\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{fontawesome} % for \faPlay \usepackage[usenames,dvipsnames]{xcolor} \usepackage{utfsym} % for the likes of \car=\usym{1F697} \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{Geneva-2206} \def\title{Cars, Interchanges, Traffic Counters, and a Pretty Darned Good Knot Invariant} \def\navigator{{ \href{\myurl}{Dror Bar-Natan}: \href{\myurl/Talks}{Talks}: \href{\myurl/Talks/\thistalk/}{\thistalk}: }} \def\thanks{{Thanks for inviting me to Geneva!}} \def\webdef{{{\greektext web}$\coloneqq$\href{http://drorbn.net/j22}{http://drorbn.net/j22/}}} \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://arxiv.org/abs/#1}{{\tiny 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\eps{\epsilon} \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\bara{{\bar a}} \def\barb{{\bar b}} \def\barT{{\bar T}} \def\bbE{{\mathbb E}} \def\bbe{\mathbbm{e}} \def\bbH{{\mathbb H}} \def\bbN{{\mathbb N}} \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\calH{{\mathcal H}} \def\calI{{\mathcal I}} \def\calK{{\mathcal K}} \def\calL{{\mathcal L}} \def\calM{{\mathcal M}} \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}} \def\car{\reflectbox{\usym{1F697}}} \def\rac{\usym{1F697}} % 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}{ \parshape 5 0in 2.6in 0in 2.6in 0in 2.6in 0in 2.6in 0in 3.95in {\red\bf Abstract.} Reporting on joint work with Roland van der Veen, I'll tell you some stories about $\rho_1$, an easy to define, strong, fast to compute, homomorphic, and well-connected knot invariant. $\rho_1$ was first studied by Rozansky and Overbay \cite{Ro, Rozansky:Burau, Rozansky:U1RCC, Overbay:Thesis}, it has far-reaching generalizations, it is dominated by the coloured Jones polynomial, and I wish I understood it. \hfill\text{{\red\bf Common misconception.} ``Dominated'' $\nRightarrow$ ``lesser''.} }}}} \def\lr{$\leftrightarrow$} \def\WeSeek{{\raisebox{2mm}{\parbox[t]{3.95in}{ {\red\bf We seek} strong, fast, homomorphic knot and tangle invariants. \par {\bf Strong.} Having a small ``kernel''. \par {\bf Fast.} Computable even for large knots (best: poly time). }}}} \def\Homomorphic{{\raisebox{0mm}{\parbox[t]{1.95in}{ {\bf Homomorphic.} Extends to tangles and behaves under tangle operations; especially gluings and doublings: }}}} \def\WhyHomomorphic{{\raisebox{2mm}{\parbox[t]{3.95in}{ {\red\bf Why care for ``Homomorphic''?} {\bf Theorem.} A knot $K$ is {\em ribbon} iff there exists a $2n$-component tangle $T$ with skeleton as below such that $\tau(T)=K$ and where $\delta(T)=U$ is the {\em untangle}: }}}} \def\Jones{{\raisebox{2mm}{\parbox[t]{2.95in}{ {\red\bf Jones:} \vskip 1mm \par Formulas stay; \par interpretations change with time. }}}} \def\FormulasA{{\raisebox{2mm}{\parbox[t]{3.125in}{ {\red\bf Formulas.