\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, datetime} \usepackage[export]{adjustbox} % Follows https://tex.stackexchange.com/questions/6073/scale-resize-large-images-graphics-that-exceed-page-margins \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{soul} % for strikeouts, \st. \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{Brisbane-250616} \def\title{The Strongest Genuinely Computable Knot Invariant in 2024} \def\navigator{{ \href{\myurl}{Dror Bar-Natan}: \href{\myurl/Talks}{Talks}: \href{\myurl/Talks/\thistalk/}{\thistalk}: }} \def\thanks{{Thanks for allowing me in Brisbane!}} \def\webdef{{{\greektext web}$\coloneqq$\href{http://drorbn.net/b25}{http://drorbn.net/b25}}} \def\web#1{{\href{\myurl/Talks/\thistalk/#1}{{\greektext web}/#1}}} \def\titleA{{\title}} \def\titleB{{\title}} \def\titleC{{\title}} \definecolor{mblue}{HTML}{E0E0FF} \definecolor{mgray}{HTML}{B0B0B0} \definecolor{morange}{HTML}{FFA50A} \definecolor{mpink}{HTML}{FFE0E0} \definecolor{myellow}{HTML}{FFFF00} \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\myellowm#1{{\setlength{\fboxsep}{0pt}\colorbox{myellow}{$#1$}}} \def\mpinkm#1{{\setlength{\fboxsep}{0pt}\colorbox{mpink}{$#1$}}} \def\mbluem#1{{\setlength{\fboxsep}{0pt}\colorbox{mblue}{$#1$}}} \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{1.2mm}{\parbox[t]{3.95in}{ \parshape 6 0in 3.35in 0in 3.35in 0in 3.35in 0in 3.35in 0in 3.35in 0in 3.95in {\red\bf Abstract.} ``Genuinely computable'' means we have computed it for random knots with over 300 crossings. ``Strongest'' means it separates prime knots with up to 15 crossings better than the less-computable HOMFLY-PT and Khovanov homology taken together. And hey, it's also meaningful and fun. Continues Rozansky, Garoufalidis, Kricker, and Ohtsuki, joint with van der Veen. \footnotesize {\bf\red Acknowledgement.} This work was supported by NSERC grants RGPIN-2018-04350 and RGPIN-2025-06718 and by the Chu Family Foundation (NYC). }}}} \def\Knots{{\raisebox{0.6mm}{\parbox[t]{3.95in}{ Tell them apart? Alternating? Bound a genus 7 surface? Complement is fibered over $S^1$? Complement is hyperbolic? Bounds a disk with only ribbon singularities? Bounds a topological / smooth non-singular disk in $B^4$? $\ldots$ }}}} \def\Strongest{{\raisebox{2mm}{\parbox[t]{3.95in}{ {\red\bf Strongest.} Testing $\Theta=(\Delta,\theta)$ on prime knots up to mirrors and reversals, counting the number of distinct values (with deficits in parenthesis): \hfill{\footnotesize ($\rho_1$: \cite{Ro, Rozansky:Burau, Rozansky:U1RCC, Overbay:Thesis, APAI}) } \centering\small\begin{tabular}[t]{c|c|c|c|c|c} & knots & $(H,Kh)$ & $(\Delta,\rho_1)$ & $\Theta=(\Delta,\theta)$ & together \\ \hline reign & & 2005-22 & 2022-24 & 2024- \\ \hline\hline xing $\leq 10$ & 249 & 248 (1) & 249 (0) & 249 (0) & 249 (0) \\ \hline xing $\leq 11$ & 801 & 771 (30) & 787 (14) & 798 (3) & 798 (3) \\ \hline xing $\leq 12$ & 2,977 & (214) & (95) & (19) & (18) \\ \hline xing $\leq 13$ & 12,965 & (1,771) & (959) & (194) & (185) \\ \hline xing $\leq 14$ & 59,937 & (10,788) & (6,253) & (1,118) & (1,062) \\ \hline xing $\leq 15$ & 313,230 & (70,245) & (42,914) & (6,758) & (6,555) \\ \end{tabular} }}}} \def\GC{{\raisebox{2mm}{\parbox[t]{3.95in}{ \parshape 1 0in 2.3in {\red\bf Genuinely Computable.