Laffite Thesis LaTeX

Laffy.tex

\documentclass{article}
\usepackage{tikz}
\usepackage{amsmath, amssymb}

\begin{document}

\section*{Homotopy Equivalence in Diagrammatic Algebra}

\textbf{Composing this with homotopy equivalence gives:}

\subsection*{First Map}

\begin{equation}
    \begin{tikzpicture}
        % First diagram
        \draw[red] (0,0) to[out=90,in=270] (1,1);
        \draw[black] (1,0) to[out=90,in=270] (0,1);
        \node at (0.5, 1.2) {\( t_{ij}^{-1} \)};
    \end{tikzpicture}
    +
    \begin{tikzpicture}
        % Second diagram
        \draw[red] (0,0) to[out=90,in=270] (1,1);
        \draw[black] (1,0) to[out=90,in=270] (0,1);
        \node at (0.5, 1.2) {\( t_{ij} t_{ij}^{-1} \)};
    \end{tikzpicture}
    =
    \begin{tikzpicture}
        % Resulting diagram
        \draw[red] (0,0) to[out=90,in=270] (0,1);
        \draw[black] (1,0) to[out=90,in=270] (1,1);
    \end{tikzpicture}
\end{equation}

\textbf{Holds on nose.}

\subsection*{Second Map}

\begin{equation}
    -V_{ij} +
    \begin{tikzpicture}
        \draw[red] (0,0) to[out=90,in=270] (1,1);
        \draw[black] (1,0) to[out=90,in=270] (0,1);
        \node at (0.5, 1.2) {\( t_{ij}^{-1} \)};
    \end{tikzpicture}
    =
    0 + t_{ij}^{-1}
\end{equation}

\textbf{Third Map:}

\begin{equation}
    \begin{tikzpicture}
        \draw[red] (0,0) to[out=90,in=270] (1,1);
        \draw[black] (1,0) to[out=90,in=270] (0,1);
        \node at (0.5, 1.2) {\( t_{ij}^{-1} \)};
    \end{tikzpicture}
    +
    \begin{tikzpicture}
        \draw[red] (0,0) to[out=90,in=270] (1,1);
        \draw[black] (1,0) to[out=90,in=270] (0,1);
        \node at (0.5, 1.2) {\( t_{ij} t_{ij} \)};
    \end{tikzpicture}
    =
    \begin{tikzpicture}
        \draw[red] (0,0) to[out=90,in=270] (0,1);
        \draw[black] (1,0) to[out=90,in=270] (1,1);
    \end{tikzpicture}
\end{equation}

\textbf{Exactly what I need.}

\subsection*{Fifth Map}

\begin{equation}
    -t_{ij}^{-1}
    \begin{tikzpicture}
        \draw[red] (0,0) to[out=90,in=270] (1,1);
        \draw[black] (1,0) to[out=90,in=270] (0,1);
    \end{tikzpicture}
    =
    -
    \begin{tikzpicture}
        \draw[red] (0,0) to[out=90,in=270] (1,1);
        \draw[black] (1,0) to[out=90,in=270] (0,1);
        \node at (0.5, 1.2) {\( t_{ij} \)};
    \end{tikzpicture}
    +
    \begin{tikzpicture}
        \draw[red] (0,0) to[out=90,in=270] (1,1);
        \draw[black] (1,0) to[out=90,in=270] (0,1);
        \node at (0.5, 1.2) {\( -t_{ij} t_{ij} \)};
    \end{tikzpicture}
\end{equation}

\end{document}