\documentclass[11pt,notitlepage]{article} \def\bare{n} \usepackage[all]{xy} \usepackage[english,greek]{babel} \usepackage{amsmath, graphicx, amssymb, datetime, multicol, stmaryrd, pifont, amscd, dbnsymb, picins, colortbl, mathtools, wasysym, needspace, import, longtable} \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={blue!50!black}, citecolor={blue!50!black}, urlcolor=blue } % 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{Indiana-1611} \def\title{A Poly-Time Knot Polynomial Via Solvable Approximation} \def\navigator{{ \href{\myurl}{Dror Bar-Natan}: \href{\myurl/Talks}{Talks}: \href{\myurl/Talks/\thistalk/}{\thistalk}: \quad(thanks for accepting my invitation!) }} \def\webdef{{{\greektext web}$\coloneqq$\url{http://drorbn.net/\thistalk/}}} \def\web#1{{\href{\myurl/Talks/\thistalk/#1}{{\greektext web}/#1}}} \def\titleA{{\title}} \def\titleB{{\title}} \def\titleC{{\title}} \def\todo#1{\text{\Huge #1}} \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\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{{\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\Abstract{{\raisebox{2mm}{\parbox[t]{3.95in}{ \parshape 10 0in 3.45in 0in 3.45in 0in 3.45in 0in 3.45in 0in 3.45in 0in 3.45in 0in 3.45in 0in 3.45in 0in 3.45in 0in 3.95in {\red Abstract.} Rozansky \cite{Rozansky:Burau} and Overbay \cite{Overbay:Thesis} described a {\red spectacular} knot polynomial that failed to attract the attention it deserved as the first poly-time-computable knot polynomial since Alexander's \cite[1928]{Alexander:TopologicalInvariants} and (in my opinion) as the second most likely knot polynomial (after Alexander's) to carry topological information. With Roland van der Veen, I will explain how to compute the Rozansky polynomial using some new commutator-calculus techniques and a Lie algebra $\frakg_1$ which is at the same time solvable and an approximation of the simple Lie algebra $sl_2$. }}}} \def\mmr{{\raisebox{2mm}{\parbox[t]{3.95in}{ \parshape 3 0in 2.45in 0in 2.45in 0in 3.95in {\red Theorem} (\cite{Bar-NatanGaroufalidis:MMR}, conjectured~\cite{MM}, elucidated~\cite{Ro}). Let $J_d(K)$ be the coloured Jones polynomial of $K$, in the $d$-dimensional representation of $sl_2$. Writing \[ \left. \frac{(q^{1/2}-q^{-1/2})J_d(K)}{q^{d/2}-q^{-d/2}} \right|_{q=e^\hbar} = \sum_{j,m\geq 0} a_{jm}(K)d^j\hbar^m, \] \parshape 5 0in 3in 0in 3in 0in 3in 0in 3in 0in 3.95in ``below diagonal'' coefficients vanish, $a_{jm}(K)=0$ if $j>m$, and ``on diagonal'' coefficients give the inverse of the Alexander polynomial: $\left(\sum_{m=0}^\infty a_{mm}(K)\hbar^m\right)\cdot A(K)(e^\hbar)=1$. ``Above diagonal'' we have {\red Rozansky's Theorem} \cite[(1.2)]{Rozansky:U1RCC}: \[ J_d(K)(q) = \frac{q^d-q^{-d}}{(q-q^{-1})A(K)(q^d)} \left(1+ \sum_{k=1}^\infty \frac{(q-1)^kR_k(K)(q^d)}{A^{2k}(K)(q^d)} \right). \] }}}} \def\Spectacular{{\raisebox{2mm}{\begin{minipage}[t]{3.95in} {\red Why ``spectacular''?