\documentclass[aspectratio=169]{beamer}
%\documentclass[11pt]{article} \usepackage{beamerarticle,hyperref}
\usepackage{pgfpages, wrapfig, mathtools, ../picins,
  stmaryrd, % for \sslash
  mathbbol % for \bbe.
}
\geometry{paperwidth=154.3mm,paperheight=90mm} % Best for PDF Annotator, full screen on 16x9.
%\pgfpagesuselayout{2 on 1}[letterpaper,border shrink=0mm]
%\pgfpagesuselayout{4 on 1}[letterpaper,border shrink=0mm,landscape]
\pgfpagesuselayout{8 on 1}[letterpaper,border shrink=0mm]

%\usepackage[breaklinks=true]{hyperref}
\hypersetup{colorlinks,
  linkcolor={green!50!black},
  citecolor={green!50!black},
  urlcolor=blue
}

\usepackage[all]{xy}

% Following http://tex.stackexchange.com/questions/23521/tabular-vertical-alignment-to-top:
\def\imagetop#1{\vtop{\null\hbox{#1}}}

\def\blue{\color{blue}}
\def\green{\color{green}}
\def\red{\color{red}}

\def\bbe{\mathbb{e}}
\def\bbN{{\mathbb N}}
\def\bbQ{{\mathbb Q}}
\def\bbZ{{\mathbb Z}}
\def\calA{{\mathcal A}}
\def\calR{{\mathcal R}}
\def\calT{{\mathcal T}}
\def\calV{{\mathcal V}}
\def\calX{{\mathcal X}}
\def\calY{{\mathcal Y}}

\def\act{{\hspace{-1pt}\sslash\hspace{-0.75pt}}}
\def\arXiv#1{{\href{http://arXiv.org/abs/#1}{arXiv:\linebreak[0]#1}}}
\def\Hom{\operatorname{Hom}}
\def\qed{{\linebreak[1]\null\hfill\text{$\Box$}}}
\def\yellowm#1{{\setlength{\fboxsep}{0pt}\colorbox{yellow}{$#1$}}}
\def\yellowt#1{{\setlength{\fboxsep}{0pt}\colorbox{yellow}{#1}}}

\def\nbpdfInput#1{\vskip 1mm\par\noindent\includegraphics{#1}}
\def\nbpdfPrint#1{\vskip 1mm\par\noindent\includegraphics{#1}}
\def\nbpdfEcho#1{\vskip 1mm\par\noindent\includegraphics{#1}}
\def\nbpdfOutput#1{\vskip 1mm\par\noindent\includegraphics{#1}}
\def\nbpdfText#1{\vskip 1mm\par\noindent\includegraphics{#1}}
\def\nbpdfSection#1{\vskip 1mm\par\noindent\includegraphics{#1}}
\def\nbpdfSubsection#1{\vskip 1mm\par\noindent\includegraphics{#1}}
\def\nbpdfSubsubsection#1{\vskip 1mm\par\noindent\includegraphics{#1}}
\def\nbpdfgraphInput#1{\vskip 1mm\par\noindent\includegraphics{#1}}
\def\nbpdfgraphOutput#1{\vskip 1mm\par\noindent\includegraphics[width=1.5in]{#1}}

\title{I Still Don't Understand the Alexander Polynomial}

\author{Dror Bar-Natan}

\institute{\url{http://drorbn.net/mo21}}

\date{Moscow by Web, April 2021}

\begin{document} \beamertemplatenavigationsymbolsempty

\begin{frame}
  %\titlepage
  \begin{center}
  \vfill
  \Large I Still Don't Understand the Alexander Polynomial
  \vfill
  \par\normalsize Dror Bar-Natan, \url{http://drorbn.net/mo21}
  \vfill
  \par\large Moscow by Web, April 2021
  \end{center}

  \vfill\normalsize
  {\bf Abstract.} As an algebraic knot theorist, I still don't understand the Alexander polynomial. There are two conventions as for how to present tangle theory in algebra: one may name the strands of a tangle, or one may name their ends. The distinction might seem too minor to matter, yet it leads to a completely different view of the set of tangles as an algebraic structure. There are lovely formulas for the Alexander polynomial as viewed from either perspective, and they even agree where they meet. But the ``strands'' formulas know about strand doubling while the ``ends'' ones don't, and the ``ends'' formulas know about skein relations while the ``strands'' ones don't. There ought to be a common generalization, but I don't know what it is.
\end{frame}

\begin{frame}
Thanks for inviting me to Moscow! As most of you have never seen it, here's a picture of the lecture room:
\[ \includegraphics[width=\textwidth]{LectureRoom.jpg} \]
\vfill If you can, please turn your video on! (And mic, whenever needed).
\end{frame}

\begin{frame}
I use talks to self-motivate; so often I choose a topic and write an abstract when I know I can do it, yet when I haven't done it yet. This time it turns out my abstract was wrong --- I'm still uncomfortable with the Alexander polynomial, but in slightly different ways than advertised two slides before.

