\documentclass[11pt,notitlepage]{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} \usepackage{tensor} \usepackage{txfonts} % for the likes of \coloneqq. \usepackage[usenames,dvipsnames]{xcolor} \usepackage[textwidth=8.5in,textheight=11in,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{LesDiablerets-1708} \def\title{The Dogma is Wrong} \def\navigator{{ \href{\myurl}{Dror Bar-Natan}: \href{\myurl/Talks}{Talks}: \href{\myurl/Talks/\thistalk/}{\thistalk}: }} \def\thanks{{Thanks for the invitation!}} \def\webdef{{{\greektext web}$\coloneqq$\href{http://drorbn.net/ld17}{http://drorbn.net/ld17/}}} \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\mgray{\color{mgray}} \def\morange{\color{morange}} \def\red{\color{red}} \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\CW{\text{\it CW}} \def\diag{\operatorname{diag}} \def\FL{\text{\it FL}} \def\IHX{\mathit{IHX}} \def\PvT{{\mathit P\!v\!T}} \def\remove{\!\setminus\!} \def\STU{\mathit{STU}} \def\TC{\mathit{TC}} \def\tr{\operatorname{tr}} \def\vT{{\mathit v\!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\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}} % 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{{\raisebox{0mm}{\parbox[t]{2.2in}{ Follows Rozansky \cite{Ro, Rozansky:Burau, Rozansky:U1RCC} and \text{Overbay} \cite{Overbay:Thesis}, joint with van der Veen. More at \web{talks}. }}}} \def\Abstract{{\raisebox{2mm}{\parbox[t]{3.95in}{ {\red Abstract.} It has long been known that there are knot invariants associated to semi-simple Lie algebras, and there has long been a dogma as for how to extract them: ``quantize and use {\blue representation theory}''. We present an alternative and better procedure: ``centrally extend, {\red approximate by solvable}, and learn how to {\red re-order exponentials} in a universal enveloping algebra''. While equivalent to the old invariants via a complicated process, our invariants are in practice stronger, faster to compute (poly-time vs. exp-time), and clearly carry topological information. }}}} \def\GWUAbstract{{\raisebox{2mm}{\parbox[t]{3.95in}{ {\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. }}}} \def\ExperimentalA{{\raisebox{3mm}{\parbox[t]{3.95in}{ {\red Experimental Analysis} (\web{Exp}). Log-log plots of computation time (sec) vs.\ crossing number, for all knots with up to 12 crossings (mean times) and for all torus knots with up to 48 crossings: }}}} \def\To12Times{{\includegraphics[width=2in]{../UNC-1610/To12Times.pdf}}} \def\TKTimes{{\includegraphics[width=2in]{../UNC-1610/TKTimes.pdf}}} \def\ExperimentalB{{\raisebox{2mm}{\parbox[t]{3.95in}{ {\red Power.} On the 250 knots with at most 10 crossings, the pair $(\omega,\rho_1)$ attains 250 distinct values, while (Khovanov, HOMFLYPT) attains only 249 distinct values. To 11 crossings the numbers are (802, 788, 772) and to 12 they are (2978, 2883, 2786). {\red Genus.