} Foremost reason: {\red\sl OBVIOUSLY.} {\footnotesize Cf.\ proving (incomputable $A$)$=$(incomputable $B$), or categorifying (incomputable $C$).} \par Also, will bound {\red genus} and may disprove {\red$\{$ribbon$\}=\{$slice$\}$}. \par {\red Genus.} \end{minipage}}}} \def\Ribbon{{\raisebox{2mm}{\parbox[t]{3.95in}{ {\red 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\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\GST{\parbox{0.5in}{\tiny Gompf, Scharlemann, Thompson \cite{GompfScharlemannThompson:Counterexample} }} \def\MetaAssoc{{\parbox{1.2in}{ (meta-associativity: $m^{ab}_x\act m^{xc}_y=m^{bc}_x\act m^{ax}_y$) }}} \def\GSTInvariants{{\raisebox{0mm}{\parbox[t]{3.95in}{\footnotesize $A^+ = -t^8+2 t^7-t^6-2 t^4+5 t^3-2 t^2-7 t+13$ $\rho_1^+ = 5t^{15}-18t^{14}+33t^{13}-32t^{12}+2t^{11}+42t^{10}-62t^9-8t^8+166t^7-242t^6+$ \newline\null\hfill$108 t^5+132 t^4-226 t^3+148 t^2-11 t-36$ }}}} \def\Gold{{\raisebox{5mm}{\begin{minipage}[t]{3.95in} \pichskip{0mm}\parpic[l]{ \includegraphics[height=0.4in]{../Greece-1607/IASLogo.png} } \picskip{2} {\red 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 \[ \begin{CD} \begin{array}{c|ccc} \omega & a & b & S \\ \hline a & \alpha & \beta & \theta \\ b & \gamma & \delta & \epsilon \\ S & \phi & \psi & \Xi \end{array} @>{\displaystyle m^{ab}_c}>{\displaystyle t_a,t_b\to t_c}> \left(\!\begin{array}{c|cc} (1-\beta)\omega & c & S \\ \hline c & \gamma+\frac{\alpha\delta}{1-\beta} & \epsilon+\frac{\delta\theta}{1-\beta} \\ S & \phi+\frac{\alpha\psi}{1-\beta} & \Xi+\frac{\psi\theta}{1-\beta} \end{array}\!\right) \end{CD} \] {\footnotesize (Roland: ``add to $A$ the product of column $b$ and row $a$, divide by $(1-A_{ab})$, delete column $b$ and row $a$''.)} \vskip 1.5mm \parshape 1 0in 3.45in For long knots, $\omega$ is Alexander, and that's the fastest Alexander algorithm I know! \hfill\text{\footnotesize Dunfield: 1000-crossing fast.} \vskip 1mm (There are also formulas for strand doubling and strand reversal). \end{minipage}}}} \def\Dunfield{{\raisebox{0mm}{\parbox[t]{3.45in}{ For long knots, $\omega$ is Alexander, and that's the fastest Alexander algorithm I know! \newline\null\hfill\text{\footnotesize Dunfield: 1000-crossing fast.} }}}} \def\EKTheorem{{\raisebox{2mm}{\begin{minipage}[t]{3.95in} {\red Theorem} \cite{EtingofKazhdan:BialgebrasI, Haviv:DiagrammaticAnalogue, Enriquez:Quantization, Severa:BialgebrasRevisited}. There is a ``homomorphic expansion'' \[ \arraycolsep=0pt \renewcommand{\arraystretch}{1} \begin{array}{c} Z\colon\left\{\parbox{0.85in}{$S$-component ($v/b$-)tangles}\right\} \to \calA^v_S \coloneqq \\ \parbox{0.45\linewidth}{\vskip 2mm \includegraphics[height=0.4in]{../../Projects/Gallery/Etingof.jpg}% \hfill\includegraphics[height=0.4in]{../../Projects/Gallery/Kazhdan.jpg}% \hfill\includegraphics[height=0.4in]{../../Projects/Gallery/Haviv.jpg}% \hfill\includegraphics[height=0.4in]{../../Projects/Gallery/Enriquez.jpg}% \hfill\includegraphics[height=0.4in]{../../Projects/Gallery/Severa.jpg} \hfill} \\ \parbox{0.45\linewidth}{\tiny Etingof \hfill Kazhdan \hfill Haviv \hfill Enriquez \hfill \v{S}evera \hfill} \end{array} \begin{array}{c}\resizebox{0.55\linewidth}{!}{\subimport{../UNC-1610/}{Av.pdf_t}}\end{array} \] \end{minipage}}}} \def\Algebras{{\raisebox{2mm}{\parbox[t]{3.95in}{ {\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: }}}} \def\Strategy{{\raisebox{2mm}{\parbox[t]{3.95in}{ {\red Good News.} In theory, enough to know $R$, {\mgray the cuaps,} and stitching/multiplication $m^{ij}_k\colon A_i\otimes A_j\to A_k$. \par{\red Problem.} Extract information out of $Z$. \par{\red Textbook Solution.} Use representation theory\hfill{\footnotesize\ldots works, slowly.} \par\parshape 1 0in 3.3in {\red Today's Solution} (with van der Veen). For some specific $\frakg$'s, work in a space of ``formulas of a specific type'' for elements of $\hat\calU(\frakg)^{\otimes S}$: \[ \left\{\parbox{1.16in}{ordered perturbed Gaussian formulas}\right\} \to \hat\calU(\frakg)^{\otimes S} \] }}}} \def\Smidgen{{\raisebox{2mm}{\parbox[t]{3.95in}{ %\parshape 2 0in 3.3in 0in 3.95in {\red 1-Smidgen $sl_2$} Let {\red $\frakg_1$} be the 4-dimensional Lie algebra $\frakg_1=\langle b,c,u,w\rangle$ over the ring $R=\bbQ[\epsilon]/(\epsilon^2=0)$, with $b$ central and with \cbox{yellow}{$[w,c]=w$, $[c,u]=u$, and $[u,w]=b-2\epsilon c$}, with CYBE $r_{ij}=(b_i-\epsilon c_i)c_j+u_iw_j$ in $\calU(\frakg_1)^{\otimes\{i,j\}}$. Over $\bbQ$, $\frakg_1$ is a {\red solvable approximation of $sl_2$}: $\frakg_1 \supset \langle b,u,w,\epsilon b,\epsilon c,\epsilon u,\epsilon w\rangle \supset \langle b,\epsilon b,\epsilon c,\epsilon u,\epsilon w\rangle \supset 0$. \hfill\text{\footnotesize(note: $\deg(b,c,u,w,\epsilon)=(1,0,1,0,1)$)} }}}} \def\ZeroSmidgen{{\raisebox{2mm}{\parbox[t]{3.95in}{ {\red 0-Smidgen $sl_2$ \smiley{}.} Let $\frakg_0$ be $\frakg_1$ at $\epsilon=0$, or $\bbQ\langle b,c,u,w\rangle/([b,\cdot]=0,\,[c,u]=u,\,[c,w]=-w,\,[u,w]=b$ with $r_{ij}=b_ic_j+u_iw_j$. It is $\frakb^\ast\rtimes\frakb$ where $\frakb$ is the 2D Lie algebra $\bbQ\langle c,w\rangle$ and $(b,u)$ is the dual basis of $(c,w)$. For topology, it is more valuable than $\frakg_1$ / $sl_2$, but topology already got by other means almost everything $\frakg_0$ gives. }}}} \def\HowArose{{\raisebox{2mm}{\parbox[t]{3.95in}{ {\red How did these arise?