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\def\myurl{http://www.math.toronto.edu/~drorbn}
\def\thistalk{LesDiablerets-2601}
\def\title{Homework}

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  \href{\myurl}{Dror Bar-Natan}:
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}}
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\newtheorem{hw}{\bf Homework Task}

%%%

\def\Abstract{{\raisebox{1.2mm}{\parbox[t]{3.95in}{
\parshape 1 0in 3.35in
{\red\bf Abstract.} I'll start with a review of my recent paper with
van der Veen, ``A Fast, Strong, Topologically Meaningful, and Fun Knot
Invariant'', and then assign some homework.

\parshape 3 0in 3.35in 0in 3.35in 0in 3.95in
\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).
}}}}

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\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 50-73 crossings:\hfill(many more at \web{DK})

\[ \includegraphics[width=0.95\linewidth]{../UBC-241004/Gallery50-73.png} \]

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

\begin{multicols*}{2}

\begin{hw} Prove all the expected symmetries of $\Theta(K)$.
\end{hw}

\begin{hw} Prove the genus bound and the Seifert condition.
\end{hw}

\begin{hw} Find a formula for $\Theta(K)$ that starts from a Seifert surface of $K$. Better if
it is completely 3D!
\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}

\begin{hw} 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}

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

\begin{hw} What happens to representation theory as $\eps\to 0$?
\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} Carthago delenda est? Every knot invariant must be categorified?
\end{hw}

\end{multicols*}

\end{document}

\endinput

