\documentclass[aspectratio=169]{beamer}
%\documentclass[11pt]{article} \usepackage{beamerarticle,hyperref}
\usepackage{pgfpages,wrapfig,mathtools,import,
  halloweenmath % for \bigskull
}
\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
}

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

\def\bbC{{\mathbb C}}
\def\bbN{{\mathbb N}}
\def\bbR{{\mathbb R}}
\def\bbZ{{\mathbb Z}}

\def\arXiv#1{{\href{http://arxiv.org/abs/#1}{{arXiv:}\linebreak[0]{#1}}}}
\def\qed{{\linebreak[1]\null\hfill\text{$\Box$}}}
\def\writhe{{\operatorname{writhe}}}

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

\def\nbpdfMessage#1{\vskip 1mm\par\noindent\includegraphics{#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{Kashaev's Signature Conjecture}

\author{Dror Bar-Natan}

\institute{\url{http://drorbn.net/kos21}}

%\date{The Fields Institute, October 2021}

\setbeamertemplate{headline}{\null\hfill\url{http://drorbn.net/cms21}}

\begin{document} \beamertemplatenavigationsymbolsempty

\begin{frame}
  %\titlepage
  \begin{center}
  \vfill
  \LARGE{\bf Kashaev's Signature Conjecture}
  \vfill
  \par\Large \href{https://winter21.cms.math.ca/}{CMS Winter 2021 Meeting}, December 4, 2021
  \vfill
  \par\normalsize Dror Bar-Natan with Sina Abbasi
  \end{center}

  \vfill\normalsize

  {\bf Agenda.} Show and tell with signatures.

  \vskip 2mm {\bf Abstract.} I will display side by side two nearly identical computer programs whose inputs are knots and whose outputs seem to always be the same. I'll then admit, very reluctantly, that I don't know how to prove that these outputs are always the same. One program I wrote mostly in Bedlewo, Poland, in the summer of 2003 and as of recently I understand why it computes the Levine-Tristram signature of a knot. The other is based on the 2018 preprint {\em On Symmetric Matrices Associated with Oriented Link Diagrams} by Rinat Kashaev (\arXiv{1801.04632}), where he conjectures that a certain simple algorithm also computes that same signature.

  \vskip 2mm If you can, please turn your video on! (And mic, whenever needed).
\end{frame}

\begin{frame}
These slides and all the code within are available at \url{http://drorbn.net/cms21}.
\vskip 10mm
(I'll post the video there too)
\end{frame}

\begin{frame}\includegraphics[width=0.498\linewidth]{Snips/BedAndKas/1.pdf}\hfill\includegraphics[width=0.498\linewidth]{Snips/BedAndKas/2.pdf}\end{frame}

\begin{frame} \null
{\bf Why am I showing you $\bigskull$code$\bigskull$?}
\begin{itemize}
\item I love code --- it's fun!
\item Believe it or not, it is more expressive than math-talk (though I'll do the math-talk as well, to confirm with prevailing norms).
\item It is directly verifiable. Once it is up and running, you'll never ask yourself ``did he misplace a sign somewhere''?
\end{itemize}
\end{frame}

\begin{frame}\includegraphics[width=0.498\linewidth]{Snips/BedAndKas/1.pdf}\hfill\includegraphics[width=0.498\linewidth]{Snips/BedAndKas/2.pdf}\end{frame}

\begin{frame} {\bf Verification.}
  \input Verification.tex
\end{frame}

\begin{frame} {\bf Label everything!}
\parbox{0.49\linewidth}{
  \[ \scalebox{0.95}{\input{K82.pdf_t}} \]
  \[ PD[X[10, 1, 11, 2], X[2, 11, 3, 12],\ldots] \]
}\hfill\parbox{0.49\linewidth}{
  \[ \scalebox{0.95}{\input{K82Labeled.pdf_t}} \]
  \[ \{X_-[-1, 11, 2, -10], X_-[-11, 3, 12, -2],\ldots\} \]
}
\end{frame}

\input RunThrough.tex

\begin{frame} {\bf Kashaev for Mathematicians.}
For a knot $K$ and a complex unit $\omega$ set $u=\Re(\omega^{1/2})$, $v=\Re(\omega)$, make an $F\times F$ matrix $A$ with contributions
\[ \input{KasXings.pdf_t} \]
and output $\frac12(\sigma(A)-w(K))$.
\end{frame}

\begin{frame} {\bf Bedlewo for Mathematicians.}
For a knot $K$ and a complex unit $\omega$ set $t=1-\omega$, $r=2\Re(t)$, make an $F\times F$ matrix $A$ with contributions
\[ \input{BedXings.pdf_t} \]
(conjugate if going against the flow) and output $\sigma(A)$.
\end{frame}

\begin{frame} {\bf Why are they equal?}
I dunno, yet note that
\vskip 10mm
\begin{itemize}
\item Kashaev is over the $\bbR$eals, Bedlewo is over the $\bbC$omplex numbers.
\item There's a factor of 2 between them, and a shift.
\end{itemize}
\vskip 10mm
\ldots so it's not merely a matrix manipulation.
\end{frame}

\begin{frame}
  \parbox[b]{0.49\textwidth}{
    {\bf Theorem.} The Bedlewo program computes the Levine-Tristram signature of $K$ at $\omega$.

    \vskip 2mm(Easy) {\bf Proof.} Levine and Tristram tell us to look at $\sigma((1-\omega)L+(1-\omega^\ast)L^T)$, where $L$ is the linking matrix for a Seifert surface $S$ for $K$: $L_{ij}=\operatorname{lk}(\gamma_i,\gamma_i^+)$ where $\gamma_i$ run over a basis of $H_1(S)$ and $\gamma_i^+$ is the pushout of $\gamma_i$. But signatures don't change if you run over and over-determined basis, and the faces make such and over-determined basis whose linking numbers are controlled by the crossings. The rest is details.
  }
  \hfill\parbox[b]{0.49\textwidth}{
    \includegraphics[width=\linewidth]{SeifertByRedelmeier.jpg}
    \newline\null\hfill\text{Art by Emily Redelmeier}
  }
\end{frame}

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

\end{document}


