\documentclass[11pt,notitlepage,a4paper]{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=210mm,textheight=297mm,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{Sydney-190916} \def\title{Algebraic Knot Theory} \def\navigator{{ \href{\myurl}{Dror Bar-Natan}: \href{\myurl/Talks}{Talks}: \href{\myurl/Talks/\thistalk/}{\thistalk}: }} \def\webdef{{{\greektext web}$\coloneqq$\href{http://drorbn.net/syd2}{http://drorbn.net/syd2/}}} \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\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\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.} This will be a very ``light'' talk: I will explain why about 13 years ago, in order to have a say on some problems in knot theory, I've set out to find tangle invariants with some nice compositional properties. In later talks in different seminars here in Sydney I will explain how such invariants were found - though they are yet to be explored and utilized. }}}} \def\MetaAssoc{{\parbox{1.1in}{ (meta-associativity: $m^{ab}_x\act m^{xc}_y=m^{bc}_x\act m^{ax}_y$) \newline (tangles are generated by $\overcrossing$ and $\undercrossing$) }}} \def\Genus{{\raisebox{2mm}{\parbox[t]{2.96in}{ {\red\bf Genus.} Every knot is the boundary of an orientable ``Seifert Surface'' (\web{SS}), and the least of their genera is the ``genus'' of the knot. {\red Claim.} The knots of genus $\leq 2$ are precisely the images of 4-component tangles via }}}} \def\Ribbon{{\raisebox{2mm}{\parbox[t]{3.96in}{ {\red\bf A Bit about Ribbon Knots.} A ``ribbon knot'' is a knot that can be presented as the boundary of a disk that has ``ribbon singularities'', but no ``clasp singularities''. A ``slice knot'' is a knot in $S^3=\partial B^4$ which is the boundary of a non-singular disk in $B^4$. Every ribbon knots is clearly slice, yet, {\red Conjecture.} Some slice knots are not ribbon. {\red Fox-Milnor.} The Alexander polynomial of a ribbon knot is always of the form $A(t)=f(t)f(1/t)$.\hfill{(also~for~slice)} }}}} \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 $\calR\coloneqq$} \def\Rb{$\kappa(\tau^{-1}(1))$} \def\AKTRibbon{{\raisebox{2mm}{\parbox[t]{3.96in}{ {\red\bf Theorem.} $K$ is ribbon iff it is $\kappa T$ for a tangle $T$ for which $\tau T$ is the untangle $U$. }}}} \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 \cite{Bar-NatanSelmani:MetaMonoids, KBH}. 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 \[ \begin{CD} \begin{array}{c|ccc} \omega & a & b & S \\ \hline a & \alpha & \beta & \theta \\ b & \gamma & \delta & \epsilon \\ S & \phi & \psi & \Xi \end{array} @>{\displaystyle m^{ab}_c}>{\displaystyle T_a,T_b\to T_c}> \left(\!\begin{array}{c|cc} (1-\beta)\omega & c & S \\ \hline c & \gamma+\frac{\alpha\delta}{1-\beta} & \epsilon+\frac{\delta\theta}{1-\beta} \\ S & \phi+\frac{\alpha\psi}{1-\beta} & \Xi+\frac{\psi\theta}{1-\beta} \end{array}\!\right) \end{CD} \] \vskip 1mm \parshape 1 0in 3.45in For long knots, $\omega$ is Alexander, and that's the fastest \text{Alexander} algorithm I know! \hfill\text{\footnotesize Dunfield: 1000-crossing fast.} \end{minipage}}}} \def\Golder{{\raisebox{-17mm}{\begin{minipage}{3.95in} {\red\bf Strand Doubling and Reversal.