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\usepackage{graphicx}
\usepackage{pinlabel}
\usepackage[ngerman]{babel}  % German umlauts
\usepackage[utf8]{inputenc}  % German umlauts
\usepackage{hyperref}  % seems to be important for multicols
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\usepackage{multicol}  % split page in two parts

\DeclareMathAlphabet{\mathpzc}{OT1}{pzc}{m}{it}
%this is for my weird f

\usepackage{concmath}
%\usepackage{helvet} %NB this was the first font I tried

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\usepackage{parskip}

 \usepackage{picins}


\setlength\columnseprule{.4pt}


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\usepackage{atbegshi}% http://ctan.org/pkg/atbegshi
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% ! This is a major hack 
% cf https://tex.stackexchange.com/questions/140168/how-to-remove-a-blank-page-before-the-title-page
% I do not know how the first blank page was being generated; this throws it out
% Need to learn how to properly 'bleed' a photo

   \begingroup
\renewcommand{\section}[2]{}%
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%%%%Extre stuff for DBN

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\def\bbZ{{\mathbb Z}}
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\def\fraka{{\mathfrak a}}
\def\frakb{{\mathfrak b}}
\def\frakh{{\mathfrak h}}
\def\frakg{{\mathfrak g}}

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\begin{document}


%----------------
% 
%\vspace*{-78mm}
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\noindent
\includegraphics[width=\linewidth]{Beehive.pdf}
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\newpage

\vspace*{0mm}

\begin{multicols}{2}
    {\normalfont \bfseries A Very Fast and Very Strong Knot Invariant}
    
 \medskip

\parpic[r]{\scriptsize\normalfont \itshape February 18, 2025.}
    {\scriptsize Dror Bar-Natan} \\[-2pt]
    {\scriptsize University of Toronto}\\[-2pt]
    {\scriptsize \normalfont \ttfamily drorbn@utoronto.ca}
    
    \medskip
 
\parpic[r]{\input{figs/SampleDiagram.pdf_t}}
%{\normalfont \itshape February 18, 2025.}
Given a knot $K$, draw it in the plane
as a long knot diagram $D$ with $n$ crossings in such a way that the
two strands intersecting at each crossing are pointing up, and so that
at its beginning and at its end the knot is oriented upward. Label
with a running index $k$
which runs from 1 to $2n+1$, and with a ``rotation number'' $\varphi_k$, the
geometric rotation number of that edge. On the right the rotation numbers for all edges are $0$
except for $\varphi_4=-1$.

Let $A$ be the $(2n+1)\times(2n+1)$ matrix with
entries in the ring $\bbZ[T^{\pm 1}]$ obtained by starting with the identity matrix $I_{2n+1}$
and adding to it one contribution per crossing as follows ($s$ is the
sign of the crossing):
\def\ip{i\!+\!1} \def\jp{j\!+\!1}
\begin{equation} \label{eq:Xings}
  \begin{array}{c}\input{figs/Xings.pdf_t}\end{array}
  \longrightarrow 
  \begin{array}{c|cccc}
    \text{add at} &   \text{col. }\ip &  \text{col. }\jp \\
    \hline
    \text{row }i & -T^s  & T^s-1 \\
    \text{row }j & 0  & -1
  \end{array}
\end{equation}
    
\columnbreak\par
    
 \parpic[r]{
 \framebox(60,75){\scriptsize %Postage ENLARGED THIS FOR A5
 }
 }
  four planar binary
  trees, each of whose roots has no left child---these four trees are shown beside the rectangulation, {\bfseries where two of the trees are single nodes (or childless roots).}
The nodes on the right correspond to the rectangles on the left. 
  
  This gives a bijective proof of the result of Asinowski and Banderier, and is part of a joint project-in-process with these authors.  


  
    
     
     \medskip
     
     \scriptsize
         \begin{thebibliography}{1}
           
\bibitem{AB}
Asinowski  \& Banderier.
\newblock {From geometry to generating functions: rectangulations and permutations}.
to appear.

\end{thebibliography}


\vfill
 \hfill {\scriptsize \, \textcopyright 2024 Mathematical Research Postcards}
    
\end{multicols} 


\end{document}