} $sl_2=\frakb^+\oplus\frakb^-/\frakh\eqqcolon sl_2^+/\frakh$, where $\frakb^+=\langle c,w\rangle/[w,c]=w$ is a Lie bialgebra with $\delta\colon\frakb^+\to\frakb^+\otimes\frakb^+$ by $\delta\colon(c,w)\mapsto(0,c\wedge w)$. Going back, $sl_2^+=\calD(\frakb^+) = (\frakb^+)^\ast\oplus\frakb^+ = \langle b,u,c,w\rangle/\cdots$. {\red Idea.} Replace $\delta\to\epsilon\delta$ over $\bbQ[\epsilon]/(\epsilon^{k+1}=0)$. At $k=0$, get $\frakg_0$. At $k=1$, get $[w,c]=w$, $[w,b']=-\epsilon w$, $[c,u]=u$, $[b',u]=-\epsilon u$, $[b',c]=0$, and $[u,w]=b'-\epsilon c$. Now note that $b'+\epsilon c$ is central, so switch to $b\coloneqq b'+\epsilon c$. This is $\frakg_1$. }}}} \def\OrderingSymbols{{\raisebox{2mm}{\parbox[t]{3.95in}{ {\red Ordering Symbols.} $\bbO\left(\text{\it poly}\mid\text{\it specs}\right)$ plants the variables of {\it poly} in $\calS(\oplus_i\frakg)$ on several tensor copies of $\calU(\frakg)$ according to {\it specs}. E.g., \[ \bbO\left( c_1^3u_1c_2e^{u_3}w_3^9|x\colon\!w_3c_1,\,y\colon\!u_1u_3c_2 \right) \!=\! w^9c^3\otimes ue^uc \in \calU(\frakg)_x\otimes\calU(\frakg)_y \] This enables the description of elements of $\hat\calU(\frakg)^{\otimes S}$ using commutative polynomials / power series. }}}} \def\ZeroInvariants{{\raisebox{2mm}{\parbox[t]{3.95in}{ {\red 0-Smidgen Invariants.} $r=Id\in\frakb^-\otimes\frakb^+$ solves the CYBE $[r_{12},r_{13}]+[r_{12},r_{23}]+[r_{13},r_{23}]=0$ in $\calU(\frakg_0)^{\otimes 3}$ and, by luck, }}}} \def\Rzero{$= R_{ij} = e^{r_{ij}} = e^{b_ic_j+u_iw_j} \in \calU(\frakg_{0,i}\oplus\frakg_{0,j})$} \def\ZeroLemma{{\raisebox{2mm}{\parbox[t]{3.95in}{ {\red Lemma.} $R_{ij} \!=\! e^{b_ic_j+u_iw_j} \!=\! \bbO\left( \exp\left(b_ic_j+\frac{e^{b_i}-1}{b_i}u_iw_j\right) |i\colon\!u_i,\,j\colon\!c_jw_j \right)$ {\red Example.} $Z(T_0)=$ \hfill $=\! \sum_{m,n} \frac{b_i^{m-n}(e^{b_i}-1)^n}{m!n!}u^n\otimes c^mw^n$. \vskip 1mm $\bbO\bigg(\exp\left( b_5c_1 \!+\! \frac{e^{b_5}-1}{b_5}u_5w_1 \!+\! b_2c_4 \!+\! \frac{e^{b_2}-1}{b_2}u_2w_4 \!-\! b_3c_6 \!+\! \frac{e^{-b_3}-1}{b_3}u_3w_6 \right)|$ \newline\null\hfill $x\colon\!c_1w_1u_2,\,y\colon\!u_3c_4w_4u_5c_6w_6\bigg) = \bbO\left(?|x\colon c_xu_xw_x,\,y\colon c_yu_yw_y\right)$ }}}} \def\ZeroGoal{{\raisebox{2mm}{\parbox[t]{3.95in}{ {\red Goal.} Write $?$ as a Gaussian: $\omega e^{L+Q}$ where $L$ bilinear in $b_i$ and $c_i$ with integer coefficients, $Q$ a balanced quadratic in $u_i$ and $w_i$ with coefficients in $R_S\coloneqq\bbQ(b_i,e^{b_i})$, and $\omega\in R_S$. }}}} \def\Big0Lemma{{\raisebox{3mm}{\begin{minipage}[t]{3.95in} {\red The Big $\frakg_0$ Lemma.} Under $[c,u]=u$, $[c,w]=-w$, and $[u,w]=b$: 1a. $N^{uc}\coloneqq\bbO(e^{\gamma c+\beta u}|uc) \overset{\to}{=} \bbO(e^{\gamma c+e^{-\gamma}\beta u}|cu)$ \hfill{\footnotesize (means $e^{\beta u}e^{\gamma c}=e^{\gamma c} e^{e^{-\gamma}\beta u}$} 1b. $N^{wc}\coloneqq\bbO(e^{\gamma c+\alpha w}|wc) \overset{\to}{=} \bbO(e^{\gamma c+e^{\gamma}\alpha w}|cw)$ \hfill{\footnotesize \ldots in the $\{ax+b\}$ group)} 2. $\bbO(e^{\alpha w+\beta u}|wu) = \bbO(e^{-b\alpha\beta+\alpha w+\beta u}|uw)$ \hfill{\footnotesize (the Weyl relations)} 3. $\bbO(e^{\delta uw}|wu)e^{\beta u} = e^{\nu\beta u}\bbO(e^{\delta uw}|wu)$, with \cbox{yellow}{$\nu=(1+b\delta)^{-1}$} \newline{\footnotesize (a. expand and crunch.