} Draw an $n$-crossing knot $K$ as on the right: all crossings face up, and the edges are marked with a running index ${k\in\{1,\ldots,2n+1\}}$ and with rotation numbers $\varphi_k$. Let $A$ be the $(2n+1)\times(2n+1)$ matrix constructed by starting with the identity matrix $I$, and adding a $2\times 2$ block for each crossing: }}}} \def\Arules{{$\displaystyle \begin{array}{c|cccc} A & \text{col }i\!+\!1 & \text{col }j\!+\!1 \\ \hline \text{row }i & -T^s & T^s-1 \\ \text{row }j & 0 & -1 \end{array} $}} \def\FormulasB{{\raisebox{0mm}{\parbox[t]{3.95in}{ Let $G=(g_{\alpha\beta})=A^{-1}$. For the trefoil example, it is: \par $\displaystyle A=\left( \begin{array}{ccccccc} 1 & -T & 0 & 0 & T-1 & 0 & 0 \\ 0 & 1 & -1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & \yellowm{-T} & 0 & 0 & \yellowm{T-1} \\ 0 & 0 & 0 & 1 & -1 & 0 & 0 \\ 0 & 0 & T-1 & 0 & 1 & -T & 0 \\ 0 & 0 & 0 & \yellowm{0} & 0 & 1 & \yellowm{-1} \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 \\ \end{array} \right), $ \par $\displaystyle G=\left( \begin{array}{ccccccc} 1 & T & 1 & T & 1 & T & 1 \\ 0 & 1 & \frac{1}{T^2-T+1} & \frac{T}{T^2-T+1} & \frac{T}{T^2-T+1} & \frac{T^2}{T^2-T+1} & 1 \\ 0 & 0 & \frac{1}{T^2-T+1} & \frac{T}{T^2-T+1} & \frac{T}{T^2-T+1} & \frac{T^2}{T^2-T+1} & 1 \\ 0 & 0 & \frac{1-T}{T^2-T+1} & \frac{1}{T^2-T+1} & \frac{1}{T^2-T+1} & \frac{T}{T^2-T+1} & 1 \\ 0 & 0 & \frac{1-T}{T^2-T+1} & -\frac{(T-1) T}{T^2-T+1} & \frac{1}{T^2-T+1} & \frac{T}{T^2-T+1} & 1 \\ 0 & 0 & 0 & 0 & 0 & 1 & 1 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 \\ \end{array} \right). $ }}}} \def\FormulasC{{\raisebox{2mm}{\parbox[t]{3.95in}{ {\bf\red Note.} The Alexander polynomial $\Delta$ is given by \[ \Delta = T^{(-\varphi-w)/2}\det(A), \qquad \text{with }\varphi = \sum_k \varphi_k,\ w \!=\! \sum_c s. \] {\red\bf Classical Topologists:} This is boring. Yawn. }}}} \def\FormulasD{{\raisebox{2mm}{\parbox[t]{3.95in}{ {\bf\red Formulas, continued.} Finally, set \[ R_1(c) \coloneqq s \left( g_{ji} \left(g_{j+1,j}+g_{j,j+1}-g_{ij}\right) -g_{ii} \left(g_{j,j+1}-1\right) -1/2 \right) \] \[ \rho_1 \coloneqq \Delta^2\left(\sum_c R_1(c) - \sum_k\varphi_k\left(g_{kk}-1/2\right)\right). \] In our example $\rho_1 = -T^2 +2T - 2 + 2T^{-1} - T^{-2}$. \par{\red\bf Theorem.} $\rho_1$ is a knot invariant.\hfill Proof: later. \par{\red\bf Classical Topologists:} Whiskey Tango Foxtrot? }}}} \def\Interpretation{{\raisebox{2mm}{\parbox[t]{4.95in}{ \parshape 5 0in 2.3in 0in 2.3in 0in 2.3in 0in 2.3in 0in 3.95in {\bf\red Cars, Interchanges, and Traffic Counters.} Cars always drive forward. When a car crosses over a bridge it goes through with (algebraic) probability $T^s\sim 1$, but falls off with probability $1-T^s\sim 0^\ast$. See also~\cite{Jones:Hecke, LinTianWang:RandomWalk}. }}}} \def\Foot{{\raisebox{0mm}{\parbox[t]{3.95in}{\footnotesize \ $^\ast$ In algebra $x\sim 0$ if for every $y$ in the ideal generated by $x$, $1-y$ is invertible. }}}} \def\dtA{{\tiny image credits:}} \def\dtB{{\tiny \href{https://diamondtraffic.com/productcategory/Portable-Counters}{diamondtraffic.com}}} \def\gab{{\raisebox{2mm}{\parbox[t]{3.95in}{ \parshape 5 0in 2.875in 0in 2.875in 0in 2.875in 0in 2.875in 0in \linewidth {\bf\red Theorem.