\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, amsthm} \usepackage[normalem]{ulem} % For \sout. \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{LesDiablerets-2601} \def\title{Homework} \def\navigator{{ \href{\myurl}{Dror Bar-Natan}: \href{\myurl/Talks}{Talks}: \href{\myurl/Talks/\thistalk/}{\thistalk}: }} \def\thanks{{OMG, thanks!}} \def\webdef{{{\greektext web}$\coloneqq$\href{http://drorbn.net/ld26}{http://drorbn.net/ld26}}} \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\magenta{\color{magenta}} \def\mgray{\color{mgray}} \def\morange{\color{morange}} \def\pink{\color{pink}} \def\purple{\color{purple}} \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\Kh{\text{\it Kh}} \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\Vol{\text{\it Vol}} \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\calB{{\mathcal B}} \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}} \def\ip{{i^+}} \def\jp{{j^+}} \def\kp{{k^+}} \def\ipp{{i^{+\!+}}} \def\jpp{{j^{+\!+}}} \def\kpp{{k^{+\!+}}} % \Gint from https://tex.stackexchange.com/questions/171415/superimpose-letter-on-integral-symbol \makeatletter \let\DOTSI\relax % amsmath support for \dots \newcommand*{\Gint}{% \DOTSI \mathop{% \mathpalette\@LetterOnInt{G}% }% \mkern-\thinmuskip % thin space is inserted between two \mathop \int } \newcommand*{\@LetterOnInt}[2]{% \sbox0{$#1\int\m@th$}% \sbox2{$% \ifx#1\displaystyle \textstyle \else \scriptscriptstyle \fi #2% \m@th$}% \dimen@=.4\dimexpr\ht0+\dp0\relax \ifdim\dimexpr\ht2+\dp2\relax>\dimen@ \sbox2{\resizebox*{!}{\dimen@}{\unhcopy2}}% \fi \dimen@=\wd0 % \ifdim\wd2>\dimen@ \dimen@=\wd2 % \fi \rlap{\hbox to \dimen@{\hfil $#1\vcenter{\copy2}\m@th$% \hfil}}% \ifdim\dimen@>\wd0 % \kern.5\dimexpr\dimen@-\wd0\relax \fi } \makeatother % From http://tex.stackexchange.com/questions/154672/how-to-get-a-medium-sized-otimes \DeclareMathOperator*{\midotimes}{\text{\raisebox{0.25ex}{\scalebox{0.8}{$\bigotimes$}}}} \newtheoremstyle{handoutstyle} % Modified from % https://tex.stackexchange.com/questions/37472/spacing-before-and-after-with-newtheoremstyle {0pt} % Space above {0pt} % Space below {\it} % Body font {} % Indent amount {\red\bf} % Theorem head font {.} % Punctuation after theorem head {.5em} % Space after theorem head {\text{\thmname{#1}\thmnumber{ #2}}} % Theorem head spec (can be left empty, meaning `normal') \theoremstyle{handoutstyle} \newtheorem{task}{Task} \def\TaskSeparator{{\vspace{-3mm}\rule{\linewidth}{1pt}\vspace{-1mm}}} %%% \def\Abstract{{\raisebox{1.2mm}{\parbox[t]{3.95in}{ \parshape 1 0in 3.3in {\red\bf Abstract.} I'll start with a review of my recent paper with van der Veen, ``A {\purple Fast}, {\purple Strong}, {\purple Topologically Meaningful}, and {\purple Fun} Knot Invariant'' \cite{Theta}, and then assign some homework. Much of what I'll say follows earlier work of Rozansky, Kricker, Garoufalidis, and Ohtsuki \cite{Rozansky:Contribution, Rozansky:Burau, Rozansky:U1RCC, Kricker:Lines, GaroufalidisRozansky:LoopExpansion, Ohtsuki:TwoLoop}. \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\A{{\raisebox{1.7mm}{\parbox[t]{3.95in}{ {\red $A$.