\documentclass[11pt,notitlepage]{article} \def\bare{n} \usepackage[all]{xy} \usepackage[english,greek]{babel} \usepackage{amsmath, graphicx, amssymb, datetime, multicol, stmaryrd, pifont, amscd, colortbl, mathtools, wasysym, needspace, import, longtable, overpic, enumitem, bbm} \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={green!50!black}, citecolor={green!50!black}, urlcolor={green!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{GWU-1612} \def\title{On Elves and Invariants} \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/\thistalk}{http://drorbn.net/\thistalk/}}} \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\ob#1{\overbracket[0.5pt][1pt]{#1}} \def\ub#1{\underbracket[0.5pt][1pt]{#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\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\frakb{{\mathfrak b}} \def\frakg{{\mathfrak g}} \def\frakh{{\mathfrak h}} \def\tilE{\tilde{E}} %%% \def\credits{{\raisebox{0mm}{\parbox[t]{2in}{ Follows Rozansky \cite{Ro, Rozansky:Burau, Rozansky:U1RCC} and \text{Overbay} \cite{Overbay:Thesis}, joint with van der Veen. }}}} \def\Abstract{{\raisebox{2mm}{\parbox[t]{3.95in}{ {\red Abstract.} Whether or not you like the formulas on this page, they describe the strongest truly computable knot invariant we know. }}}} \def\ThreeSteps{{\raisebox{2mm}{\parbox[t]{2.25in}{ {\red Three steps} to the computation of $\rho_1$: \newline{\red 1.~Preparation.} Given $K$, results \[ \langle\text{\sl long word}\,\|\,\text{\sl simple formulas}\rangle. \] {\red 2.~Rewrite rules.} Make the {\sl word} simpler and the {\sl formulas} more complicated, until the {\sl word} ``$\mathit{elf}$'' is reached. \newline{\red 3.~Readout.} The invariant $\rho_1$ is read from the last {\sl formulas}. }}}} \def\prepres{$\langle \mathit{elf}\ldots\mathit{elf}\,\|\, \omega_0;L_0;Q_0;P_0\rangle$} \def\endres{$\langle \mathit{elf}\,\|\, \omega;-;-;P\rangle$} \def\rhoone{$\rho_1(K)=\rho_1(\omega,P)$} \def\Preparation{{\raisebox{2mm}{\parbox[t]{3.95in}{ \parshape 8 0in 2.95in 0in 2.95in 0in 2.95in 0in 2.95in 0in 2.95in 0in 2.95in 0in 2.95in 0in 3.95in {\red Preparation.} Draw $K$ using a 0-framed 0-rotation planar diagram $D$ where all crossings are pointing up. Walk along $D$ labeling features by $1,\ldots,m$ in order: over-passes, under-passes, and right-heading cups and caps (``$\pm$-cuaps''). If $x$ is a xing, let $i_x$ and $j_x$ be the labels on its over/under strands, and let $s_x$ be $0$ if it right-handed and $-1$ otherwise. If $c$ is a cuap, let $i_c$ be its label and $s_c$ be its sign. Set \begin{align*} (L;Q;P) &= \hspace{-1.3mm}\sum_{x\colon(i,j,s)}\hspace{-1mm}(-)^s\left( l_j;\, t^se_if_j;\, (-t)^se_il_{(1+s)i-sj}f_j+l_il_j+\frac{t^{2s}e_i^2f_j^2}{4} \right) \\ &+ \sum_{c\colon(i,s)}\hspace{-1mm}\left(0;\,0;\,s\cdot l_i\right). \end{align*} This done, output $\displaystyle \langle e_1l_1f_1e_2l_2f_2\cdots e_ml_mf_m \,\|\, 1;L;Q;P\rangle$. }}}} \def\InFormulas{{\raisebox{2mm}{\parbox[t]{3.95in}{ {\red In {\sl formulas},} $L$ is always $\bbZ$-linear in $\{l_i\}$, $Q$ is an $R$-linear combination of $\{e_if_j\}$ where $R\coloneqq\bbQ[t^{\pm 1}]$, and $P$ is an $R$-linear combination of $\{1,\, l_i,\, l_il_j,\, e_if_j,\, e_il_jf_k,\, e_ie_jf_kf_l\}$.