} Up to 12 xings, always $\rho_1$ is symmetric under $t\leftrightarrow t^{-1}$. With $\rho_1^+$ denoting the positive-degree part of $\rho_1$, always $\deg \rho_1^+\leq 2g-1$, where $g$ is the 3-genus of $K$ (equality for 2530 knots). This gives a lower bound on $g$ in terms of $\rho_1$ (conjectural, but undoubtedly true). This bound is often weaker than the Alexander bound, yet for 10 of the 12-xing Alexander failures it does give the right answer. }}}} \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 $\calR\coloneqq\kappa(\tau^{-1}(1))$} \def\Vo{{\raisebox{0mm}{\parbox[t]{0.74in}{\footnotesize\raggedright \cite{Vo:Thesis}: Works for Alexander! }}}} \def\GSTInvariants{{\raisebox{0mm}{\parbox[t]{3.95in}{\footnotesize \hfill$A^+ = -t^8+2 t^7-t^6-2 t^4+5 t^3-2 t^2-7 t+13$ $\rho_1^+ = 5t^{15}-18t^{14}+33t^{13}-32t^{12}+2t^{11}+42t^{10}-62t^9-8t^8+166t^7-242t^6+$ \newline\null\hfill$108 t^5+132 t^4-226 t^3+148 t^2-11 t-36$ }}}} \def\LoyalOpposition{{\raisebox{2.5mm}{\parbox[t]{3.95in}{ \parshape 2 0in 3.95in 0in 1.375in {\red The Loyal Opposition.} For certain algebras, work in a homomorphic poly-dimensional ``space of formulas''. }}}} \def\mijk{$m^{ij}_k$} \def\FS{$\{\calF_S\}$} \def\AS{$\{A^{\otimes S}\}$} \def\Moduli{{\raisebox{2mm}{\parbox[t]{1.95in}{ {\red 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\WhySolv{{\raisebox{2mm}{\parbox[t]{3.95in}{ {\red Why are ``solvable algebras'' any good?} Contrary to common beliefs, computations in semi-simple Lie algebras are just awful: \vskip 10mm Yet in solvable algebras, exponentiation is fine and even BCH, $z=\log(\bbe^x\bbe^y)$, is bearable: }}}} \def\mmr{{\raisebox{2mm}{\parbox[t]{3.95in}{ \parshape 3 0in 2.45in 0in 2.45in 0in 3.95in {\red 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}: \[ J_d(K)(q) = \frac{q^d-q^{-d}}{(q-q^{-1})\omega(K)(q^d)} \left(1+ \sum_{k=1}^\infty \frac{(q-1)^k\rho_k(K)(q^d)}{\omega^{2k}(K)(q^d)} \right). \] }}}} \def\MMG{\parbox{0.6in}{\scriptsize\raggedright Melvin, Morton, Garoufalidis }} \def\technique{{\raisebox{2mm}{\parbox[t]{2.75in}{ {\red The Yang-Baxter Technique.} Given an algebra $A$ (typically $\hat\calU(\frakg)$ or $\hat\calU_q(\frakg)$) and elements \[ R=\sum a_i\otimes b_i\in A\otimes A \quad\text{and}\quad C\in A, \] form \[ Z=\sum_{i,j,k}Ca_ib_ja_kC^2b_ia_jb_kC. \] {\red Problem.} Extract information from $Z$. {\red The Dogma.} Use representation theory. In principle finite, but {\em slow}. }}}} \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 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$[h_i,f_{jk}] \!=\! \epsilon(\delta_{ij}-\delta_{ik})f_{jk}$ }}}} \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{Dogma1.pdftex_t}\vfill\null \end{center} \eject \newgeometry{textwidth=8in,textheight=10.5in} \begin{multicols}{2} \raggedcolumns {\red The $sl_2$ Example.} Let $\frakg^\epsilon = \langle h,e,l,f\rangle / ([h,\cdot]=0,\,[e,l]=-e,\,[f,l]=f,\, [e,f]=h-2\epsilon l)$ and let $\frakg_k = \frakg^\epsilon/(\epsilon^{k+1}=0)$. {\red The Main $\frakg_k$ Theorem.} The $\frakg_k$-invariant of any $S$-component tangle $T$ can be written in the form \[ Z(T)=\bbO\left(\omega\bbe^{L+Q+P}\colon\bigotimes_{i\in S}e_il_if_i\right), \] where $\omega$ is a scalar (meaning, a rational function in the variables $h_i$ and their exponentials $t_i\coloneqq\bbe^{h_i}$), where $L=\sum a_{ij}h_il_j$ is a balanced quadratic in the variables $h_i$ and $l_j$ with integer coefficients, where $Q=\sum b_{ij}e_if_j$ is a balanced quadratic in the variables $e_i$ and $f_j$ with scalar coefficients $b_{ij}$, and where $P$ is a polynomial in $\{\epsilon,e_i,l_i,f_i\}$ (with scalar coefficients) whose $\epsilon^d$-term is of degree at most $2d+2$ in $\{e_i,\sqrt{l_i},f_i\}$. Furthermore, after setting $h_i=h$ and $t_i=t$ for all $i$, the invariant $Z(T)$ is poly-time computable. {\red The Main $\frakg_k$ Lemma.