} With $T$ an indeterminate, start from a presentation matrix $A$ for the Alexander module of $K$, coming from the Wirtinger presentation of $\pi_1(K)$: $A\coloneqq I_{2n+1}+\sum_c A_c$, where \vskip -4mm \parshape 1 0in 3.125in \[ \begin{array}{c}\input{figs/Xings.pdf_t}\end{array} \ \rightarrow\ \begin{array}{c|cccc} A_c & i+1 & j+1 \\ \hline i & -T^s & T^s-1 \\ j & 0 & -1 \end{array} \] \[ A=\left( \begin{array}{ccccccc} 1 & \mbluem{-T} & 0 & 0 & \mbluem{T-1} & 0 & 0 \\ 0 & 1 & \mpinkm{-1} & 0 & 0 & \mpinkm{\ 0\ } & 0 \\ 0 & 0 & 1 & \myellowm{-T} & 0 & 0 & \myellowm{T-1} \\ 0 & \mbluem{\ 0\ } & 0 & 1 & \mbluem{-1} & 0 & 0 \\ 0 & 0 & \mpinkm{T-1} & 0 & 1 & \mpinkm{-T} & 0 \\ 0 & 0 & 0 & \myellowm{\ 0\ } & 0 & 1 & \myellowm{-1} \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 \\ \end{array} \right) \] }}}} \def\G{{\raisebox{2mm}{\parbox[t]{3.95in}{ {\red $G$.} Let $G = (g_{\alpha\beta}) \coloneqq A^{-1}$, the ``two point function'': $G=\left( \arraycolsep=3.5pt \begin{array}{ccccccc} 1 & T & 1 & T & 1 & T & 1 \\ 0 & 1 & \frac{1}{T^2-T+1} & \frac{T}{T^2-T+1} & \frac{T}{T^2-T+1} & \frac{T^2}{T^2-T+1} & 1 \\ 0 & 0 & \frac{1}{T^2-T+1} & \frac{T}{T^2-T+1} & \frac{T}{T^2-T+1} & \frac{T^2}{T^2-T+1} & 1 \\ 0 & 0 & \frac{1-T}{T^2-T+1} & \frac{1}{T^2-T+1} & \frac{1}{T^2-T+1} & \frac{T}{T^2-T+1} & 1 \\ 0 & 0 & \frac{1-T}{T^2-T+1} & -\frac{(T-1) T}{T^2-T+1} & \frac{1}{T^2-T+1} & \frac{T}{T^2-T+1} & 1 \\ 0 & 0 & 0 & 0 & 0 & 1 & 1 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 \\ \end{array} \right)$ Let $T_1$ and $T_2$ be new indeteminates, let $T_3=T_1T_2$, and let $G_\nu=(g_{\nu\alpha\beta})$ be $G$ with $T\to T_\nu$, for $\nu=1,2,3$. }}}} \def\thet{{\raisebox{3mm}{\parbox[t]{2.25in}{ \[ {\red\theta} \sim \Delta_1\Delta_2\Delta_3\sum_{c_0,c_1}g_{1i_0i_1}g_{2i_0i_1}g_{3i_1i_0} + \text{l.o.} \] \[ {\red\Theta} = (\Delta,\theta)\in\bbZ[T^{\pm1}]\times\bbZ[T_1^{\pm1},T_2^{\pm1}] \] }}}} \def\random{{\raisebox{1mm}{\parbox[t]{1.8in}{ A random 300 xing knot from \cite{DHOEBL:Random}. For most invariants, 300 is science fiction. }}}} \def\Strong{{\raisebox{2mm}{\parbox[t]{4in}{ {\bf\purple Strong.} $\Theta$ vs.\ a slew of other reasonably-computable invariants (deficits shown): \!\resizebox{4.02in}{!}{\def\s{$\sim$} \begin{tabular}{c|c|c|c|c|c|c} \hline $n$& $\leq 10$& $\leq 11$& $\leq 12$& $\leq 13$& $\leq 14$& $\leq 15$ \\ \hline knots& 249& 801& 2,977& 12,965& 59,937& 313,230 \\ \hline $\Delta$& (38)& (250)& (1,204)& (7,326)& (39,741)& (236,326) \\ \hline $\sigma_{LT}$& (108)& (356)& (1,525)& (7,736)& (40,101)& (230,592) \\ \hline $J$& (7)& (70)& (482)& (3,434)& (21,250)& (138,591) \\ \hline $\Kh$& (6)& (65)& (452)& (3,226)& (19,754)& (127,261) \\ \hline $H$& (2)& (31)& (222)& (1,839)& (11,251)& (73,892) \\ \hline $\Vol$& (\s6)& (\s25)& (\s113)& (\s1,012)& (\s6,353)& (\s43,607) \\ \hline $(\Kh,H,\Vol)$& (\s0)& (\s14)& (\s84)& (\s911)& (\s5,917)& (\s41,434) \\ \hline $(\Delta,\rho_1)$& (0)& (14)& (95)& (959)& (6,253)& (42,914) \\ \hline $(\Delta,\rho_1,\rho_2)$& (0)& (14)& (84)& (911)& (5,926)& (41,469) \\ \hline $(\rho_1,\rho_2,\Kh,H,\Vol)$& (0)& (\s14)& (\s84)& (\s911)& (\s5,916)& (\s41,432) \\ \hline \rowcolor{yellow} $\Theta$& (0)& (3)& (19)& (194)& (1,118)& (6,758) \\ \hline $(\Theta,\rho_2)$& (0)& (3)& (10)& (169)& (982)& (6,341) \\ \hline $(\Theta,\sigma_{LT})$& (0)& (3)& (19)& (194)& (1,118)& (6,758) \\ \hline $(\Theta,\Kh)$& (0)& (3)& (18)& (185)& (1,062)& (6,555) \\ \hline $(\Theta,H)$& (0)& (3)& (18)& (185)& (1,064)& (6,563) \\ \hline $(\Theta,\Vol)$& (0)& (\s3)& (\s10)& (\s169)& (\s973)& (\s6,308) \\ \hline $(\Theta,\rho_2,\Kh,H,\Vol)$& (0)& (\s3)& (\s10)& (\s169)& (\s972)& (\s6,304) \\ \hline \end{tabular}} }}}} \def\TopMean{{\raisebox{2mm}{\parbox[t]{4in}{ {\bf\purple Topologically Meaningful.