\hfill\text{\footnotesize(The key to computability!)} }}}} \def\RewriteRules{{\raisebox{2mm}{\parbox[t]{3.95in}{ {\red Rewrite Rules.} Manipulate $\langle\text{\sl word}\,\|\,\text{\sl formulas}\rangle$ expressions using the rewrite rules below, until you come to the form ${\langle e_1l_1f_1\,\|\, \omega;-;-;P\rangle}$. Output $(\omega,P)$. {\red Rule 1, Deletions.} If a letter appears in {\sl word} but not in {\sl formulas}, you can delete it. {\red Rule 2, Merges.} In {\sl word}, you can replace {\em adjacent} $v_iv_j$ with $v_k$ (for $v\in\{e,l,f\}$) while making the same changes in {\sl formulas} (provided $k$ creates no naming clashes). E.g., \[ \langle\ldots e_ie_j\ldots \|\, Z\rangle \to \langle\ldots e_k\ldots\|\, Z|_{e_i,e_j\to e_k}\rangle. \] {\red Rule 3, $\mathit{le}$ Sorts.} Provided $k$ introduces no clashes, given $\langle\ldots l_je_i\ldots\|\,\omega;L;Q;P\rangle$, decompose $L=\lambda l_j+L'$, $Q=\alpha e_i+Q'$, write $P=P(e_i,l_j)$ (with messy coefficients), set $q=\bbe^\gamma\beta e_k+\gamma l_k$, and output \[ \left\langle \ldots e_kl_k\ldots\|\,\omega; L|_{l_j\to l_k}; t^\lambda\alpha e_k+Q'; \bbe^{-q}P(\partial_\beta,\partial_\gamma)\bbe^q|_{\beta\to\alpha/\omega,\,\gamma\to\lambda\log t} \right\rangle. \] {\red Rule 4, $\mathit{fl}$ Sorts.} Provided $k$ introduces no clashes, given $\langle\ldots f_il_j\ldots\|\,\omega;L;Q;P\rangle$, decompose $L=\lambda l_j+L'$, $Q=\alpha f_i+Q'$, write $P=P(f_i,l_j)$ (with messy coefficients), set $q=\bbe^\gamma\beta f_k+\gamma l_k$, and output \[ \left\langle \ldots l_kf_k\ldots\|\,\omega; L|_{l_j\to l_k}; t^\lambda\alpha f_k+Q'; \bbe^{-q}P(\partial_\beta,\partial_\gamma)\bbe^q|_{\beta\to\alpha/\omega,\,\gamma\to\lambda\log t} \right\rangle. \] }}}} \def\RuleFive{{\raisebox{2mm}{\parbox[t]{3.95in}{ {\red Rule 5, $\mathit{fe}$ Sorts.} Provided $k$ introduces no clashes, given $\langle\ldots f_ie_j\ldots\|\,\omega;L;Q;P\rangle$, decompose $Q=Q_{fe}f_ie_j+Q_ff_i+Q_ee_j+Q'$ write $P=P(f_i,e_j)$ (with messy coefficients), set $\mu=1+(t-1)\delta$ and $q=((1-t)\alpha\beta+\beta e_k+\alpha f_k+\delta e_kf_k)/\mu$, and output \[ \left.\left\langle \ldots e_kf_k\ldots \left\|\,{ \mu\omega; L;\,\mu\omega q+\mu Q';\atop \omega^4\Lambda_k+\bbe^{-q}P(\partial_\alpha,\partial_\beta)(\bbe^q) }\right. \right\rangle\right|_{\alpha\to Q_f/\omega,\,\beta\to Q_e/\omega,\atop\delta\to Q_{fe}/\omega}, \] where $\Lambda_k$ is the {\greektext L'ogos}, ``a principle of order and knowledge'': \begin{align*} \Lambda_k = \frac{t+1}{4}\bigg(& -\delta (\mu +1) \left(\beta ^2 e_k^2+\alpha ^2 f_k^2\right) -\delta^3 (3 \mu +1)e_k^2f_k^2 \\ & -2 \left(\beta e_k+\alpha f_k\right)\left( \alpha\beta+2\delta\mu+\delta^2(2\mu+1)e_kf_k+2\delta\mu^2l_k \right) \\ & -4 (\alpha \beta +\delta \mu ) \left(\delta (\mu +1) e_k f_k+\mu ^2 l_k\right) -4 \delta ^2 \mu ^2 e_k f_k l_k \\ & +(t-1) \left(2 (\alpha \beta +\delta \mu )^2-\alpha ^2 \beta ^2\right)\bigg). \end{align*} }}}} \def\ElfMerges{{\raisebox{3mm}{\parbox[t]{3.95in}{ {\red $\mathit{elf}$ merges,} $m^{ij}_k$, are defined as compositions \begin{multline*} e_il_i\ob{f_ie_j}l_jf_j \xrightarrow{S^{f_ie_j}_x} e_i\ob{l_ie_x}\ob{f_xl_j}f_j \xrightarrow{S^{l_ie_x}_x\act S^{f_xl_j}_x} \ob{e_ie_x}\ob{l_xl_x}\ob{f_xf_j} \\ \xrightarrow{i,j,x\to k} e_kl_kf_k \end{multline*} }}}} \def\Readout{{\raisebox{2mm}{\parbox[t]{3.25in}{ {\red Readout.