} Here's $\Theta$ on a random 300 crossing knot (from \cite{DHOEBL:Random}). For almost every other invariant, that's science fiction. \parshape 6 0in 2.3in 0in 2.3in 0in 2.3in 0in 2.3in 0in 2.3in 0in \linewidth {\red\bf Fun.} There's so much more to see in 2D pictures than in 1D ones! Yet almost nothing of the patterns you see we know how to prove. We'll have fun with that over the next few years. Would you join? {\bf\red Meaningful.} $\theta$ gives a genus bound (unproven yet with confidence). We hope (with reason) it says something about ribbon knots. {\bf\red Conventions.} $T$, $T_1$, and $T_2$ are indeterminates and $T_3\coloneqq T_1T_2$. }}}} \def\Preparation{{\raisebox{2mm}{\parbox[t]{3in}{ {\red\bf Preparation.} Draw an $n$-crossing knot $K$ as a diagram $D$ 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 $\varphi_k$. }}}} \def\TrafficRules{{\raisebox{2mm}{\parbox[t]{3in}{ \parshape 4 0in 3in 0in 3in 0in 3in 0.72in 2.28in {\bf\red Model $T$ Traffic Rules.} Cars always drive forward. When a car crosses over a sign-$s$ bridge it goes through with (algebraic) probability $T^s\sim 1$, but falls off with probability $1-T^s\sim 0$. At the very end, cars fall off and disappear. On various edges {\em traffic counters} are placed. See also~\cite{Jones:Hecke, LinTianWang:RandomWalk}. }}}} \def\dtA{{\tiny image credits:}} \def\dtB{{\tiny \href{https://diamondtraffic.com/productcategory/Portable-Counters}{diamondtraffic.com}}} \def\DallEA{{\tiny image credits:}} \def\DallEB{{\tiny \href{https://labs.openai.com/}{Dall-E}}} \def\gab{{\raisebox{2mm}{\parbox[t]{3.95in}{ \parshape 4 0in 2.9in 0in 2.9in 0in 2.9in 0in 3.95in {\bf\red Definition.} The {\em traffic function} $G=(g_{\alpha\beta})$ (also, the {\em Green function} or the {\em two-point function}) 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). There are also model-$T_\nu$ traffic functions $G_\nu=(g_{\nu\alpha\beta})$ for $\nu=1,2,3$. \hfill{\bf\red Example.} }}}} \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\Theorem{{\raisebox{2mm}{\parbox[t]{3.95in}{ \parshape 4 0in 2.875in 0in 2.875in 0in 2.875in 0in 3.95in {\bf\red Theorem} \cite{Theta}. With $c=(s,i,j)$, $c_0=(s_0,i_0,j_0)$, and $c_1=(s_1,i_1,j_1)$ denoting crossings, there is a quadratic $F_1(c)\in\bbQ(T_\nu)[g_{\nu\alpha\beta}:\alpha,\beta\in\{i,j\}]$, a cubic $F_2(c_0,c_1) \in \bbQ(T_\nu)[g_{\nu\alpha\beta}:\alpha,\beta\in\{i_0,j_0,i_1,j_1\}]$, and a linear $F_3(\varphi,k)$ such that $\theta$ is a knot invariant: \[ \theta(D) \coloneqq \underbrace{\Delta_1\Delta_2\Delta_3}_{\parbox{0.66in}{\scriptsize\centering normalization, see later }} \left(\sum_c F_1(c) + \sum_{c_0,c_1} F_2(c_0,c_1) + \sum_kF_3(\varphi_k,k)\right), \] \vskip 22mm If these pictures remind you of Feynman diagrams, it's because they are Feynman diagrams~\cite{IType}. }}}} \def\LemmaA{{\raisebox{2mm}{\parbox[t]{3.95in}{ {\bf\red Lemma 1.} The traffic function $g_{\alpha\beta}$ is a ``relative invariant'': }}}} \def\egA{{e.g. $g_{2ii} g_{3jj}$}} \def\egB{{e.g. $g_{3j_0i_1}g_{1j_1i_0}g_{2i_1i_0}$}} \def\egC{{e.g. $g_{3kk}$}} \def\messA{{$(1\!-\!T)^2\!+\!T(1\!-\!T)$}} \def\messB{{$(1\!-\!T)T$}} \def\messC{{$T(1\!-\!T)$}} \def\messD{{$1\!