\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, ../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=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{Toronto-1912} \def\title{Geography vs.\ Identity} \def\navigator{{ \href{\myurl}{Dror Bar-Natan}: \href{\myurl/Talks}{Talks}: \href{\myurl/Talks/\thistalk/}{\thistalk}: }} \def\thanks{{Thanks for inviting me to the {\em Topology} session!}} \def\webdef{{{\greektext web}$\coloneqq$\href{http://drorbn.net/to19}{http://drorbn.net/to19/}}} \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\PvB{{\mathit P\!v\!B}} \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\vB{{\mathit v\!B}} \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\Abstract{{\raisebox{2mm}{\parbox[t]{3.95in}{ {\red\bf Abstract.} Which is better, an emphasis on where things happen or on who are the participants? I can't tell; there are advantages and disadvantages either way. Yet much of quantum topology seems to be heavily and unfairly biased in favour of geography. }}}} \def\geog{{so {\red $x$} is $\gamma_2$.}} \def\iden{{At {\red $x$} strand 1 crosses strand 3, so {\red $x$} is $\sigma_{13}$.}} \def\Geographers{{\raisebox{2mm}{\parbox[t]{3.95in}{ \parshape 5 0in 2.7in 0in 2.7in 0in 2.7in 0in 2.7in 0in 3.95in {\red\bf Geographers} care for placement; for them, braids and tangles have ends at some distinguished points, hence they form categories whose objects are the placements of these points. For them, the basic operation is a binary ``stacking of tangles''. They are lead to monoidal categories, braided monoidal categories, representation theory, and much or most of we call ``quantum topology''. }}}} \def\Identiters{{\raisebox{2mm}{\parbox[t]{3.95in}{ \parshape 5 0in 2.1in 0in 2.1in 0in 2.1in 0in 2.1in 0in 3.95in {\red\bf Identiters} believe that strand identity persists even if one crosses or is being crossed. The key operation is a unary stitching operation $m^{ab}_c$, and one is lead to study meta-monoids, meta-Hopf-algebras, etc. See \web{reg}, \web{kbh}. }}}} \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 (\web{mac}). 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 \[ \begin{CD} \begin{array}{c|ccc} \omega & a & b & S \\ \hline a & \alpha & \beta & \theta \\ b & \gamma & \delta & \epsilon \\ S & \phi & \psi & \Xi \end{array} @>{\displaystyle m^{ab}_c}>{\displaystyle T_a,T_b\to T_c}> \left(\!\begin{array}{c|cc} (1-\beta)\omega & c & S \\ \hline c & \gamma+\frac{\alpha\delta}{1-\beta} & \epsilon+\frac{\delta\theta}{1-\beta} \\ S & \phi+\frac{\alpha\psi}{1-\beta} & \Xi+\frac{\psi\theta}{1-\beta} \end{array}\!\right) \end{CD} \] \end{minipage}}}} \def\Braids{{\raisebox{0mm}{\parbox[t]{3.95in}{ \def\neg{\hspace{-1.5mm}} Geography:\hfill\text{(better topology!)} \[ GB\coloneqq\langle\gamma_i\rangle\left/\left(\neg\begin{array}{c} \gamma_i\gamma_k=\gamma_k\gamma_i \text{ when }|i-k|>1 \\ \gamma_i\gamma_{i+1}\gamma_i=\gamma_{i+1}\gamma_i\gamma_{i+1} \end{array}\neg\right)\right. = B. \] Identity:\hfill\text{(captures quantum algebra!)