\hfill b. use $w=b\hat{x}$, $u=\partial_x$. \hfill c. use ``scatter and glow''.)} 4. $\bbO(e^{\delta uw}|wu) = \bbO(\nu e^{\nu\delta uw}|uw)$ \hfill{\footnotesize (same techniques)} 5. $N^{wu}\coloneqq\bbO(e^{\beta u+\alpha w+\delta uw}|wu) \overset{\to}{=} \bbO(\nu e^{-b\nu\alpha\beta+\nu\alpha w+\nu\beta u+\nu\delta uw} |uw)$ 6. $N^{c_ic_j}_k\coloneqq\bbO(\zeta|c_ic_j)\overset{\to}{=}\bbO(\zeta/(c_i,c_j\to c_k)|c_k)$ {\red Sneaky.} $\alpha$ may contain (other) $u$'s, $\beta$ may contain (other) $w$'s. \end{minipage}}}} \def\ZeroStitching{{\raisebox{3mm}{\parbox[t]{3.95in}{ {\red Strand Stitching,} $m^{ij}_k$, is defined as the composition \begin{multline*} c_iu_i\ob{w_ic_j}u_jw_j \xrightarrow{N^{w_ic_j}_k} c_i\ob{u_ic_k}\ob{w_ku_j}w_j \xrightarrow{N^{u_ic_k}_k\act N^{w_ku_j}_k} \ob{c_ic_k}\ob{u_ku_k}\ob{w_kw_j} \\ \xrightarrow{N^{c_ic_k}_k\act --\act N^{w_kw_j}_k} c_ku_kw_k \end{multline*} }}}} \def\OneInvariants{{\raisebox{3mm}{\parbox[t]{4in}{ {\red 1-Smidgen Invariants.} Much is the same: {\red The Big $\frakg_1$ Lemma.} Parts 1 and 6 are the same, yet 5. \cbox{yellow}{$\bbO\left(e^{\alpha w+\beta u+\delta uw}|wu\right) = \bbO\left(\nu (1+\epsilon\nu\Lambda) e^{\nu(-b\alpha\beta+\alpha w+\beta u+\delta uw)}|cuw\right)$} Here $\Lambda$ is for {\greektext L'ogos}, ``a principle of order and knowledge'', a balanced quartic in $\alpha$, $\beta$, $c$, $u$, and $w$: \begin{align*} \Lambda = & - b\nu\left(\nu^2\alpha^2\beta^2+4\delta\nu\alpha\beta+2\delta^2\right)/2 - \delta\nu^3(3b\delta+2)\beta^2u^2/2 \\ & - b\delta^4\nu^3u^2w^2/2 - \delta^2\nu^3(2b\delta+1)\beta u^2w \\ & - \nu^2(2b\delta+1)(\nu\alpha\beta+2\delta)\beta u - 2b\delta^2\nu^2(\nu\alpha\beta+\delta)uw \\ & + \delta\nu^3(b\delta+2)\alpha^2w^2/2 + 2(\nu\alpha\beta+\delta)c + 2\delta\nu\beta cu + 2\delta^2\nu cuw \\ & + 2\delta\nu \alpha cw + \delta^2\nu^3\alpha uw^2 + \nu^2(\nu\alpha\beta+2\delta)\alpha w. \end{align*} {\red Proof.} A brutal hell. {\red Problem.} We now need to normal-order perturbed Gaussians! {\red Solution.} Borrow some tactics from QFT: \begin{multline*} \bbO(\epsilon P(c,u)e^{\gamma c+\beta u}|uc) = \bbO(\epsilon P(\partial_\gamma,\partial_\beta)e^{\gamma c+\beta u}|uc) = \\ \bbO(\epsilon P(\partial_\gamma,\partial_\beta)e^{\gamma c+e^{-\gamma}\beta u}|cu), \end{multline*} \vskip -4mm and likewise \newline\null\hfill\cbox{yellow}{$\displaystyle \bbO\left(\epsilon P(u,w)e^{\alpha w+\beta u+\delta uw}|wu\right) \!