} The Green function $g_{\alpha\beta}$ is the reading of a traffic counter at $\beta$, if car traffic is injected at $\alpha$ (if $\alpha=\beta$, the counter is {\em after} the injection point). \par{\bf\red Example.} \vskip 0.625in \parshape 5 0in 3.5in 0in 3.5in 0in 3.5in 0in 3.5in 0in \linewidth {\bf\red Proof.} Near a crossing $c$ with sign $s$, incoming upper edge~$i$ and incoming lower edge $j$, both sides satisfy the {\em $g$-rules}: \[ g_{i\beta} = \delta_{i\beta}+T^sg_{i+1,\beta}+(1-T^s)g_{j+1,\beta}, \quad g_{j\beta} = \delta_{j\beta}+g_{j+1,\beta}, \] and always, $g_{\alpha,2n+1} = 1$: use common sense and $AG=I\ (=GA)$. \par{\bf\red Bonus.} Near $c$, both sides satisfy the further {\em $g$-rules}: \[ g_{\alpha i} = T^{-s}(g_{\alpha,i+1}-\delta_{\alpha,i+1}), \quad g_{\alpha j} = g_{\alpha,j+1} - (1-T^s)g_{\alpha i} - \delta_{\alpha,j+1}. \] }}}} \def\kinkA{{$\sum_{p\geq 0}(1\!-\!T)^p=T^{-1}$}} \def\kinkG{{$G=\begin{pmatrix}1&T^{-1}&1\\0&T^{-1}&1\\0&0&1\end{pmatrix}$}} \def\InvarianceA{{\raisebox{2mm}{\parbox[t]{3.95in}{ {\bf\red Invariance of $\rho_1$.} We start with the hardest, Reidemeister 3: }}}} \def\InvarianceB{{\raisebox{0mm}{\parbox[t]{3.95in}{ $\Rightarrow$ Overall traffic patterns are unaffected by Reid3! \newline $\Rightarrow$ Green's $g_{\alpha\beta}$ is unchanged by Reid3, provided the cars injection site $\alpha$ and the traffic counters $\beta$ are away. \parshape 6 0in 2.5in 0in 2.5in 0in 2.5in 0in 2.5in 0in 2.5in 0in \linewidth $\Rightarrow$ Only the contribution from the $R_1$ terms within the Reid3 move matters, and using $g$-rules the relevant $g_{\alpha\beta}$'s can be pushed outside of the Reid3 area: \input{Invariance.tex} }}}} \def\messA{{$(1\!-\!T)^2\!+\!T(1\!-\!T)$}} \def\messB{{$(1\!-\!T)T$}} \def\messC{{$T(1\!-\!T)$}} \def\messD{{$1\!-\!T$}} \def\TopologyHat{{\raisebox{2mm}{\parbox[t]{3.2in}{ {\bf\red Wearing my Topology hat} the formula for $R_1$, and even the idea to look for $R_1$, remain a complete mystery to me. }}}} \def\QuantumAlgebraHat{{\raisebox{2mm}{\parbox[t]{3.95in}{ \parshape 4 0in 3.45in 0in 3.45in 0in 3.45in 0in \linewidth {\bf\red Wearing my Quantum Algebra hat,} I spy a Heisenberg algebra $\bbH=A\langle p,x\rangle/([p,x]=1)$: \[ \text{cars}\leftrightarrow p \qquad \text{traffic counters}\leftrightarrow x \] {\bf\red Where did it come from?