} $\theta$ is near $\Delta$ and we dream that anything $\Delta$ can do, $\theta$ does too (sometimes better). The following two conjectures are verified for knots with $\leq 13$ crossings: \par{\bf\red Conjecture 1.} $\deg_{T_1}\theta(K)\leq 2g(K)$. \par{\bf\red Conjecture 2.} If $K$ is a fibered knot and $d$ is the degree of $\Delta(K)$ (the highest power of $T$), then the coefficient of $T_2^{2d}$ in $\theta(K)$, which is a polynomial in $T_1$, is an integer multiple of $T_1^d\Delta(K)|_{T\to T_1}$. \par{\bf\red Dream.} $\theta$ has something to say about ribbon knots. }}}} \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{HWT1.pdftex_t}\vfill\null \end{center} \newpage \newgeometry{textwidth=8in,textheight=10.5in} \parbox[t]{5.45in}{ The 132-crossing torus knot $T_{22/7}$:\hfill(many more at \web{TK}) \newline\includegraphics[width=\linewidth]{../Toronto-241030/T227Plot.pdf} } \hfill \parbox[t]{2.45in}{ Random knots from \cite{DHOEBL:Random} with 51 -- 75 crossings: (many more at \web{DK}) \newline \vskip -7mm\includegraphics[height=\linewidth,angle=-90]{figs/Beehive.pdf} } %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 51 -- 68 %crossings:\hfill(many more at \web{DK}) % %\[ \includegraphics[width=0.95\linewidth]{figs/Beehive.pdf} \] \vspace{-21mm} {\bf\red Moral.} We\newline must figure out\newline all that we don't\newline yet know about $\Theta$! \vspace{-2mm} %\parshape 4 0in 0.75in 0in 0.85in 0in 0.95in 0in 1.05in %{\bf\red Moral.} We must figure out all that we don't yet know about $\Theta$! %{\bf\red Moral.} We\newline must come to\newline terms with $\Theta$! \begin{multicols}{2} \begin{task} Make the ``data'' formulas human friendly. \end{task} \TaskSeparator \begin{task} Prove the hexagonal symmetry of $\theta(K)$, and that $\theta(K) = \theta(-K) = -\theta(\bar{K})$. \end{task} That's harder than it seems! The formulas don't naively show any of that. $\Delta$ has a palindromic symmetry first conjectured in Alexander's original paper~\cite{Alexander:TopologicalInvariants} --- it is invariant under $T\to T^{-1}$. Proving this took a few years, and the proof starting from the Wirtinger presentation is quite involved (e.g.~\cite[Chapter IX]{CrowellFox:KnotTheory}). \TaskSeparator %\begin{task} Show that $\theta$ dominates the %Rozansky-Overbay invariant $\rho_1$ \cite{Rozansky:Contribution, %Rozansky:Burau, Rozansky:U1RCC, Overbay:Thesis, APAI}. %Precisely, show that $\rho_1 = -\theta|_{T_1\to T, T_2\to 1}$. %\end{task} \begin{task} With $\rho_1$ the Rozansky-Overbay invariant \cite{Rozansky:Contribution, Rozansky:Burau, Rozansky:U1RCC, Overbay:Thesis, APAI}, show that $\rho_1 = -\theta|_{T_1\to T, T_2\to 1}$. \end{task} This one should be easy with techniques from~\cite[Section 4.2]{Theta}. \TaskSeparator \begin{task} Is $\theta$ related to