=\! \bbO\left(\epsilon P(\partial_\beta,\partial_\alpha)\nu e^{\nu(-b\alpha\beta+\alpha w+\beta u+\delta uw)}|cuw\right) $}\hfill\null \vskip 3pt {\red Note.} Strand stitching requires a tiny extra step. {\red Finally,} the values of the generators $\overcrossing$, $\undercrossing$, $\overrightarrow{n}$, $\overleftarrow{n}$, $\underrightarrow{u}$, and $\underleftarrow{u}$, are set by brutally solving many equations, non-uniquely. }}}} \def\Pragmatic{{\raisebox{3mm}{\parbox[t]{3.95in}{ {\red Pragmatic Simplifications.} Get rid of $\zeta=(e^b-1)/b$ factors by rescaling $u\to\bar{u}=\zeta u$. Complement this with $\beta\to\bar{\beta}=\zeta^{-1}\beta$, $\delta\to\bar{\delta}=\zeta^{-1}\delta$, $\epsilon\to\bar{\epsilon}=\zeta^{-1}\epsilon$. Simplify further by naming $e^b\to t$; e.g., $\nu\to\bar{\nu}=(1+(t-1)\delta)^{-1}$. Get confused by renaming $(\bar{u},\bar\beta,\bar\delta,\bar\nu)\to(u,\beta,\delta,\nu)$, and more confused by working with $\mu=\nu^{-1}$ and $\bbE(\omega,L,Q,P) \coloneqq \omega^{-1}(1+\epsilon\omega^{-4}P)e^{L+\omega^{-1}Q}$, where $\omega\in R\coloneqq \bbQ(t_k)$, $L=\sum l_{ij}b_ic_j$ with $l_{ij}\in\bbZ$, $Q=\sum q_{ij}u_iw_j$ with $q_{ij}\in R$, and $P$ is a balanced quartic polynomial in $c_i$, $u_i$, and $w_i$ with coefficients in $R$. Magically, all coefficients are now Laurent polynomials in the $t_k$'s. }}}} \def\Complexity{{\raisebox{15pt}{\begin{minipage}[t]{3.95in}{%\footnotesize \parpic[r]{$ \ub{n}_A \ob{ \sum\nolimits_{\scriptstyle d=0}^4 \ub{w^{4-d}}_E\ub{w^d}_F }^B \ub{n^2}_G = n^3w^4\in[n^5,n^7] $} {\red Rough complexity estimate,} after $t_k\to t$. $n$: xing number; $w$: width, maybe ${\sim\sqrt{n}}$. $A$: go over stitchings in order. $B$: multiplication ops per $N^{u_iw_j}$. $d$: deg of $u_i,w_j$ in $P$. $E$: $\#$terms of deg $d$ in $P$. $F$: ops per term. $G$: cost per polynomial multiplication op. }\end{minipage}}}} \def\MMG{\parbox{0.6in}{\scriptsize\raggedright Melvin, Morton, Garoufalidis }} \def\ExperimentalA{{\raisebox{3mm}{\parbox[t]{3.95in}{ {\red Experimental Analysis} (\web{Exp}). Log-log plot 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\ExperimentalB{{\raisebox{3mm}{\parbox[t]{3.95in}{ {\red Conjecture} (checked on the same collections). Given a knot $K$ with Alexander polynomial $A$, there is a polynomial $\rho_1$ such that \[ P = A^2\left((t-2+t^{-1})\rho_1 + t AA'\left( \frac{(4+t-t^2)(uw+(t-1)c)}{2(t-1)}-1 \right)\right). \] Furthermore, $A$ and $\rho_1$ are symmetric under $t\to t^{-1}$, so let $A^+$ and $\rho_1^+$ be their ``positive parts'', so e.g., $\rho_1(t)=\rho_1^+(t)+\rho_1^+(t^{-1})-\rho_1^+(0)$. {\red Power.