-\!T$}} \def\i{{$i$}} \def\j{{$j$}} \def\k{{$k$}} \def\m{{$m$}} \def\n{{$n$}} \def\s{{$s$}} \def\ip{{$i^+$}} \def\jp{{$j^+$}} \def\kp{{$k^+$}} \def\ipp{{$i^{+\!+}$}} \def\jpp{{$j^{+\!+}$}} \def\kpp{{$k^{+\!+}$}} \def\LemmaB{{\raisebox{2mm}{\parbox[t]{3.95in}{ \parshape 3 0in 3in 0in 3in 0in 3.95in {\bf\red Lemma 2.} With $k^+\coloneqq k+1$, the ``$g$-rules'' hold near a crossing $c=(s,i,j)$: \[ g_{j\beta} = g_{j^+\beta} + \delta_{j\beta} \quad g_{i\beta} = T^sg_{i^+\beta} + (1-T^s)g_{j^+\beta} + \delta_{i\beta} \quad g_{2n^+,\beta} = \delta_{2n^+,\beta} \] \[ g_{\alpha i^+} = T^sg_{\alpha i} + \delta_{\alpha i^+} \quad g_{\alpha j^+} = g_{\alpha j} + (1-T^s)g_{\alpha i} + \delta_{\alpha j^+} \quad g_{\alpha,1} = \delta_{\alpha,1} %\quad g_{\alpha,2n^+} = 1 \] {\bf\red Corollary 1.} $G$ is easily computable, for $AG=I$ ($=GA$), with $A$ the $(2n+1)\times(2n+1)$ identity matrix with additional contributions: \newline\null\hfill$ c=(s,i,j) \mapsto \begin{array}{c|cccc} A & \text{col }i^+ & \text{col }j^+ \\ \hline \text{row }i & -T^s & T^s-1 \\ \text{row }j & 0 & -1 \end{array} $ \vskip -2mm For the trefoil example, we have: \[ A=\left( \begin{array}{ccccccc} 1 & \mbluem{-T} & 0 & 0 & \mbluem{T-1} & 0 & 0 \\ 0 & 1 & \mpinkm{-1} & 0 & 0 & \mpinkm{\ 0\ } & 0 \\ 0 & 0 & 1 & \myellowm{-T} & 0 & 0 & \myellowm{T-1} \\ 0 & \mbluem{\ 0\ } & 0 & 1 & \mbluem{-1} & 0 & 0 \\ 0 & 0 & \mpinkm{T-1} & 0 & 1 & \mpinkm{-T} & 0 \\ 0 & 0 & 0 & \myellowm{\ 0\ } & 0 & 1 & \myellowm{-1} \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 \\ \end{array} \right), \] \[ G=\left( \begin{array}{ccccccc} 1 & T & 1 & T & 1 & T & 1 \\ 0 & 1 & \frac{1}{T^2-T+1} & \frac{T}{T^2-T+1} & \frac{T}{T^2-T+1} & \frac{T^2}{T^2-T+1} & 1 \\ 0 & 0 & \frac{1}{T^2-T+1} & \frac{T}{T^2-T+1} & \frac{T}{T^2-T+1} & \frac{T^2}{T^2-T+1} & 1 \\ 0 & 0 & \frac{1-T}{T^2-T+1} & \frac{1}{T^2-T+1} & \frac{1}{T^2-T+1} & \frac{T}{T^2-T+1} & 1 \\ 0 & 0 & \frac{1-T}{T^2-T+1} & -\frac{(T-1) T}{T^2-T+1} & \frac{1}{T^2-T+1} & \frac{T}{T^2-T+1} & 1 \\ 0 & 0 & 0 & 0 & 0 & 1 & 1 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 \\ \end{array} \right) \] }}}} \def\Alexander{{\raisebox{2mm}{\parbox[t]{3.95in}{ {\bf\red Note.} The Alexander polynomial $\Delta$ is given by \newline\null \hfill$\Delta = T^{(-\varphi-w)/2}\det(A)$, \hfill with $\varphi = \sum_k \varphi_k$, $w = \sum_c s$. \hfill\null \newline We also set $\Delta_\nu\coloneqq\Delta(T_\nu)$ for $\nu=1,2,3$. }}}} \def\Conjectures{{\raisebox{2mm}{\parbox[t]{3.95in}{ {\red\bf Questions, Conjectures, Expectations, Dreams.} {\bf Question 1.} What's the relationship between $\Theta$ and the Garoufalidis-Kashaev invariants \cite{GaroufalidisKashaev:Multivariable, GaroufalidisLi:Patterns}? {\bf Conjecture 2.} On classical (non-virtual) knots, $\theta$ always has hexagonal ($D_6$) symmetry. {\bf Conjecture 3.} $\theta$ is the $\epsilon^1$ contribution to the ``solvable approximation'' of the $sl_3$ universal invariant, obtained by running the quantization machinery on the double $\calD(\frakb,b,\eps\delta)$, where $\frakb$ is the Borel subalgebra of $sl_3$, $b$ is the bracket of $\frakb$, and $\delta$ the cobracket. See \cite{PG, DPG, Schaveling:Thesis} {\bf Conjecture 4.