the invariants studied by Garoufalidis, Kashaev, and Li \cite{GaroufalidisKashaev:Multivariable, GaroufalidisLi:Patterns}? \end{task} They have the same \sout{hexagonal symmetry} and genus bound. \TaskSeparator \begin{task} Explain the ``Chladni patterns''. Are there ``dominant modes'' of $\theta$ that can be \newline computed in isolation? \vspace{-8mm} \end{task} \parbox[b]{1.8in}{ \tiny left: \copyright\ \href{https://www.whipplemuseum.cam.ac.uk/explore-whipple-collections/acoustics/ernst-chladni-physicist-musician-and-musical-instrument-maker}{Whipple Museum of the History of Science, University of Cambridge}; right: \href{https://creativecommons.org/licenses/by-sa/4.0/deed.en}{CC-BY-SA 4.0} / \href{https://en.wikipedia.org/wiki/Chladni\%27s_law}{Wikimedia} / Matemateca (IME USP) / Rodrigo Tetsuo Argenton } \hfill\includegraphics[height=0.7in]{../../Projects/Theta/figs/chladni_plate_at_whipplemuseum.cam.ac.uk.jpg} \hfill\includegraphics[height=0.7in]{../../Projects/Theta/figs/Chladni_plate_10_cropped_Wikimedia_Image.jpg} \TaskSeparator \begin{task} Prove the genus bound of Conjecture~1. \end{task} This is probably coming. One can bound the degree of $\Delta=\det(A)$ in terms of $g(K)$ using the Seifert presentation of the Alexander module. Pushing further, likely one can bound the degree of $(g_{\alpha\beta}) = A^{-1}$ in terms of $g(K)$, and that's probably enough. %\TaskSeparator \begin{task} Find a 3D interpretation of the $g_{\alpha\beta}$'s. \end{task} They must be closely related to the equivariant linking numbers of \cite{KojimaYamasaki:NewInvariants, GaroufalidisKricker:RationalNoncommutative, GaroufalidisTeichner:TrivialAlexander, Ohtsuki:EquivariantLinkingMatrices, Lescop:Equivariant}. \TaskSeparator \begin{task} \label{task:calF} Find a formula $\calF$ for $\Theta(K)$ that starts from a Seifert surface $\Sigma$ of $K$. Better if $\calF$ is completely 3D! Assuming Task~\ref{task:TwoLoop}, it is known that $\Theta$ depends only of invariants of type $\leq 3$ of $\Sigma$. Maybe $\calF$ is about configuration space integrals / chopstick towers? \hfill\text{\rm See CS: \cite{Thurston:IntegralExpressions, Lescop:GraphConfigurations, Tennessee-1103}, BF: \cite{CattaneoRossi:WilsonSurfaces, Vienna-1402}% } \end{task} \resizebox{0.4\linewidth}{!}{\input{figs/CosmicCoincidences.pdf_t}} \resizebox{0.6\linewidth}{!}{\input{figs/BFDiagram.pdf_t}} \TaskSeparator \begin{task} Is there an intrinsic theory of finite type invariants for Seifert surfaces? For task~\ref{task:Integration}, does its gr map to functions on $H_1$? \end{task} \parpic[r]{\scalebox{0.64}{\input{figs/TrefoilSeifert.pdf_t}}} My current best understanding of finite \text{type} \text{invariants} for Seifert surfaces goes through \text{thick} graphs. \TaskSeparator \begin{task} \label{task:fibered} Prove the fibered condition of Conjecture~2. \end{task} If $K$ is fibered, $\deg\Delta(K)=g(K)$ and $\Delta(K)$ is monic. Indeed, $K$ is then the mapping cylinder of a diffeomorphism $f\colon\Sigma\to\Sigma$. The Alexander module of $K$ is generated by $H_1(\Sigma)$ with relations $\{\gamma = Tf_*\gamma\colon\gamma\in H_1(\Sigma)\}$. Thus the highest monomial in $\Delta$ is $T^g\det(f_*)$ and $\det(f_*)=\pm 1$ as $f_*$ preserves the intersection pairing. If only we had a formula for $\theta$ in terms of $f$\ldots %\TaskSeparator \begin{task} In general, find a formula for $\Theta$ corresponding to each known presentation of the Alexander module. \end{task} Wirtinger is $2\{\text{xings}\}\to\{\text{edges}\}$. Dehn is $\{\text{xings}\}\to\{\text{faces}\}$. Co-Dehn is $\{\text{faces}\}\to\{\text{xings}\}$. Burau is $\{\text{braid strands}\}\to\{\text{braid strands}\}$. Seifert is $H_1(\Sigma)\to H_1(\Sigma)$, and so is the presentation from Task \ref{task:fibered}. Grid diagrams lead to $\{\text{grid number}\}\to\{\text{grid number}\}$ (may relate to HFK). There's more! \TaskSeparator \begin{task} \label{task:Integration} Write up the integration story. \end{task} {\bf Claim} (e.g., \cite{Bonn-2505}). Cutting corners and with $\eps^2=0$, \[ \frac{1}{\Delta_1\Delta_2\Delta_3}\exp\left( \eps\cdot\frac{\theta}{\Delta_1\Delta_2\Delta_3} \right) \sim\Gint_{\prod_e\bbR^6_{p_{1e},p_{2e},p_{3e},x_{1e},x_{2e},x_{3e}}} \prod_c\bbe^{L_c}, \] where $\Gint$ denotes perturbed formal Gaussian integration (i.e., ``Feynman Diagrams'') and $L_c$ is \includegraphics[width=\linewidth]{figs/Lagrangian.pdf} In fact, we first found $L_c$ using the method of undetermined coefficients, and then derived $F_1$ and $F_2$ from it. \TaskSeparator \begin{task} Find a similar perturbed Gaussian integral formula for $\theta$, but with integration over $6H_1(\Sigma)$. The quadratic $Q$ will be the same as in the Seifert-Alexander formula (but repeated 3 times, for each $T_\nu$). The perturbation $P_\eps$ will be given by low-degree finite type invariants of curves on $\Sigma$ (possibly also dependent on the intersection points of such curves, or on other information coming from $\Sigma$). \end{task} \TaskSeparator \begin{task} \label{task:TwoLoop} Prove that $\theta$ is equal to the two-loop contribution $Z^{(2)}$ to the Kontsevich integral $Z$. \end{task} Composed with the inverse PBW isomorphism $\chi^{-1}$, $\chi^{-1}\circ Z$ takes values in unitrivalent Jacobi diagrams, $\calB=\{\fourwheel\twowheel\ldots\}/IHX$. Rozansky conjectured~\cite{Rozansky:RationalStructure, GaroufalidisRozansky:LoopExpansion} and Kricker proved~\cite{Kricker:Lines} that \[ \log(\chi^{-1}\circ Z) = \begin{array}{c} \input{figs/LoopExpansion.pdf_t} \end{array} \] where $\begin{array}{c}\input{figs/HairDef.pdf_t}\end{array}$, $f_1\in\bbQ\llbracket t\rrbracket$, and $f_2\in\bbQ\llbracket t_1,t_2\rrbracket$ satisfy $f_1 = \frac12\log\frac{\sinh(t/2)}{t\Delta(\bbe^t)/2}$ and $f_2 = Z^{(2)}(\bbe^{t_1},\bbe^{t_2})\left/ \Delta(\bbe^{t_1})\Delta(\bbe^{t_1})\Delta(\bbe^{t_1+t_2}) \right.$ where $Z^{(2)}\in\bbZ[T_1^{\pm 1},T_2^{\pm 1}]$ is the ``two loop polynomial''. Ohtsuki~\cite{Ohtsuki:TwoLoop} studied $Z^{(2)}$ extensively, and almost certainly, $Z^{(2)}=\theta$. Prove that! \vskip 8cm %\TaskSeparator \begin{task} \label{task:geps} Complete and write up the $\frakg^+_\eps$ story. \end{task} %\vskip -6mm %\[ % \def\bracket{$b({\red\uppertriang})=b\colon{\red\uppertriang}\otimes\!{\red\uppertriang} \to{\red\uppertriang}$} %\def\cobracket{$b({\blue\lowertriang})\leadsto\delta\colon{\red\uppertriang} \to{\red\uppertriang}\otimes\!