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%\parbox[b]{4in}{
%  {\tiny Dror Bar-Natan: MAT 1350 Knot Theory: \LARGE\bf Tentative Plan}
%  }
\parbox[b]{1.5in}{\tiny
  \href{http://www.math.toronto.edu/~drorbn}{Dror Bar-Natan}: \href{http://www.math.toronto.edu/~drorbn/classes}{Classes}:
  \newline \href{http://www.math.toronto.edu/~drorbn/classes/\#2122}{2021-22}: MAT 1350 Knot Theory:
}~{\LARGE\bf Tentative Plan}
\hfill\parbox[b]{3.2in}{\tiny
  \null\hfill\sheeturl
  \newline\null\hfill initiated July 1, 2021; modified \today, \ampmtime
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  \qrcode[height=1.2em,level=L,nolink]{drorbn.net/ap/Classes/21-1350-KnotTheory}
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\begin{multicols*}{2}

First class on Friday September 10. Reading week is after Lecture 24. {\blue Regrets in blue}, {\red gaps in red}.

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\vspace{-5mm}

Topics in Algebraic Topology I: Algebraic Knot Theory and Computation.

The destination will be ``a poly-time computable strong knot invariant with good algebraic properties''. But you will be taking the course for the journey, not for the destination: What are knots and what are some of the problems around them? Why care about ``invariants with good algebraic properties''? What is the ``Yang-Baxter equation''? What are ``virtual tangles''? What are ``Hopf algebras''? Why would a topologist care about computations in Heisenberg algebras more than most physicists? How does Gaussian integration, and how do Feynman diagrams, arise in pure algebra? What is the ``Drinfel'd Double Procedure''? Are we there yet?

The professor for this class does not believe anything that he does unless it is coded and the code runs. A useful life skill you will learn here is that even the incredibly abstract can become a computer program, often with no loss to its beauty.

\begin{lecture} Course Introduction.
\end{lecture}

\begin{lecture} Knots and the Kauffman bracket. \regref{Khovanov}.
\end{lecture}

\begin{lecture} Mathematica and implementing the Kauffman bracket.
\end{lecture}

{\footnotesize Following \url{http://drorbn.net/syd3}.}

\begin{lecture} A faster Kauffman bracket program.
\end{lecture}

{\footnotesize Following \url{http://drorbn.net/syd3}, with {\tt ThinPosition} from {\tt WG.nb}.}

\begin{lecture} Tangles and planar algebras. \regref{KhovanovTangles}, \regref{OtherAlgebraicStructures}.
\end{lecture}

\begin{lecture} Three basic problems: unknotting, genus, ribbon knots. Display a list of ribbon knots.
\end{lecture}

\begin{lecture} Aside: the Seifert algorithm.
\end{lecture}

\begin{lecture} Tangles and the three basic problems.
\end{lecture}

\begin{lecture} The Yang-Baxter approach and the $WG$ algebras.
\end{lecture}

\begin{lecture} {\red Implementation.}
\end{lecture}

\begin{lecture} Virtual tangles and rotational virtual tangles. Meta monoids.
\end{lecture}

\begin{lecture} Quasi-triangular Hopf algebras and the Kerler algebra.
\end{lecture}

\begin{lecture} {\red Implementation.}
\end{lecture}

\begin{lecture} The Heisenberg algebra $\bbH$, $hR_0$, and the PBW principle.
\end{lecture}

\begin{lecture} Generating functions and $hm$.
\end{lecture}

\begin{lecture} Gaussians and compositions.
\end{lecture}

\begin{lecture} Implementation, testing.
\end{lecture}

\begin{lecture} Yang-Baxter. {\red An aside on the harmonic oscillator.}
\end{lecture}

\begin{lecture} $\Gamma$ calculus and the Alexander polynomial. \regref{Alexander}.
\end{lecture}

\begin{lecture} $hR_\eps$. Gaussian integration.
\end{lecture}

\begin{lecture} Perturbation theory for Gaussian integration.
\end{lecture}

\begin{lecture} Perturbation theory for Gaussian compositions.
\end{lecture}

\begin{lecture} Implementation.
\end{lecture}

\begin{lecture} The Rozansky-Overbay invariants.
\end{lecture}

\begin{lecture} $CU_0$, $QU_0$, and the $QU_0$-calculus.
\end{lecture}

\begin{lecture} Genus using $QU_0$.
\end{lecture}

\begin{lecture} {\red Fox-Milnor using $QU_0$?}

\begin{lecture} $CU_\eps$, Wigner contractions, solvable approximation.
\end{lecture}
\end{lecture}

\begin{lecture} OU tangles and the Drinfel'd Double procedure. {\red A finite-dimensional example.}
\end{lecture}

\begin{lecture} $QU_\eps$ and $P$.
\end{lecture}

\begin{lecture} The rest of the $QU_\eps$ structure.
\end{lecture}

\begin{lecture} Implementation, verification, computation.
\end{lecture}

\begin{lecture} {\red $QU_\eps$ and genus.}
\end{lecture}

\begin{lecture} {\red From $QU_\eps$ to $hR_\eps$.}
\end{lecture}

\begin{lecture} ?
\end{lecture}

\begin{lecture} ?
\end{lecture}

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\begin{regret} \label{reg:Khovanov} Khovanov homology.
\end{regret}

\begin{regret} \label{reg:KhovanovTangles} Khovanov homology for tangles.
\end{regret}

\begin{regret} \label{reg:OtherAlgebraicStructures} Other algebraic structures near knot theory: (monoidal) categories, braid groups. Also mention contraction (circuit) algebras, meta-monoids, meta-Hopf-algebras, quandles, \ldots
\end{regret}

\begin{regret} \label{reg:Alexander} Oh there's so much more on the Alexander polynomial!!!
\end{regret}

\end{multicols*}

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