{\bf My discomfort.}
\begin{itemize}
\item I can compute the multivariable Alexander polynomial real fast:
\[ \begin{array}{c}\includegraphics[width=20mm]{WikipediaBorromeanRings.png}\end{array}
  \longrightarrow (uvw)^{-1/2}(u-1)(v-1)(w-1).
\]
\item But I can only prove ``skein relations'' real slow:
\[ \scalebox{0.8}{\input{figs/C3R.pdf_t}} \]
\end{itemize}

\end{frame}

\begin{frame}

This talk is to a large extent an elucidation of the Ph.D.\ theses of my former students Jana Archibald and Iva Halacheva. See \cite{Archibald:Thesis, Halacheva:Thesis, Halacheva:AlexanderType}.

\[ \includegraphics[height=24mm]{../../Projects/Gallery/Archibald.jpg}
  \qquad\qquad
  \includegraphics[height=24mm]{../../Projects/Gallery/Halacheva.jpg}
\]
Also thanks to Roland van der Veen for comments.

\vskip 2mm
{\bf A technicality.} There's supposed to be fire alarm testing in my building today. Don't panic!
\end{frame}

\begin{frame}
{\LARGE 1. Virtual Skein Theory Heaven}

{\bf Definition.} A ``Contraction Algebra'' assigns a set $\calT(\calX,X)$ to any pair of finite sets $\calX=\{\xi\ldots\}$ and $X=\{x,\ldots\}$ provided $|\calX|=|X|$, and has operations

\begin{itemize}
\item ``Disjoint union'' $\sqcup\colon\calT(\calX,X)\times\calT(\calY,Y)\to\calT(\calX\sqcup\calY,X\sqcup Y)$, provided $\calX\cap\calY=X\cap Y=\emptyset$.
\item ``Contractions'' $c_{x,\xi}\colon\calT(\calX,X)\to\calT(\calX\setminus\xi,X\setminus x)$, provided $x\in X$ and $\xi\in\calX$.
\item Renaming operations $\sigma^\xi_\eta\colon\calT(\calX\sqcup\{\xi\},X)\to\calT(\calX\sqcup\{\eta\},X)$ and $\sigma^x_y\colon\calT(\calX,X\sqcup\{x\})\to\calT(\calX,X\sqcup\{y\})$.
\end{itemize}
Subject to axioms that will be specified right after the two examples in the next three slides.

If $R$ is a ring, a contraction algebra is said to be ``$R$-linear'' if all the $\calT(\calX,X)$'s are $R$-modules, if the disjoint union operations are $R$-bilinear, and if the contractions $c_{x,\xi}$ and the renamings $\sigma^\cdot_\cdot$ are $R$-linear.

(Contraction algebras with some further ``unit'' properties are called ``wheeled props'' in~\cite{MarklMerkulovShadrin:WheeledPROPs, DancsoHalachevaRobertson:CircuitAlgebras})

\end{frame}

\begin{frame}
\[ \input{figs/calT.pdf_t} \]
{\bf Example 1.} Let $\calT(\calX,X)$ be the set of virtual tangles with incoming ends (``tails'') labeled by $\calX$ and outgoing ends (``heads'') labeled by $X$, with $\sqcup$ and $\sigma^\cdot_\cdot$ the obvious disjoint union and end-renaming operations, and with $c_{x,\xi}$ the operation of attaching a head $x$ to a tail $\xi$ while introducing no new crossings.

{\bf Note 1.} $\calT$ can be made linear by allowing formal linear combinations.

