\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{Macquarie-261125} \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 Macquarie University!}} \def\webdef{{{\greektext web}$\coloneqq$\href{http://drorbn.net/mac26}{http://drorbn.net/mac26/}}} \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.} When you describe one manifold embedded in another (e.g., when you talk about knots) you can do it from the perspective of the ambient manifold or from the perspective of the embedded manifold. For reason I will explain in my talk, the former is ``geography'' and the latter is ``identity''. After making the distinction, I will say that almost all appearances of category theory within low dimensional topology are within the ``geography'' perspective. But then I will tell a few stories from the ``identity'' perspective, in which some generalized algebraic structures appear, but they are not, generally speaking, categories. So in some sense my talk will be, ``here is category theory; I'm talking about something else''. }}}} \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]{../Toronto-1912/Snips/GvIExamples/P-\the\snip.pdf} \ifnum \snip<1 \repeat \newcount\snip \snip=0\loop \advance \snip 1 \par\needspace{0mm}\includegraphics[scale=\cellscale]{../Toronto-1912/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]{../Toronto-1912/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]{../Toronto-1912/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