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\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}}}
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\newtheorem{hw}{Homework Task}

%%%

\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 by 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]{3.125in}{
{\red $A$.} Let $T$, $T_1$, and $T_2$ be indeteminates, and let $T_3=T_1T_2$.
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
\[
  \begin{array}{c}\input{figs/Xings.pdf_t}\end{array}
  \ \rightarrow\ 
  \begin{array}{c|cccc}
    A_c &   i+1  &  j+1 \\
    \hline
    i & -T^s  & T^s-1 \\
    j & 0  & -1
  \end{array}
\]
\[
A=\left(
\begin{array}{ccccccc}
 1 & \mbluem{-T} & 0 & 0 & \mbluem{T-1} & 0 & 0 \\
 0 & 1 & \mpinkm{-1} & 0 & 0 & \mpinkm{\ 0\ } & 0 \\
 0 & 0 & 1 & \myellowm{-T} & 0 & 0 & \myellowm{T-1} \\
 0 & \mbluem{\ 0\ } & 0 & 1 & \mbluem{-1} & 0 & 0 \\
 0 & 0 & \mpinkm{T-1} & 0 & 1 & \mpinkm{-T} & 0 \\
 0 & 0 & 0 & \myellowm{\ 0\ } & 0 & 1 & \myellowm{-1} \\
 0 & 0 & 0 & 0 & 0 & 0 & 1 \\
\end{array}
\right)
\]
}}}}

\def\G{{\raisebox{2mm}{\parbox[t]{3.95in}{
{\red $G$.} Let $G = (g_{\alpha\beta}) \coloneqq A^{-1}$:

$G=\left(
\begin{array}{ccccccc}
 1 & T & 1 & T & 1 & T & 1 \\
 0 & 1 & \frac{1}{T^2-T+1} & \frac{T}{T^2-T+1} & \frac{T}{T^2-T+1} &
\frac{T^2}{T^2-T+1} &
   1 \\
 0 & 0 & \frac{1}{T^2-T+1} & \frac{T}{T^2-T+1} & \frac{T}{T^2-T+1} &
\frac{T^2}{T^2-T+1} &
   1 \\
 0 & 0 & \frac{1-T}{T^2-T+1} & \frac{1}{T^2-T+1} & \frac{1}{T^2-T+1} &
\frac{T}{T^2-T+1} &
   1 \\
 0 & 0 & \frac{1-T}{T^2-T+1} & -\frac{(T-1) T}{T^2-T+1} & \frac{1}{T^2-T+1} &
   \frac{T}{T^2-T+1} & 1 \\
 0 & 0 & 0 & 0 & 0 & 1 & 1 \\
 0 & 0 & 0 & 0 & 0 & 0 & 1 \\
\end{array}
\right)$

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}{
\[
  \theta \sim \Delta_1\Delta_2\Delta_3\sum_{c_0,c_1}g_{1i_0i_1}g_{2i_0i_1}g_{3i_1i_0}
  + \text{l.o.}
\]
\[ \Theta=(\Delta,\theta)\in\bbZ[T^{\pm1}]\times\bbZ[T_1^{\pm1},T_2^{\pm1}] \]
}}}}

\def\random{{\raisebox{1mm}{\parbox[t]{1.85in}{
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 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
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\begin{center} \null\vfill\input{HWT1.pdftex_t}\vfill\null \end{center}

\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 51 -- 68
crossings:\hfill(many more at \web{DK})

\[ \includegraphics[width=0.95\linewidth]{figs/Beehive.pdf} \]

\newgeometry{textwidth=8in,textheight=10.5in}

\begin{multicols*}{2}

\begin{hw} Prove the hexagonal symmetry of $\theta(K)$, and that $\theta(K) =
\theta(-K) = -\theta(\bar{K})$.
\end{hw}

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}).

\parpic[r]{\parbox{2.1in}{\begin{center}
  \includegraphics[height=0.7in]{../../Projects/Theta/figs/chladni_plate_at_whipplemuseum.cam.ac.uk.jpg}
  \includegraphics[height=0.7in]{../../Projects/Theta/figs/Chladni_plate_10_cropped_Wikimedia_Image.jpg}
  \newline \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
\end{center}}}
\begin{hw} Explain the ``Chladni patterns''. Are there ``dominant parts'' of $\theta$
that can be computed in isolation? 
\end{hw}

\begin{hw} Prove the genus bound of Conjecture~1.
\end{hw}

\begin{hw} 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?
\end{hw}

\begin{hw} Is there an intrinsic theory of finite type invariants for Seifert surfaces?
Does its gr map to functions on $H_1$?
\end{hw}