{\bf Note 2.} $\calT$ is finitely presented, with generators the positive and negative crossings, and with relations the Reidemeister moves! (If you want, you can take this to be the definition of ``virtual tangles'').
\end{frame}

\begin{frame} \null
{\bf Note 3.} A contraction algebra morphism out of $\calT$ is an invariant of virtual tangles (and hence of virtual knots and links) and would be an ideal tool to prove Skein Relations:
\[ \input{figs/C3VR.pdf_t} \]
\end{frame}

\begin{frame} \null
{\bf Example 2.} Let $V$ be a finite dimensional vector space and set $\calV(\calX,X)\coloneqq(V^\ast)^{\otimes\calX}\otimes V^{\otimes X}$, with $\sqcup=\otimes$, with $\sigma^\cdot_\cdot$ the operation of renaming a factor, and with $c_{x,\xi}$ the operation of contraction: the evaluation of tensor factor $\xi$ (which is a $V^\ast$) on tensor factor $x$ (which is a $V$).
\end{frame}

\begin{frame} \null
{\bf Axioms.} One axiom is primary and interesting,
\begin{itemize}
\item Contractions commute! Namely, $c_{x,\xi}\act c_{y,\eta} = c_{y,\eta}\act c_{x,\xi}$ (or in old-speak, $c_{y,\eta}\circ c_{x,\xi} = c_{x,\xi}\circ c_{y,\eta})$.
\end{itemize}
And the rest are just what you'd expect:
\begin{itemize}
\item $\sqcup$ is commutative and associative, and it commutes with $c_{\cdot,\cdot}$ and with $\sigma^\cdot_\cdot$ whenever that makes sense.
\item $c_{\cdot,\cdot}$ is ``natural'' relative to renaming: $c_{x,\xi}=\sigma^x_y\act\sigma^\xi_\eta\act c_{y,\eta}$.
\item $\sigma^\xi_\xi=\sigma^x_x=Id$, $\sigma^\xi_\eta\act\sigma^\eta_\zeta=\sigma^\xi_\zeta$, $\sigma^x_y\act\sigma^y_z=\sigma^x_z$, and renaming operations commute where it makes sense.
\end{itemize}
% In fancy-speak, we could say that $\calT$ is a monoidal functor from finite sets into sets (or into $R$-mod), and that
\end{frame}

\begin{frame} \null {\bf Comments.}
\begin{itemize}
\item We can relax $|\calX|=|X|$ at no cost.
\item We can lose the distinction between $\calX$ and $X$ and get ``circuit algebras''.
\item There is a ``coloured version'', where $\calT(\calX,X)$ is replaced with $\calT(\calX,X,\lambda,l)$ where $\lambda\colon\calX\to C$ and $l\colon X\to C$ are ``colour functions'' into some set $C$ of ``colours'', and contractions $c_{x,\xi}$ are allowed only if $x$ and $\xi$ are of the same colour, $l(x)=\lambda(\xi)$. In the world of tangles, this is ``coloured tangles''.
\end{itemize}
\end{frame}

\begin{frame}
{\LARGE 2. Heaven is a Place on Earth}
(A version of the main results of Archibald's thesis,~\cite{Archibald:Thesis}).
\vskip 2mm

Let us work over the base ring $\calR=\bbQ[\{T^{\pm 1/2}\colon T\in C\}]$. Set
\[ \calA(\calX,X)\coloneqq \{w\in\Lambda(\calX\sqcup X)\colon \deg_\calX w=\deg_X w\} \]
(so in particular the elements of $\calA(\calX,X)$ are all of even degree). The union operation is the wedge product, the renaming operations are changes of variables, and $c_{x,\xi}$ is defined as follows. Write $w\in\calA(\calX,X)$ as a sum of terms of the form $uw'$ where $u\in\Lambda(\xi,x)$ and $w'\in\calA(\calX\setminus\xi,X\setminus x)$, and map $u$ to $1$ if it is $1$ or $x\xi$ and to $0$ is if is $\xi$ or $x$:
\[ 1w'\mapsto w', \qquad \xi w'\mapsto 0, \qquad xw'\mapsto 0, \qquad x\xi w'\mapsto w'. \]
{\bf Proposition.} $\calA$ is a contraction algebra.
\end{frame}

