algebras, with funny products / co-products. \item (3m) The DO category. \item (5m) Example: the Abelian case. \item (3m) Example: k=0. \item (5m) Example: sl2. \item (10m) Zipping and composing. The zipping theorem. \item (5m) Docility and polynomiallity. \item (5m) Solvable Approximation. \item (8m) Mention Rozansky-Overbay, mention computations. \end{enumerate} }}}} \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{CWOR1.pdftex_t}\vfill\null \null\vfill\input{CWOR2.pdftex_t}\vfill\null \end{center} \eject \newgeometry{textwidth=8in,textheight=10.5in} {\Large{\red\bf Computation without Representation}}\hfill {\small {\magenta Anhui University, Hefei,} November 2018}\hfill \parbox[b]{1.3in}{\tiny \null\hfill\url{http://drorbn.net/hef18} %\newline\null\hfill modified \today } \vskip -3mm \rule{\textwidth}{1pt} \vspace{-8mm} \begin{multicols}{2} \raggedcolumns {\red\bf The Algebras $H$ and $H^\ast$.} Let ${q=\bbe^{\epsilon}}$ and set $H=\langle a,x\rangle/({[a,x]= x})$ with \[ {A=\bbe^{-\epsilon a}}, \quad xA=qAx, \quad S_{\!H}(a,A,x)=(-a, A^{-1}, -A^{-1}x), \] \[ \Delta_H(a,A,x)=(a_1+a_2, A_1A_2, x_1+A_1x_2) \] and dual $H^\ast=\langle b, y\rangle/({[b,y]=-\epsilon y})$ with \[ {B=\bbe^{- b}}, \quad By=qyB, \quad S_{\!H^\ast}(b,B,y)=(-b, B^{-1}, -yB^{-1}), \] \[ \Delta_{H^\ast}(b,B,y)=(b_1+b_2, B_1B_2, y_1B_2+y_2). \] Pairing by $(a,x)^\ast=(b,y)$ ($\Rightarrow\langle B,A\rangle=q$) making $\langle y^lb^i,a^jx^k\rangle = \delta_{ij}\delta_{kl}j![k]_q!$ so $R=\sum\frac{y^kb^j\otimes a^jx^k}{j![k]_q!}$. {\red\bf The Algebra $QU$.} By the Drinfel'd double procedure, $QU=H^{\ast\text{\it cop}}\otimes H$ with $(\phi f)(\psi g) = \langle \psi_1S^{-1}f_3\rangle \langle \psi_3,f_1\rangle(\phi\psi_2)(f_2g)$ and \[ S(y,b,a,x) = (-B^{-1}y, -b, -a, -A^{-1}x),\] \[ \Delta(y,b,a,x) = (y_1+y_2B_1, b_1+b_2, a_1+a_2, x_1+A_1x_2).\] {\red\bf The 2D Lie Algebra.} One may show$^\ast$ 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 A Full Implementation at $\epsilon^2=0$.} \newcount\snip \snip=0\loop \advance \snip 1 \par\needspace{20mm}\includegraphics[scale=\cellscale]{../StonyBrook-1805/Snips/ybax-\the\snip.pdf} \ifnum \snip<30 \repeat {\red\bf 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 \[ \log\bbe_q^x = \sum_{k\geq 1}\frac{(1-q)^kx^k}{k(1-q^k)} = x + \frac{(1-q)^2x^2}{2(1-q^2)} + \ldots . \] {\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 \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 {\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 Extend to $sl_3$ and beyond. \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} \null\hfill{\red\bf Further References.} {\footnotesize %\par\vspace{-3mm} \renewcommand{\section}[2]{}% \begin{thebibliography}{} \setlength{\parskip}{0pt} \setlength{\itemsep}{0pt plus 0.3ex} \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[Fa]{Faddeev:ModularDouble} L.~Faddeev, {\em Modular Double of a Quantum Group,} \arXiv{math/9912078}. \bibitem[Qu]{Quesne:Jackson} C.~Quesne, {\em Jackson's $q$-Exponential as the Exponential of a Series,} \arXiv{math-ph/0305003}. \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} \end{document} \endinput