{\red\uppertriang}$} % \input{figs/Triangles.pdf_t} %\] %\vskip -4mm %\vskip -4mm %\parpic[r]{ % \def\bracket{$b({\red\uppertriang})=b\colon{\red\uppertriang}\otimes\!{\red\uppertriang} \to{\red\uppertriang}$} %\def\cobracket{$b({\blue\lowertriang})\leadsto\delta\colon{\red\uppertriang} \to{\red\uppertriang}\otimes\!{\red\uppertriang}$} % \input{figs/Triangles.pdf_t} %} { \def\Double{{\calD}} \def\Double{{\operatorname{Double}}} Let $\frakg$ be a semisimple Lie algebra, let $\frakh$ be its Cartan subalgebra, and let $\frakb^u$ and $\frakb^l$ be its upper and lower Borel subalgebras. Then $\frakb^u$ has a bracket $\beta$, and as the dual of $\frakb^l$, $\frakb^u$ also has a cobracket $\delta$, and in fact, $\frakg\oplus\frakh \equiv \Double(\frakb^u,\beta,\delta)$. Let $\frakg^+_\eps \coloneqq \Double(\frakb^u,\beta,\eps\delta)$ (mod $\eps^{d+1}$ it is solvable for any $d$). We expect that $\Theta$ is the universal invariant (in the sense of Lawrence and Ohtsuki \cite{Lawrence:UniversalUsingQG, Ohtsuki:QuantumInvariants}) corresponding to $sl^+_{3,\eps}$, computed modulo $\eps^2$ (in fact, that's how we guessed it). See \cite{DPG, PG}. } \TaskSeparator \begin{task} Go beyond $sl_3$ and the first power of $\eps$! \end{task} This sounds very appealing, and you will surely get stronger and stronger invariants. But they will be less and less computable \frownie. \TaskSeparator \begin{task} Find a w-style charaterization of $\Theta$. \end{task} \vskip -5mm \parpic[r]{\input{figs/OCTC.pdf_t}} Compare with \cite{HabiroKanenobuShima:R2K, HabiroShima:R2KII, WKO1}, where $\Delta$ is charaterized {\em on w-knots} by the overcrossings / tails commute relation. Similarly it should be possible to characterize $\Theta$ on rotational virtual knots by some ``overcrossings / tails nearly commute'' relation. Assuming Task~\ref{task:TwoLoop}, there is a characterization of $\Theta$ in terms of \cite{GaroufalidisRozansky:LoopExpansion}'s ``null filtration''. I find it too complicated to work with. \TaskSeparator \begin{task} Relate the $\frakg^+_\eps$ story with (rotational) virtual knots \cite{Kauffman:RotationalVirtualKnots}, with $\vec{\calA}$ \cite{Polyak:ArrowDiagrams}, and with quantization of Lie bialgebras \cite{EtingofKazhdan:BialgebrasI, EtingofKazhdan:BialgebrasII, Enriquez:Quantization, Severa:BialgebrasRevisited} \end{task} $\xymatrix@R=0mm@C=5mm{ \calK_S \ar[r]^Z \ar[dd]^a & \calA_S \ar[rd] \ar[dd]^\alpha & \\ & & \calU_S(\frakg^+_\eps) \\ \calK_S^{rv} \ar[r]^{Z^{rv}} & \calA_S^{rv} \ar[ru] & }$ \hfill $\xymatrix@R=-1.75mm@C=4mm{ \calK_S/\text{\cite{GaroufalidisRozansky:LoopExpansion}}_{k+2} \ar[r]^Z \ar[dd]^a & \calA_S/{(k+1)-\atop\text{loops}} \ar[rd] \ar[dd]^\alpha & \\ & & \frac{\calU_S(\frakg^+_\eps)}{\eps^{k+1}} \\ \calK_S^{rv}/O\!C^{k+1} \ar[r]^{Z^{rv}} & \calA_S^{rv}/T\!C^{k+1} \ar[ru] & }$ We expect that there is a commutative diagram as on the left, which descends to the one at the right, with $\Theta$ corresponding to $\frakg=sl_3$ and $k=1$. But we're missing $Z^{rv}$ which may be hidden inside \cite{EtingofKazhdan:BialgebrasI, EtingofKazhdan:BialgebrasII, Enriquez:Quantization, Severa:BialgebrasRevisited}. \TaskSeparator \begin{task} Understand Chern-Simons theory with gauge group $\frakg^+_\eps$. \end{task} Is