} Consider $\frakg_\eps\coloneqq sl_{2+}^\eps\coloneqq L\langle y,b,a,x\rangle$ with relations \[ [b,x]=\eps x, \quad [b,y]=-\eps y, \quad [b,a]=0, \] \[ [a,x]=x, \quad [a,y]=-y, \quad [x,y]=b+\eps a. \] At invertible $\eps$, it is isomorphic to $sl_2$ plus a central factor, and it can be quantized \`a la Drinfel'd~\cite{Drinfeld:QuantumGroups} much like $sl_2$ to get an algebra $QU=A\langle y,b,a,x\rangle$ subject to (with $q=\bbe^{\hbar\eps})$: \[ [b,a]=0, \quad [b,x]=\eps x,\quad [b,y]=-\eps y, \] \[ [a,x]=x, \quad [a,y]=-y, \quad xy-qyx=\frac{1-\bbe^{-\hbar(b+\eps a)}}{\hbar}. \] Now $QU$ has an $R$-matrix solving Yang-Baxter (meaning Reid3), \[ R=\sum_{m,n\geq 0}\frac{y^nb^m\otimes (\hbar a)^m(\hbar x)^n}{m![n]_q!}, \quad([n]_q!\text{ is a ``quantum factorial''}) \] and so it has an associated ``universal quantum invariant'' \`a la \text{Lawrence} and Ohtsuki~\cite{Lawrence:UniversalUsingQG, Ohtsuki:QuantumInvariants}, $Z_\eps(K)\in QU$. Now $QU\cong\calU(\frakg_\eps)$ (only as algebras!) and $\calU(\frakg_\eps)$ represents into $\bbH$ via \[ y\to -tp-\eps\cdot xp^2,\quad b\to t+\eps\cdot xp,\quad a\to xp,\quad x\to x, \] (abstractly, $\frakg_\eps$ acts on its Verma module \[ \calU(\frakg_\eps)/(\calU(\frakg_\eps)\langle y,a,b-\eps a-t\rangle)\cong\bbQ[x] \] by differential operators, namely via $\bbH$), so $R$ can be pushed to $\calR\in\bbH\otimes\bbH$. Everything still makes sense at $\eps=0$ and can be expanded near $\eps=0$ resulting with $\calR=\calR_0(1+\eps \calR_1+\cdots)$, with $\calR_0=\bbe^{t(xp\otimes 1-x\otimes p)}$ and $\calR_1$ a quartic polynomial in $p$ and $x$. So $p$'s and $x$'s get created along $K$ and need to be pushed around to a standard location (``normal ordering''). This is done using \begin{align*} (p\otimes 1)\calR_0 &= \calR_0(T(p\otimes 1)+(1-T)(1\otimes p)), \\ (1\otimes p)\calR_0 &= \calR_0(1\otimes p), \end{align*} and when the dust settles, we get our formulas for $\rho_1$. But $QU$ is a quasi-triangular Hopf algebra, and hence {\red\bf $\rho_1$ is homomorphic}. Read more at~\cite{PP1, PG} and hear more at \web{SolvApp}, \web{Dogma}, \web{DoPeGDO}, \web{FDA}, \web{AQDW}. \parshape 1 0in 3.25in Also, we can (and know how to) look at higher powers of $\eps$ and we can (and more or less know how to) replace $sl_2$ by arbitrary semi-simple Lie algebra (e.g.,~\cite{Schaveling:Thesis}). So {\red\bf $\rho_1$ is not alone!} These constructions are very similar to Rozansky-Overbay \cite{Ro, Rozansky:Burau, Rozansky:U1RCC, Overbay:Thesis} and hence to the ``loop expansion'' of the Kontsevich integral and the coloured Jones polynomial. }}}} \def\Insanity{{\raisebox{0mm}{\parbox[t]{3.95in}{ If this all reads like {\bf\red insanity} to you, it should (and you haven't seen half of it). Simple things should have simple explanations. Hence, {\bf\red Homework.} Explain $\rho_1$ with no reference to quantum voodoo and find it a topology home (large enough to house generalizations!). Make explicit the homomorphic properties of $\rho_1$. Use them to do topology! }}}} \def\refs{{\raisebox{5mm}{\parbox[t]{3.95in}{ %{\red\bf References.