} \newline \resizebox{\linewidth}{!}{$\displaystyle \begin{CD} \begin{array}{c|cc} \omega & a & S \\ \hline a & \alpha & \theta \\ S & \phi & \Xi \end{array} @>q\Delta^a_{bc}>{\mu\coloneqq T_a-1\atop{\nu\coloneqq \alpha-\sigma_a\atop T_a\mapsto T_bT_c}}> \left(\begin{array}{c|ccc} \omega & b & c & S \\ \hline b & (\sigma_a-\alpha T_a-\nu T_c)/\mu & (T_b-1)T_c\nu/\mu & (T_b-1)T_c\theta/\mu \\ c & (T_c-1)\nu/\mu & (\alpha-\sigma_a T_a-\nu T_c)/\mu & (T_c-1)\theta/\mu \\ S & \phi & \phi & \Xi \end{array}\right) \\ @VdS^aVT_a\to T_a^{-1}V \\ \hspace{0mm}\left(\begin{array}{c|cc} \alpha\omega/\sigma_a & a & S \\ \hline a & 1/\alpha & \theta/\alpha \\ S & -\phi/\alpha & (\alpha\Xi-\phi\theta)/\alpha \end{array}\right)\hspace{-33mm} @. \raisebox{-2mm}{\hspace{16mm}\parbox{81mm}{ Where $\sigma$ assigns to every $a\in S$ a Laurent monomial $\sigma_a$ in $\{t_b\}_{b\in S}$ subject to $\sigma\left( \tensor[_a]{\overcrossing}{_b},\tensor[_b]{\undercrossing}{_a} \right) = (a\to1,\,b\to t_a^{\pm 1})$, $\sigma(T_1\sqcup T_2)=\sigma(T_1)\sqcup\sigma(T_2)$, and $\sigma\act m^{ab}_c = \left. (\sigma\setminus\{a,b\})\cup(c\to\sigma_a\sigma_b) \right|_{t_a,t_b\to t_c}$. }} \end{CD} $} \end{minipage}}}} \def\Vo{{\raisebox{2mm}{\parbox[t]{3.5in}{ {\red\bf Vo's Thesis \cite{Vo:Thesis}.} A proof of the Fox-Milnor theorem for ribbon knots using this technology (and more). }}}} \def\Implementation{{\raisebox{2.5mm}{\parbox[t]{2in}{ \parshape 2 0in 2in 0.1in 1.5in {\red\bf Implementation} key idea: \newline $\left(\omega,A=(\alpha_{ab})\right)\leftrightarrow$ \newline $\left(\omega,\lambda=\sum\alpha_{ab}t_ah_b\right)$ }}}} \def\Comments{{\raisebox{2mm}{\parbox[t]{2.75in}{ {\bf\red Fact.} $\Gamma$ is better viewed as an invariant of a certain class of 2D knotted objects in $\bbR^4$ \cite{WKO1, KBH}. {\bf\red Fact.} $\Gamma$ is the ``0-loop'' part of an invariant that generalizes to ``$n$-loops'' (1D tangles only, see further talks and future publications with van der Veen). {\bf\red Speculation.} Stepping stones to categorification? }}}} \def\refs{{\raisebox{4mm}{\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[BN]{KBH} D.~Bar-Natan, {\em Balloons and Hoops and their Universal\newline Finite Type Invariant, BF Theory, and an Ultimate Alexander Invariant,} \web{KBH}, \arXiv{1308.1721}. \bibitem[BND]{WKO1} D.~Bar-Natan and Z.~Dancso, {\em Finite Type Invariants of W-Knotted Objects I: $w$-Knots and the Alexander Polynomial,} Alg.\ and Geom.\ Top.\ {\bf 16-2} (2016) 1063--1133, \arXiv{1405.1956}, \web{WKO1}. \bibitem[BNS]{Bar-NatanSelmani:MetaMonoids} D.~Bar-Natan and S.~Selmani, {\em Meta-Monoids, Meta-Bicrossed Products, and the Alexander Polynomial,} J.\ of Knot Theory and its Ramifications {\bf 22-10} (2013), \arXiv{1302.5689}. \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[Vo]{Vo:Thesis} H.~Vo, {\em Alexander Invariants of Tangles via Expansions,} University of Toronto Ph.D.\ thesis, \web{Vo}. \end{thebibliography}} }}}} \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\scalebox{0.98}{\input{AKT1.pdftex_t}}\vfill\null\end{center} \AtEndDocument{\includepdf[pages={1}, noautoscale=true, offset=0in 0in]{TangleCharacterizationOfRibbon.pdf}} \end{document} \endinput