} On the 250 knots with at most 10 crossings, the pair $(A,\rho_1)$ 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). }}}} \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{PPSA1.pdftex_t}\vfill\null\eject \null\vfill\input{PPSA2.pdftex_t}\vfill\null\eject \end{center} \eject \newgeometry{textwidth=8in,textheight=10.5in} \def\cellscale{0.65} \begin{multicols}{2} \raggedcolumns %\noindent\includegraphics[width=0.5\linewidth]{../UNC-1610/To12Times.png}% % \includegraphics[width=0.5\linewidth]{../UNC-1610/TKTimes.png} {\red Genus.} Up to 12 crossings, 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-crossing Alexander failures it does give the right answer. {\large\red Demo Programs for 0-Co.}\hfill\web{Demo} \vskip -2mm \begin{mdframed}[hidealllines=true,backgroundcolor=yellow!20,innerleftmargin=3pt,innerrightmargin=3pt,leftmargin=-3pt,rightmargin=-3pt] \hfill{\red The $R$-matrices} \vskip -3mm\includegraphics[scale=\cellscale]{Snips/0R.pdf} \end{mdframed} \vskip -2mm \vskip -2mm \begin{mdframed}[hidealllines=true,backgroundcolor=green!10,innerleftmargin=3pt,innerrightmargin=3pt,leftmargin=-3pt,rightmargin=-3pt] \hfill{\red Utilities} \vskip -3mm\includegraphics[scale=\cellscale]{Snips/0Util.pdf} \hfill{\red Normal Ordering Operators} \vskip -3mm\includegraphics[scale=\cellscale]{Snips/0NO.pdf} \hfill{\red Stitching} \vskip -3mm\includegraphics[scale=\cellscale]{Snips/0m.pdf} \end{mdframed} \vskip -2mm \hfill{\red Some calculations for $T_0$} \vskip -3mm\includegraphics[scale=\cellscale]{Snips/T0.pdf} \includegraphics[scale=\cellscale]{Snips/ZT0.pdf} \hfill{\red Verifying meta-associativity} \includegraphics[scale=\cellscale]{Snips/0Q0.pdf} %\includegraphics[scale=\cellscale]{Snips/0NODemo.pdf} %\includegraphics[scale=\cellscale]{Snips/0mDemo.pdf} \includegraphics[scale=\cellscale]{Snips/0MetaAssoc.pdf} \Needspace{2cm} \hfill{\red Testing R3} \vskip -3mm\includegraphics[scale=\cellscale]{Snips/0R3Left.pdf} \includegraphics[scale=\cellscale]{Snips/0R3.pdf} \hfill{\red $8_{17}$} \vskip -4mm \hfill\resizebox{!}{38mm}{\subimport{../LesDiablerets-1608/}{figs/817.pdf_t}}\hfill\null \includegraphics[scale=\cellscale]{Snips/0817.pdf} {\large\red Demo Programs for 1-Co.}\hfill\web{Demo} \vskip -2mm \begin{mdframed}[hidealllines=true,backgroundcolor=yellow!20,innerleftmargin=3pt,innerrightmargin=3pt,leftmargin=-3pt,rightmargin=-3pt] \includegraphics[scale=\cellscale]{Snips/Logos.pdf} \vskip -6mm \hfill{\red The {\greektext L'ogos}} \end{mdframed} \vskip -2mm \vskip -2mm \begin{mdframed}[hidealllines=true,backgroundcolor=yellow!20,innerleftmargin=3pt,innerrightmargin=3pt,leftmargin=-3pt,rightmargin=-3pt] \hfill{\red The Generators} \vskip -3mm\includegraphics[scale=\cellscale]{Snips/1Gens.pdf} \end{mdframed} \vskip -2mm \begin{mdframed}[hidealllines=true,backgroundcolor=green!10,innerleftmargin=3pt,innerrightmargin=3pt,leftmargin=-3pt,rightmargin=-3pt] \hfill{\red Differential Polynomials} \includegraphics[scale=\cellscale]{Snips/1DP.pdf} \hfill{\red Utilities} \vskip -3mm\includegraphics[scale=\cellscale]{Snips/1Util.pdf} \hfill{\red Normal Ordering Operators} \includegraphics[scale=\cellscale]{Snips/1NOc.pdf} \includegraphics[scale=\cellscale]{Snips/1NOuw.pdf} \hfill{\red Stitching} \vskip -3mm\includegraphics[scale=\cellscale]{Snips/1m.pdf} \end{mdframed} \vskip -2mm \parpic[r]{\resizebox{0.94in}{!