} {\scriptsize %\par\vspace{-3mm} \renewcommand{\section}[2]{}% \begin{thebibliography}{} \setlength{\parskip}{0pt} \setlength{\itemsep}{0pt plus 0.3ex} \bibitem[BV1]{PP1} D.~Bar-Natan and R.~van~der~Veen, {\em A Polynomial Time Knot\newline Polynomial,} Proc.\ Amer.\ Math.\ Soc.\ {\bf 147} (2019) 377--397, \arXiv{1708.04853}. \bibitem[BV2]{PG} D.~Bar-Natan and R.~van~der~Veen, {\em Perturbed Gaussian Generating Functions for Universal Knot Invariants,} \arXiv{2109.02057}. \bibitem[Dr]{Drinfeld:QuantumGroups} V.~G.~Drinfel'd, {\em Quantum Groups,} Proc.\ Int.\ Cong.\ Math., 798--820, Berkeley, 1986. \bibitem[Jo]{Jones:Hecke} V.~F.~R.~Jones, {\em Hecke Algebra Representations of Braid Groups and Link Polynomials,} Annals Math., {\bf 126} (1987) 335-388. \bibitem[La]{Lawrence:UniversalUsingQG} R.~J.~Lawrence, {\em Universal Link Invariants using Quantum Groups,} Proc\. XVII Int.\ Conf.\ on Diff.\ Geom.\ Methods in Theor.\ Phys., Chester, England, August 1988. World Scientific (1989) 55--63. \bibitem[LTW]{LinTianWang:RandomWalk} X-S.~Lin, F.~Tian, and Z.~Wang, {\em Burau Representation and Random Walk on String Links,} Pac.\ J.\ Math., {\bf 182-2} (1998) 289--302, \arXiv{q-alg/9605023}. \bibitem[Oh]{Ohtsuki:QuantumInvariants} T.~Ohtsuki, {\em Quantum Invariants,} Series on Knots and Everything {\bf 29}, World Scientific 2002. \bibitem[Ov]{Overbay:Thesis} A.~Overbay, {\em Perturbative Expansion of the Colored Jones Polynomial,} Ph.D.\ thesis, University of North Carolina, August 2013, \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[Sch]{Schaveling:Thesis} S.~Schaveling, {\em Expansions of Quantum Group Invariants,} Ph.D.\ thesis, Universiteit Leiden, September 2020, \web{Scha}. \end{thebibliography}} }}}} \def\nbpdfInput#1{\vskip 1mm\par\noindent\includegraphics{#1}} \def\nbpdfEcho#1{\vskip 1mm\par\noindent\includegraphics{#1}} \def\nbpdfPrint#1{\vskip 1mm\par\noindent\includegraphics{#1}} \def\nbpdfText#1{\vskip 1mm\par\noindent\includegraphics{#1}} \def\nbpdfMessage#1{} \def\nbpdfOutput#1{\vskip 1mm\par\noindent\includegraphics{#1}} \def\nbpdfSubsection#1{\vskip 1mm\par\noindent\includegraphics{#1}} \def\nbpdfSubsubsection#1{\vskip 1mm\par\noindent\includegraphics{#1}} \def\nbpdfgraphInput#1{\vskip 1mm\par\noindent\includegraphics{#1}} \def\nbpdfgraphOutput#1{\vskip 1mm\par\noindent\includegraphics[width=1.5in]{#1}} \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{Cars1.pdftex_t}\vfill\null \end{center} \newgeometry{textwidth=8in,textheight=10.5in} \begin{multicols*}{2} \input{Rho1.tex} \end{multicols*} \newgeometry{textwidth=8.5in,textheight=11in,centering} \begin{center} \null\vfill\input{Cars2.pdftex_t}\vfill\null \end{center} \AtEndDocument{\includepdf[pages={-}]{../../2012-01/Infrastructure.pdf}} \end{document} \endinput