\begin{frame} \null
{\bf Alternative Formulations.}
\begin{itemize}
\item\hfill$\displaystyle c_{x,\xi}w = \iota_\xi\iota_x\bbe^{x\xi}w,$\hfill\null
where $\iota_\cdot$ denotes interior multiplication.
\item Using Fermionic integration, \hfill$\displaystyle c_{x,\xi}w = \int\bbe^{x\xi}w\,d\xi dx.$\hfill\null
\item $c_{x,\xi}$ represents composition in exterior algebras! With $X^\ast\coloneqq\{x^\ast\colon x\in X\}$, we have that $\Hom(\Lambda X,\Lambda Y)\cong\Lambda(X^\ast\sqcup Y)$ and the following square commutes:
    \[ \xymatrix{
      \Hom(\Lambda X,\Lambda Y)\otimes\Hom(\Lambda Y,\Lambda Z) \ar[r]^<>(0.5)\act \ar@{<->}[d] &
      \Hom(\Lambda X,\Lambda Z) \ar@{<->}[d] \\
      \Lambda(X^\ast\sqcup Y\sqcup Y^\ast\sqcup Z) \ar[r]^<>(0.5){\prod_{y\in Y}c_{y,y^\ast}} &
      \Lambda(X^\ast,Z)
    } \]
\item Similarly, $\Lambda(\calX\sqcup X)\cong(H^\ast)^{\otimes\calX}\otimes H^{\otimes X}$ where $H$ is a 2-dimensional ``state space'' and $H^\ast$ is its dual. Under this identification, $c_{x,\xi}$ becomes the contraction of an $H$ factor with an $H^\ast$ factor.
\end{itemize}
\end{frame}

\begin{frame}
We construct a morphism of coloured contraction algebras $\calA\colon\calT\to\calA$ by declaring
\begin{eqnarray*}
  X_{ijkl}[S,T] & \mapsto & T^{-1/2}\exp\left(
    \begin{pmatrix}\xi_l&\xi_i\end{pmatrix}\begin{pmatrix}1&1-T\\0&T\end{pmatrix}\begin{pmatrix}x_j\\x_k\end{pmatrix}
  \right) \\
  \bar{X}_{ijkl}[S,T] & \mapsto & T^{1/2}\exp\left(
    \begin{pmatrix}\xi_i&\xi_j\end{pmatrix}\begin{pmatrix}T^{-1}&0\\1-T^{-1}&1\end{pmatrix}\begin{pmatrix}x_k\\x_l\end{pmatrix}
  \right)\\
  P_{ij}[T] & \mapsto & \exp(\xi_ix_j)
\end{eqnarray*}
with
\[ \def\X{$X_{ijkl}[S,T]$} \def\Xbar{$\bar{X}_{ijkl}[S,T]$}  \def\P{$P_{ij}[T]$}
  \input{figs/Gens.pdf_t}
\]
(Note that the matrices appearing in these formulas are the Burau matrices).
\end{frame}

\begin{frame}{\bf Theorem.}
If $D$ is a classical link diagram with $k$ components coloured $T_1,\ldots,T_k$ whose first component is open and the rest are closed, if $MVA$ is the multivariable Alexander polynomial of the closure of $D$ (with these colours), and if $\rho_j$ is the counterclockwise rotation number of the $j$th component of $D$, then
\[ \calA(D) = T_1^{-1/2}(T_1-1)\left(\prod_jT_j^{\rho_j/2}\right)\cdot MVA\cdot(1+\xi_{\text{in}}\wedge x_{\text{out}}). \]
($\calA$ vanishes on closed links).
\end{frame}

\input{Alpha.tex}

\begin{frame}
{\LARGE 5. Some Problems in Heaven}
Unfortunately, $\dim\calA(\calX,X)=\dim\Lambda(\calX,X)=4^{|X|}$ is big. Fortunately, we have the following theorem, a version of one of the main results in Halacheva's thesis,~\cite{Halacheva:Thesis, Halacheva:AlexanderType}:

{\bf Theorem.} Working in $\Lambda(\calX\cup X)$, if $w=\omega\bbe^\lambda$ is a balanced Gaussian (namely, a scalar $\omega$ times the exponential of a quadratic $\lambda=\sum_{\zeta\in\calX,z\in X}\alpha_{\zeta,z}\zeta z$), then generically so is $c_{x,\xi}\bbe^\lambda$.

(This is great news! The space of balanced quadratics is only $|\calX||X|$-dimensional!)
\end{frame}

\begin{frame}\null
{\bf Proof.} Recall that $c_{x,\xi}\colon(1,\xi,x,x\xi)w'\mapsto(1,0,0,1)w'$, write $\yellowm{\lambda=\mu+\eta x+\xi y+\alpha\xi x}$, and ponder $\bbe^\lambda =$
\[ \ldots + \frac{1}{k!}
  \underbrace{(\mu+\eta x+\xi y+\alpha\xi x)(\mu+\eta x+\xi y+\alpha\xi x)\cdots(\mu+\eta x+\xi y+\alpha\xi x)}_{k\text{ factors}}
  + \ldots .
\]
Then $c_{x,\xi}\bbe^\lambda$ has three contributions:
\begin{itemize}
\item $\bbe^\mu$, from the term proportional to $1$ (namely, independent of $\xi$ and $x$) in $\bbe^\lambda$
\item $-\alpha\bbe^\mu$, from the term proportional to $x\xi$, where the $x$ and the $\xi$ come from the same factor above.
\item $\eta y\bbe^\mu$, from the term proportional to $x\xi$, where the $x$ and the $\xi$ come from different factors above.
\end{itemize}
So $\yellowm{c_{x,\xi}\bbe^\lambda} = \bbe^\mu(1-\alpha+\eta y) = (1-\alpha)\bbe^\mu(1+\eta y/(1-\alpha)) = (1-\alpha)\bbe^\mu\bbe^{\eta y/(1-\alpha)} = \yellowm{(1-\alpha)\bbe^{\mu+\eta y/(1-\alpha)}}$. \qed
\end{frame}