there a gauge that leads to the formula $\calF$ of Task~\ref{task:calF}? \TaskSeparator \begin{task} What happens to representation theory as $\eps\to 0$? Is there any fun in continuous morphisms $\frakg^+_\eps\to gl^+_{n,\eps}$? \end{task} \TaskSeparator \begin{task} Study $\theta$ on links. \end{task} Does it make sense even if $\Delta=0$? Does it depend on the choice of the cut component? \TaskSeparator \begin{task} Does $\Theta$ extend to knots in $\bbZ HS$ / $\bbQ HS$? \hfill {\rm $Z$ and $Z^{(2)}$ do.} \end{task} \TaskSeparator \begin{task} Is there a surgery formula for $\Theta$? \hfill {\rm $Z$ and $Z^{(2)}$ have.} \end{task} \TaskSeparator \begin{task} Extend $\Theta$ to tangles and figure out how it behaves under strand doubling. \end{task} $Z$ and $Z^{(2)}$ extend but their extensions depend on parenthesizations. From Task~\ref{task:geps} we expect that $\Theta$ will extend without the need for parenthesizations, yet with an asymmetry built into the doubling operations. Note that tangles and strand doubling are keys to ``algebraic knot theory'' \cite{AKT}. %\TaskSeparator \begin{task} Make Kricker / Ohtsuki \cite{Kricker:Lines, Ohtsuki:TwoLoop} more computable! \end{task} \TaskSeparator \begin{task} Find a multi-variable version of $\theta$ for links, like there is a multi-variable Alexander for links (e.g.~\cite[Chapter~7]{Kawauchi:Srvery}). \end{task} It is suggested by both $\frakg^+_\eps$ consideration and the loop expansion. \TaskSeparator \begin{task} \label{task:last} Find a ribbon condition satisfied by $\Theta$. \end{task} \parpic[r]{ \includegraphics[height=0.6in]{../Pitzer-250308/RibbonSingularity.pdf} \includegraphics[height=0.6in]{../Pitzer-250308/RibbonKnot.pdf} \includegraphics[height=0.6in]{../Pitzer-250308/Seifert4Ribbon.pdf} } For a ribbon knot $K$, one may find a Seifert surface $\Sigma$ half of whose homology is generated by the components of an unlink embedded in $\Sigma$. This makes for a presentation matrix $A$ of the Alexander module of $K$ that has big blocks of zeros, and this leads to the Fox-Milnor condition \cite{FoxMilnor:CobordismOfKnots}, $\Delta \doteq \det(A) \doteq f(T)f(T^{-1})$ for some $f\in\bbZ[T^{\pm 1}]$. If $\det A$ is constrained for ribbon knots, perhaps so is $A^{-1}$ and therefore $\Theta$? \TaskSeparator \parpic[r]{\parbox{1in}{ \includegraphics[height=0.6in]{../../Projects/Gallery/Khovanov.jpg} \includegraphics[height=0.6in]{figs/CatoTheElder.jpg} }} {\bf\red Bonus Task.} Carthago delenda est and \text{every} knot polynomial must be categorified.\hfill\text{\tiny M. Khovanov \& Cato the Elder} \vskip 2mm \TaskSeparator \input refs.tex \end{multicols} \vfill \begin{center} \includegraphics[width=0.715\linewidth]{../../Projects/Theta/Theta_16up_1.pdf} \parbox[b]{0.23\linewidth}{\begin{center} \includegraphics[width=0.92\linewidth]{figs/P41.pdf} \newline A $(2,41,-41)$ pretzel for \newline dessert \end{center}}% \end{center}% \newpage \begin{center} \includegraphics[height=\textheight, page=1]{../../Projects/Theta/Theta_16up_2.pdf} \newpage\includegraphics[height=\textheight, page=2]{../../Projects/Theta/Theta_16up_2.pdf} \end{center} %\includepdf[pages=-]{../../Projects/Theta/Theta_16up_2.pdf} \end{document} \endinput