} $\theta$ is equal to the ``two-loop contribution to the Kontsevich Integral'', as studied by Garoufalidis, Rozansky, Kricker, and in great detail by Ohtsuki \cite{GaroufalidisRozansky:LoopExpansion, Ro, Rozansky:Burau, Rozansky:U1RCC, Kricker:Lines, Ohtsuki:TwoLoop}. {\bf Fact 5.} $\theta$ has a perturbed Gaussian integral formula, with integration carried out over over a space $6E$, consisting of 6 copies of the space of edges of a knot diagram $D$. See \cite{IType}. {\bf Conjecture 6.} For any knot $K$, its genus $g(K)$ is bounded by the $T_1$-degree of $\theta$: $2g(K) \geq \deg_{T_1}\theta(K)$. {\bf Conjecture 7.} $\theta(K)$ has another perturbed Gaussian integral formula, with integration carried out over over the space $6H_1$, consisting of 6 copies of $H_1(\Sigma)$, where $\Sigma$ is a Seifert surface for $K$. {\bf Expectation 8.} There are many further invariants like $\theta$, given by Green function formulas and/or Gaussian integration formulas. One or two of them may be stronger than $\theta$ and as computable. \parshape 1 0in 3in {\bf Dream 9.} These invariants can be explained by something less foreign than semisimple Lie algebras. \vskip 3mm \parshape 1 0in 1.65in {\bf Dream 10.} With Conjecture 7 in mind, $\theta$ will have something to say about ribbon knots. }}}} \def\refs{{\raisebox{4mm}{\resizebox{3.95in}{!}{\parbox[t]{4.666666in}{ %{\red\bf References.} {\footnotesize \def\bysame{{---}} %\par\vspace{-3mm} \renewcommand{\section}[2]{}% \begin{thebibliography}{} \setlength{\parskip}{0pt} \setlength{\itemsep}{0pt plus 0.3ex} \import{../PhuQuoc-2506}{refs.tex} \end{thebibliography}} }}}}} \def\In{\rlap{\protect\makebox[-4mm]{\smiley}}} \def\Out{\rlap{\protect\makebox[-4mm]{\footnotesize\faLaptop}}} \def\nbpdfInput#1{\vskip 1mm\par\noindent\In\includegraphics[valign=t,max width=\linewidth]{#1}} \def\nbpdfEcho#1{\vskip 1mm\par\noindent\Out\includegraphics[valign=t]{#1}} %\def\nbpdfPrint#1{\vskip 1mm\par\noindent\Out\includegraphics[valign=t]{#1}} \def\nbpdfPrint#1{} \def\nbpdfText#1{\vskip 1mm\par\noindent\includegraphics[valign=t]{#1}} \def\nbpdfMessage#1{} \def\nbpdfOutput#1{\vskip 1mm\par\noindent\Out\includegraphics[valign=t,max width=\linewidth]{#1}} \def\nbpdfSubsection#1{\vskip 1mm\par\noindent\includegraphics[valign=t]{#1}} \def\nbpdfSubsubsection#1{\vskip 1mm\par\noindent\includegraphics[valign=t]{#1}} \def\nbpdfgraphInput#1{\vskip 1mm\par\noindent\includegraphics[valign=t]{#1}} \def\nbpdfgraphOutput#1{\vskip 1mm\par\noindent\Out\includegraphics[width=1.5in valign=t]{#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{SGC24T1.pdftex_t}\vfill\null \end{center} \begin{center} \null\vfill\import{../PhuQuoc-2506}{SGC24T2.pdftex_t}\vfill\null \end{center} \newgeometry{textwidth=8in,textheight=10.5in} \begin{multicols*}{2} {\bf\red Corollary 2.} Proving invariance is easy: \resizebox{\linewidth}{!}{\import{../Toronto-241030}{R3.pdftex_t}} \import{../PhuQuoc-2506}{Theta.tex} \end{multicols*} \newpage \newgeometry{textwidth=8in,textheight=10.5in} The 132-crossing torus knot $T_{22/7}$:\hfill(many more at \web{TK}) \[ \includegraphics[width=0.9\linewidth]{../Toronto-241030/T227Plot.pdf} \] Random knots from \cite{DHOEBL:Random}, with 50-73 crossings:\hfill(many more at \web{DK}) \[ \includegraphics[width=0.95\linewidth]{../UBC-241004/Gallery50-73.png} \] \end{document} \endinput