\begin{frame}{\bf $\Gamma$-calculus.}
Thus we have an almost-always-defined ``$\Gamma$-calculus'': a contraction algebra morphism $\calT(\calX,X)\to R\times(\calX\otimes_{R/R}X)$ whose behaviour under contractions is given by
\[ c_{x,\xi}(\omega,\lambda=\mu+\eta x+\xi y+\alpha\xi x) = ((1-\alpha)\omega, \mu+\eta y/(1-\alpha)). \]
($\Gamma$ is fully defined on pure tangles -- tangles without closed components -- and hence on long knots).
\end{frame}

\input{Gamma.tex}

\begin{frame} {\bf What I still don't understand.}
\begin{itemize}
\item What becomes of $c_{x,\xi}\bbe^\lambda$ if we have to divide by $0$ in order to write it again as an exponentiated quadratic? Does it still live within a very small subset of $\Lambda(\calX\sqcup X)$?
\item How do cablings and strand reversals fit within $\calA$?
\item Are there ``classicality conditions'' satisfied by the invariants of classical tangles (as opposed to virtual ones)?
\end{itemize}
\end{frame}

\newpage \null \vskip 5mm {\LARGE References}

\begin{thebibliography}{DHR}

\bibitem[Ar]{Archibald:Thesis} J.~Archibald,
  {\em The Multivariable Alexander Polynomial on Tangles,}
  University of Toronto Ph.D.\ thesis, 2010, \url{http://drorbn.net/mo21/AT}.

\bibitem[Co]{Conway:Enumeration} J.~H.~Conway,
  {\em An Enumeration of Knots and Links, and some of their Algebraic Properties,} Computational Problems in Abstract Algebra (Proc.\ Conf.,
Oxford, 1967), Pergamon, Oxford, 1970, 329-–358.

\bibitem[DHR]{DancsoHalachevaRobertson:CircuitAlgebras} Z.~Dancso, I.~Halacheva, and M.~Robertson,
  {\em Circuit Algebras are Wheeled Props,}
  J.\ Pure and Appl.\ Alg., to appear, \arXiv{2009.09738}.

\bibitem[Ha1]{Halacheva:Thesis} I.~Halacheva,
  {\em Alexander Type Invariants of Tangles, Skew Howe Duality for Crystals and The Cactus Group,}
  University of Toronto Ph.D.\ thesis, 2016, \url{http://drorbn.net/mo21/HT}.

\bibitem[Ha2]{Halacheva:AlexanderType} I.~Halacheva,
  {\em Alexander Type Invariants of Tangles,}
  \arXiv{1611.09280}.

\bibitem[MMS]{MarklMerkulovShadrin:WheeledPROPs} M. Markl, S. Merkulov, and S. Shadrin,
  {\em Wheeled PROPs, Graph Complexes and the Master Equation,}
  J.\ Pure and Appl.\ Alg.\ {\bf 213-4} (2009) 496--535, \arXiv{math/0610683}.

\bibitem[Mu]{MurakamiJ:StateModel} J.~Murakami,
  {\em A State Model for the Multivariable Alexander Polynomial,}
  Pacific J.\ Math.\ {\bf 157-1} (1993) 109-–135.

\bibitem[NS]{NaikStanford:Move} S.~Naik and T.~Stanford,
  {\em A Move on Diagrams that Generates S-Equivalence of Knots,}
  J.\ Knot Theory Ramifications {\bf 12-5} (2003) 717-–724, \arXiv{math/9911005}.

\bibitem[Wo]{Wolfram:Mathematica}
  {\em Wolfram Language \& System Documentation Center,}
  \url{https://reference.wolfram.com/language/}.

\end{thebibliography}

\begin{frame}
    Thank You!
\end{frame}

\end{document}