} \[ IB\coloneqq\langle\sigma_{ij}\rangle\left/\left(\neg\begin{array}{c} \sigma_{ij}\sigma_{kl}=\sigma_{kl}\sigma_{ij} \text{ when }|\{i,j,k,l\}|=4 \\ \sigma_{ij}\sigma_{ik}\sigma_{jk}=\sigma_{jk}\sigma_{ik}\sigma_{ij} \text{ when }|\{i,j,k\}|=3 \end{array}\neg\right)\right. = \PvB. \] {\red Theorem.} Let $S=\{\tau\}$ be the symmetric group. Then $\vB$ is both \[ \PvB\rtimes S \cong B\ast S\left/\left( \gamma_i\tau=\tau\gamma_j \text{ when }\tau i=j,\,\tau(i+1)=(j+1) \right)\right. \] (and so $\PvB$ is ``bigger'' then $B$, and hence quantum algebra doesn't see topology very well). {\red Proof.} Going left, $\gamma_i\mapsto\sigma_{i,i+1}(i\ i+1)$. Going right, if $ij$ use $\sigma_{ij}\mapsto(j\ j\!+\!1\ \ldots\ i)\gamma_j(i\ i\!-\!1\ \ldots\ j\!+\!1)$. }}}} \def\dts{{$\!\cdots$}} \def\Burau{{\raisebox{2mm}{\parbox[t]{3.95in}{ \parshape 4 0in 3.35in 0in 3.35in 0in 3.35in 0in 3.95in {\red\bf The Burau Representation} of $\PvB_n$ acts on $R^n\coloneqq\bbZ[t^{\pm 1}]^n = R\langle v_1,\ldots,v_n\rangle$ by \[ \sigma_{ij}v_k=v_k+\delta_{kj}(t-1)(v_j-v_i). \] \newcount\snip \snip=0\loop \advance \snip 1 \par\needspace{0mm}\includegraphics[scale=\cellscale]{Snips/GvIExamples/P-\the\snip.pdf} \ifnum \snip<1 \repeat \newcount\snip \snip=0\loop \advance \snip 1 \par\needspace{0mm}\includegraphics[scale=\cellscale]{Snips/GvIExamples/B-\the\snip.pdf} \ifnum \snip<7 \repeat \parshape 1 0in 2.95in $S_n$ acts on $R^n$ by permuting the $v_i$ so the Burau representation extends to $\vB_n$ and restricts to $B_n$. With this, $\gamma_i$ maps $v_i\mapsto v_{i+1}$, $v_{i+1}\mapsto tv_i\!+\!(1\!-\!t)v_{i+1}$, and otherwise $v_k\mapsto v_k$. }}}} \def\Gassner{{\raisebox{2mm}{\parbox[t]{3.95in}{ \parshape 4 0in 3.35in 0in 3.35in 0in 3.35in 0in 3.95in {\red\bf The Gassner Representation} of $\PvB_n$ acts on $V=R^n\coloneqq\bbZ[t_1^{\pm 1},\ldots,t_n^{\pm 1}]^n = R\langle v_1,\ldots,v_n\rangle$ by \[ \sigma_{ij}v_k=v_k+\delta_{kj}(t_i-1)(v_j-v_i). \] \newcount\snip \snip=0\loop \advance \snip 1 \par\needspace{0mm}\includegraphics[scale=\cellscale]{Snips/GvIExamples/G-\the\snip.pdf} \ifnum \snip<3 \repeat $S_n$ acts on $R^n$ by permuting the $v_i$ {\em and} the $t_i$, so the Gassner representation extends to $\vB_n$ and then restricts to $B_n$ as a $\bbZ$-linear $\infty$-dimensional representation. It then descends to ${\mathit P\!B}_n$ as a finite-rank $R$-linear representation, with lengthy non-local formulas. {\red Geographers:} Gassner is an obscure partial extension of Burau. {\red Identiters:} Burau is a trivial silly reduction of Gassner. }}}} \def\TurboGassner{{\raisebox{2mm}{\parbox[t]{3.95in}{ \parshape 4 0in 3.35in 0in 3.35in 0in 3.35in 0in 3.95in {\red\bf The Turbo-Gassner Representation.} With the same $R$ and $V$, $TG$ acts on $V\oplus(R^n\otimes V)\oplus (\calS^2V\otimes V^\ast) = R\langle v_k, v_{lk}, u_iu_jw_k \rangle$ by \newcount\snip \snip=0\loop \advance \snip 1 \par\needspace{0mm}\includegraphics[scale=\cellscale]{Snips/GvIExamples/TG-\the\snip.pdf} \ifnum \snip<3 \repeat \vskip -5.5mm \parshape 1 0.75in 2.95in Like Gassner, $TG$ is also a representation of ${\mathit P\!B}_n$. {\red I have no idea where it belongs!} }}}} \def\cellscale{0.645} \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{GvI1.pdftex_t}}\vfill\null\end{center} \end{document} \endinput