} Given $\langle\mathit{elf}\,\|\,\omega;-;-;P\rangle$, output \[ \rho_1(K)\coloneqq \frac{t(P|_{e,l,f\to 0}-t\omega'\omega^3)}{(t-1)^2\omega^2}. \] {\footnotesize ($\omega$ is the Alexander polynomial, $L$ and $Q$ are not interesting).} }}}} \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 $K$ (equallity 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\WhyWorks{{\raisebox{2mm}{\parbox[t]{3.95in}{ {\red Why Works?} The Lie algebra $\frakg_1$ (below) is a ``solvable approximation of $sl_2$''. {\red Theorem.} The map (as defined below) \[ \langle w\,\|\,\omega;L;Q;P \rangle \mapsto \bbO\left(\omega^{-1}\bbe^{L\log t+\omega^{-1}Q}(1+\epsilon\omega^{-4}P)\colon\,w\right) \in\hat\calU(\frakg_1) \] is well defined modulo the sorting rules. It maps the initial preparation to a product of ``$R$-matrices'' and ``cuap values'' satisfying the usual moves for Morse knots (R3, etc.). (And hence the result is a ``quantum invariant'', except computed very differently; no representation theory!). }}}} \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{Elves1.pdftex_t}\vfill\null\eject \end{center} \eject \newgeometry{textwidth=8in,textheight=10.5in} \def\cellscale{0.65} \begin{multicols}{2} \raggedcolumns {\red 1-Smidgen $sl_2$} Let {\red $\frakg_1$} be the 4-dimensional Lie algebra $\frakg_1=\langle h,e',l,f\rangle$ over the ring $R=\bbQ[\epsilon]/(\epsilon^2=0)$, with $h$ central and with $[f,l]=f$, $[e',l]=-e'$, and $[e',f]=h-2\epsilon l$. Over $\bbQ$, $\frakg_1$ is a {\red solvable approximation of $sl_2$}: $\frakg_1 \supset \langle h,e',f,\epsilon h,\epsilon e',\epsilon l,\epsilon f\rangle \supset \langle h,\epsilon h,\epsilon e',\epsilon l,\epsilon f\rangle \supset 0$. Pragmatics: declare $\deg(h,e',l,f,\epsilon)=(1,1,0,0,1)$ and set $t\coloneqq \bbe^h$ and $e\coloneqq (t-1)e'/h$. {\red How did it arise?} $sl_2=\frakb^+\oplus\frakb^-/\frakh\eqqcolon sl_2^+/\frakh$, where $\frakb^+=\langle l,f\rangle/[f,l]=f$ is a Lie bialgebra with $\delta\colon\frakb^+\to\frakb^+\otimes\frakb^+$ by $\delta\colon(l,f)\mapsto(0,l\wedge f)$. Going back, $sl_2^+=\calD(\frakb^+) = (\frakb^+)^\ast\oplus\frakb^+ = \langle h',e',l,f\rangle/\cdots$. {\red Idea.} Replace $\delta\to\epsilon\delta$ over $\bbQ[\epsilon]/(\epsilon^{k+1}=0)$. At $k=1$, get $[f,l]=f$, $[f,h']=-\epsilon f$, $[l,e']=e'$, $[h',e']=-\epsilon e'$, $[h',l]=0$, and $[e',f]=h'-\epsilon l$. Now note that $h'+\epsilon l$ is central, so switch to $h\coloneqq h'+\epsilon l$. This is $\frakg_1$. {\red Ordering Symbols.} $\bbO\left(\text{\it poly}\mid\text{\it specs}\right)$ plants the variables of {\it poly} in $\hat\calS(\oplus_i\frakg)$ along $\hat\calU(\frakg)$ according to {\it specs}. E.g., \[ \bbO\left(e_1\bbe^{e_3}l_1^3l_2f_3^9\,|\,f_3l_1e_1e_3l_2\right) = f^9l^3e\bbe^{e}l \in \hat\calU(\frakg). \] This enables the description of elements of $\hat\calU(\frakg)$ using commutative polynomials / power series. In $\frakg_1$, no need to specify $h$~/~$t$. {\red Algebras and Invariants.