}{\input{figs/Trefoil.pdf_t}}} \picskip{1} \includegraphics[scale=\cellscale]{Snips/131.pdf} %\vfill\rule{\linewidth}{1pt}%\vspace{-1mm} %\includegraphics[height=0.6in]{../../Projects/Gallery/Kronecker.jpg} %\hfill %\parbox[b]{2.7in}{ % ``God created the knots; all else in % \newline topology is the work of mortals.'' % \vskip 1.5mm % {\footnotesize Leopold Kronecker (modified)\hfill\href{http://katlas.org}{katlas.org}} %} %\hfill %\includegraphics[height=0.6in]{../../Projects/Gallery/The_Knot_Atlas.png} {\red Questions and To Do List.} $\bullet$~Clean up and write up. $\bullet$~Implement well, compute for everything in sight. $\bullet$~Why are our quantities polynomials rather than just rational functions? $\bullet$~Bounds on their degrees? $\bullet$~Their integrality ($\bbZ$) properties? $\bullet$~Can everything be re-stated using integrals ($\int$)? $\bullet$~Find the 2-variable version (for knots). How complex is it? $\bullet$~What about links / closed components? $\bullet$~Fully digest the ``expansion'' theorem; include cuaps. $\bullet$~Explore the \text{(non-)dependence} on $R$. $\bullet$~Is there a canonical $R$? $\bullet$~What does ``group like'' mean? $\bullet$~Strand removal? Strand doubling? Strand reversal? $\bullet$~Say something about knot genus. $\bullet$~Find the EK/AT/KV ``vertex''. $\bullet$~Use as a playground to study associators/braidors. $\bullet$~Restate in topological language. $\bullet$~Study the associated (v-)braid representations. $\bullet$~Study mirror images and the $\frakb^+\leftrightarrow\frakb^-$ involution. $\bullet$~Study ribbon knots. $\bullet$~Make precise the relationship with $\Gamma$-calculus and Alexander. $\bullet$~Relate to the coloured Jones polynomial. $\bullet$~Relate with ``ordinary'' $q$-algebra. $\bullet$~$k$-smidgen $sl_n$, etc. $\bullet$~Are there ``solvable'' CYBE algebras not arising from semi-simple algebras? $\bullet$~Categorify and appease the Gods. \hfill\text{\red Help Needed!} {\red Disclaimer.} This is all quite new. The overall picture is correct, but many pieces are certainly not in their final form yet. \columnbreak {\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} %\eject \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 $A_+$ \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=24.5pt]{../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 \subimport{../UNC-1610/}{table.tex} \end{longtable}} \end{document} \endinput