\begin{hw} Prove the the fibered condition of Conjecture~2.
\end{hw}

\begin{hw} In general, find a formula for $\Theta$ corresponding to each known formula for the
Alexander polynomial.
\end{hw}

\begin{hw} Write up the integration story.
\end{hw}

\vskip 5cm

\begin{hw} \label{task:TwoLoop} Prove that $\Theta$ is equal to the two-loop contribution to the Kontsevich integral.
\end{hw}

\begin{hw} Complete and write up the $\frakg_\eps$ story.
\end{hw}

\vskip 5cm

\begin{hw} Understand Chern-Simons theory with gauge group $\frakg_\eps$.
\end{hw}

\begin{hw} What happens to representation theory as $\eps\to 0$? Is
there any fun in continuous morphisms $\frakg_\eps\to gl_{n,\eps}^+$?
\end{hw}

\begin{hw} Does $\Theta$ extend to knots in $\bbZ HS$/$\bbQ HS$?
\end{hw}

\begin{hw} Is there a surgery formula for $\Theta$?
\end{hw}

\begin{hw} Find a ribbon condition satisfied by $\Theta$.
\end{hw}

\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 $\Theta$?

\begin{hw} Carthago delenda est? Must $\Theta$ be categorified?
\end{hw}

\hfill{\normalsize\red\bf References.}

%\par\vspace{-6mm}
\renewcommand{\section}[2]{}%
\begin{thebibliography}{}
\setlength{\parskip}{0pt}
\setlength{\itemsep}{0pt plus 0.3ex}
{\footnotesize

\bibitem[Al]{Alexander:TopologicalInvariants} J.~W.~Alexander,
  {\em Topological invariants of knots and link,}
  Trans.\ Amer.\ Math.\ Soc.\ {\bf 30} (1928) 275--306.

\bibitem[BV]{Theta} D.~Bar-Natan and R.~van der Veen,
  {\em A Fast, Strong, Topologically Meaningful, and Fun Knot Invariant,}
  \web{Theta} and \arXiv{2509.18456}.

\bibitem[CF]{CrowellFox:KnotTheory} R.~H.~Crowell and R.~H.~Fox,
  {\em Introduction to Knot Theory,}
  Springer-Verlag GTM {\bf 57} (1963).

\bibitem[DHOEBL]{DHOEBL:Random} N.~Dunfield, A.~Hirani, M.~Obeidin, A.~Ehrenberg,
  S.~Bhattacharyya, D.~Lei, and others,
  {\em Random Knots: A Preliminary Report,} lecture notes at \web{DHOEBL}. Also a
  data file at \web{DD}.

\bibitem[FM]{FoxMilnor:CobordismOfKnots} R.~H.~Fox and J.~W.~Milnor,
  {\em Singularities of 2-Spheres in 4-Space and Cobordism of Knots,}
  Osaka J.\ Math.\ {\bf 3-2} (1966) 257--267.

\bibitem[GR]{GaroufalidisRozansky:LoopExpansion} S.~Garoufalidis and L.~Rozansky,
  {\em The Loop Expansion of the Kontsevich Integral, the Null-Move,
  and $S$-Equivalence,}
  \arXiv{math.GT/0003187}.

\bibitem[Kr]{Kricker:Lines} A.~Kricker,
  {\em The Lines of the Kontsevich Integral and Rozansky's Rationality Conjecture,}
  \arXiv{math/0005284}.

\bibitem[Oh]{Ohtsuki:TwoLoop} T.~Ohtsuki,
  {\em On the 2-Loop Polynomial of Knots,}
  Geometry \& Topology {\bf 11} (2007) 1357--1475.

\bibitem[Ro1]{Rozansky:Contribution} L.~Rozansky,
  {\em A Contribution of the Trivial Flat Connection to the Jones
  Polynomial and Witten's Invariant of 3D Manifolds, I,}
  Comm.\ Math.\ Phys.\ {\bf 175-2} (1996) 275--296, \arXiv{hep-th/9401061}.

\bibitem[Ro2]{Rozansky:Burau} L.~Rozansky,
  {\em The Universal $R$-Matrix, Burau Representation and the Melvin-Morton
    Expansion of the Colored Jones Polynomial,}
  Adv.\ Math.\ {\bf 134-1} (1998) 1--31, \arXiv{q-alg/9604005}.

\bibitem[Ro3]{Rozansky:U1RCC} L.~Rozansky,
  {\em A Universal $U(1)$-RCC Invariant of Links and Rationality Conjecture,}
  \arXiv{math/0201139}.

}
\end{thebibliography}

A $(2,41,-41)$ pretzel for dessert:\hfill\includegraphics[width=2in,valign=t]{figs/P41.pdf}

\end{multicols*}

\end{document}

\endinput