} The following ``re-ordering relations'' hold: \[ \bbO\left(\bbe^{\gamma l+\beta e}\colon le\right) = \bbO\left(\bbe^{\gamma l+\bbe^\gamma \beta e}\colon el\right) \quad\text{(and similarly for $fl\to lf$)}, \] \[ \bbO\left(\bbe^{\beta e+\alpha f+\delta ef}\colon fe\right) = \bbO\left(\nu\bbe^{ \nu(-\alpha\beta h+\beta e+\alpha f+\delta ef) + \lambda_k(\epsilon,e,l,f,\alpha,\beta,\delta) }\colon elf\right), \] with $\nu=(1+h\delta)^{-1}$ and where $\lambda_k(\epsilon,e,l,f,\alpha,\beta,\delta)$ is some fixed polynomial of degree at most $2k+2$ in $\epsilon,e,\sqrt{l},f,\alpha,\beta,\delta$, with scalar coefficients. \def\cellscale{0.65} {\large\red Demo Programs.}\hfill\web{Demo} \begin{overpic}[scale=\cellscale]{../Toulouse-1705/Snips/CF.pdf} \put(91,11.5){\red Formatting} \end{overpic} \hfill{\red Preparation} \vskip -12pt\includegraphics[scale=\cellscale]{../Toulouse-1705/Snips/Preparation.pdf} %\begin{overpic}[scale=\cellscale]{../Toulouse-1705/Snips/Prep31.pdf} %\put(91,11.5){\red Preparing the Trefoil} %\end{overpic} \hfill{\red Preparing the Trefoil} \newline\vskip -8mm\null\hfill\includegraphics[width=0.7in]{../Toulouse-1705/31.pdf} \newline\vskip -35mm\includegraphics[scale=\cellscale]{../Toulouse-1705/Snips/Prep31.pdf} \hfill{\red Differential Polynomials} \newline\null\hfill\text{\footnotesize (Implementing $P(\partial_\alpha,\partial_\beta)(f)$)} \vskip -26pt\includegraphics[scale=\cellscale]{../Toulouse-1705/Snips/DP.pdf} \hfill{\red $\mathit{le}$ and $\mathit{fl}$ Sorts} \vskip -12pt\includegraphics[scale=\cellscale]{../Toulouse-1705/Snips/Slx.pdf} \vskip 2mm \begin{overpic}[scale=\cellscale]{../Toulouse-1705/Snips/Logos.pdf} \put(91,11.5){\red The {\greektext L'ogos}} \end{overpic} %\hfill{\red The {\greektext L'ogos}} %\newline\includegraphics[scale=\cellscale]{../Toulouse-1705/Snips/Logos.pdf} \hfill{\red $\mathit{fe}$ Sorts} \vskip -12pt\includegraphics[scale=\cellscale]{../Toulouse-1705/Snips/Sfe.pdf} \hfill{\red Elf Merges} \vskip -12pt\includegraphics[scale=\cellscale]{../Toulouse-1705/Snips/m.pdf} \hfill{\red Rewriting the Trefoil} \newline\null\hfill\text{\footnotesize (by merging 16 elves)} \vskip -26pt\includegraphics[scale=\cellscale]{../Toulouse-1705/Snips/E31.pdf} \hfill{\red Readout} \newline\includegraphics[scale=\cellscale]{../Toulouse-1705/Snips/rho1.pdf} \hfill{\red $\rho_1(3_1)$} \vskip -12pt\includegraphics[scale=\cellscale]{../Toulouse-1705/Snips/rho131.pdf} {\red References.}{\footnotesize %\par\vspace{-3mm} \renewcommand{\section}[2]{}% \begin{thebibliography}{} \setlength{\parskip}{0pt} \setlength{\itemsep}{0pt plus 0.3ex} \bibitem[BN]{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[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[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, University of Toronto Ph.D.\ thesis, in preparation. \end{thebibliography}} \end{multicols} \begin{multicols}{2} \begin{center} \includegraphics[height=0.7in]{../Indiana-1611/Spring2.png} \includegraphics[height=0.7in]{../Indiana-1611/Spring2.png} \end{center} \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]{3.24in}{ {\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