} Given any unital algebra $A$ (even better if $A$ is Hopf; typically, $A\sim\hat{\calU}(\frakg)$), appropriate {\morange orange} $R\in A\otimes A$, {\mgray and appropriate cuaps $\in A$,} get an $A^{\otimes S}$-valued invariant of pure $S$-component tangles: \resizebox{\linewidth}{!}{\input{figs/AlgAndInvts.pdf_t}} {\red What we didn't say} (more, including videos, in \web{Talks}). \begin{itemize}[leftmargin=*,labelindent=0pt,itemsep=-2pt,topsep=0pt] \item $\rho_1$ is ``line'' in the coloured Jones polynomial; related to Melvin-Morton-Rozansky. \item $\rho_1$ extends to ``rotational virtual tangles'' and is a projection of the universal finite type invariant of such. \item $\rho_1$ seems to have a better chance than anything else we know to detect a counterexample to slice$=$ribbon. \item $\rho_1$ leads to many questions and a very long to-do list. Years of work, many papers ahead. Have fun! \end{itemize} {\large\red Demo Programs.}\hfill\web{Demo} \begin{overpic}[scale=\cellscale]{Snips/CF.pdf} \put(91,11.5){\red Formatting} \put(90.08,8.25){\scriptsize (prints differ~\frownie)} \end{overpic} \hfill{\red Preparation} \vskip -12pt\includegraphics[scale=\cellscale]{Snips/Preparation.pdf} \hfill{\red Preparing the Trefoil} \vskip -12pt\includegraphics[scale=\cellscale]{Snips/Prep31.pdf} \hfill{\red Differential Polynomials} \newline\null\hfill\text{\footnotesize (Implementing $P(\partial_\alpha,\partial_\beta)(f)$)} \vskip -26pt\includegraphics[scale=\cellscale]{Snips/DP.pdf} \hfill{\red $\mathit{le}$ and $\mathit{fl}$ Sorts} \vskip -12pt\includegraphics[scale=\cellscale]{Snips/Slx.pdf} \vskip 2mm \begin{overpic}[scale=\cellscale]{Snips/Logos.pdf} \put(91,11.5){\red The {\greektext L'ogos}} \end{overpic} %\hfill{\red The {\greektext L'ogos}} %\newline\includegraphics[scale=\cellscale]{Snips/Logos.pdf} \hfill{\red $\mathit{fe}$ Sorts} \vskip -12pt\includegraphics[scale=\cellscale]{Snips/Sfe.pdf} \hfill{\red Elf Merges} \vskip -12pt\includegraphics[scale=\cellscale]{Snips/m.pdf} \hfill{\red Rewriting the Trefoil} \newline\null\hfill\text{\footnotesize (by merging 16 elves)} \vskip -26pt\includegraphics[scale=\cellscale]{Snips/E31.pdf} \hfill{\red Readout} \newline\includegraphics[scale=\cellscale]{Snips/rho1.pdf} \hfill{\red $\rho_1(3_1)$} \vskip -12pt\includegraphics[scale=\cellscale]{Snips/rho131.pdf} \vskip 1mm {\red References.}{\footnotesize %\par\vspace{-3mm} \renewcommand{\section}[2]{}% \begin{thebibliography}{} \setlength{\parskip}{0pt} \setlength{\itemsep}{0pt plus 0.3ex} \input{refs.tex} \end{thebibliography}} \end{multicols} \vskip -5mm \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]{3in}{ {\blue $#2$}\quad $#3$\hfill $#5$ / #7 \\ {\red $#4$} \hfill $#6$ / #8 }} {\footnotesize \begin{longtable}{|cl|cl|cl|} \hline \headcell & \headcell \\ \endhead \hline \input table.tex \end{longtable}} \end{document} \endinput