added graphics for the state of the art

graphics/1-to-3_pachner_move/1-to-3_pachner_move.tex

+\documentclass{article}
+
+\usepackage{amsmath, tikz}
+
+
+\usepackage[graphics,tightpage,active]{preview}
+\PreviewEnvironment{tikzpicture}
+\PreviewEnvironment{equation}
+\PreviewEnvironment{equation*}
+\newlength{\imagewidth}
+\newlength{\imagescale}
+\pagestyle{empty}
+\thispagestyle{empty}
+
+\tikzstyle{vertex} = [circle, fill = white, draw = black]
+
+
+\begin{document}
+\begin{tikzpicture}
+  \pgftransformcm{1}{0}{0}{sqrt(3)/2}{\pgfpoint{0cm}{0cm}}
+  \draw (1, .75) node[left, rotate=30] {1} -- (2, 1.5);
+  \draw (3, .75) node[right, rotate=-30] {2} -- (2, 1.5);
+  \draw (2, 2.75) node[above] {0} -- (2, 1.5);
+  \node [vertex] at (2, 1.5) {} ;
+
+  \draw[->] (4,1.5) -- (5,1.5) ;
+
+  \draw (6 + 0, 0) -- (6 + 1, .75) node[above right] {1} ;
+  \draw (6 + 4, 0) -- (6 + 3, .75) node[above left]  {2} ;
+  \draw (6 + 2, 4) -- (6 + 2, 2.75) node[below] {0} ;
+
+  \draw (6 + 0, 0) -- (6 + 1, 2) node[above left] {2} -- (6 + 2, 4) -- (6 + 3, 2) node[above right] {1} -- (6 + 4, 0) -- (6 + 2, 0) node[below] {0} -- cycle ;
+  \node [vertex] at (6 + 0, 0) {} ;
+  \node [vertex] at (6 + 4, 0) {} ;
+  \node [vertex] at (6 + 2, 4) {} ;
+\end{tikzpicture}
+\end{document}

graphics/discrete_3_dim_sphere/discrete_3_dim_sphere.tex

+\documentclass{article}
+
+\usepackage{tikz}
+\usepackage{amsmath}
+
+\usepackage[graphics,tightpage,active]{preview}
+\PreviewEnvironment{tikzpicture}
+\PreviewEnvironment{equation}
+\PreviewEnvironment{equation*}
+\newlength{\imagewidth}
+\newlength{\imagescale}
+\pagestyle{empty}
+\thispagestyle{empty}
+
+\tikzstyle{vertex} = [circle, fill = white, draw = black]
+
+\begin{document}
+\begin{tikzpicture}
+  \pgftransformcm{1}{0}{0}{sqrt(3)/2}{\pgfpoint{0cm}{0cm}}
+  \draw (0, 0) -- (1, .75) node[above] {1} -- (2, 1.5);
+  \draw (4, 0) -- (3, .75) node[above] {2} -- (2, 1.5);
+  \draw (2, 4) -- (2, 2.75) node[left] {0} -- (2, 1.5);
+
+  \draw (0, 0) -- (1, 2) node[above left] {2} -- (2, 4) -- (3, 2) node[above right] {1} -- (4, 0) -- (2, 0) node[below] {0} -- cycle ;
+  \node [vertex] at (0, 0) {} ;
+  \node [vertex] at (4, 0) {} ;
+  \node [vertex] at (2, 4) {} ;
+  \node [vertex] at (2, 1.5) {} ;
+\end{tikzpicture}
+\end{document}

graphics/discrete_manifold_graph_duality/discrete_manifold_graph_duality.tex

+\documentclass{article}
+
+\usepackage{tikz, amsmath}
+
+\usepackage[graphics,tightpage,active]{preview}
+\PreviewEnvironment{tikzpicture}
+\PreviewEnvironment{equation}
+\PreviewEnvironment{equation*}
+\newlength{\imagewidth}
+\newlength{\imagescale}
+\pagestyle{empty}
+\thispagestyle{empty}
+
+\tikzstyle{vertex} = [circle, fill = white, draw = black]
+
+
+\begin{document}
+\begin{tikzpicture}
+  \pgftransformcm{1}{0}{0}{sqrt(3)/2}{\pgfpoint{0cm}{0cm}}
+  \draw (0,0) -- (.5, 1) node[left] {2} -- (1.5, 3) node[left] {2} -- (2, 4) -- (2.5, 3) node[right] {1} -- (3.5,  1) node[right] {1} -- (4,0) -- (3,0) node[below] {0} -- (1,0) node[below] {0} -- cycle ;
+  \draw[fill = gray!30] (1, 2) -- (2, 2) node[below] {0} -- (3, 2) -- (2.5, 1) node[left] {2} -- (2,0) -- (1.5, 1) node[right] {1} -- cycle ;
+  \node [below] at (1, 1) {} ;
+  \node [below] at (3, 1) {} ;
+  \node [below] at (2, 3) {} ;
+  \node [above] at (2, 1) {} ;
+
+  \draw[<->] (5, 2) -- (6,2) ;
+
+  \draw (8 + 0, 0) -- (8 + 1, .75) node[above, rotate = 30] {1} -- (8 + 2, 1.5);
+  \draw (8 + 4, 0) -- (8 + 3, .75) node[above, rotate = -30] {2} -- (8 + 2, 1.5);
+  \draw (8 + 2, 4) -- (8 + 2, 2.75) node[left] {0} -- (8 + 2, 1.5);
+
+  \draw (8 + 0, 0) -- (8 - 1, .75) node[left, rotate = -30] {2} ;
+  \draw (8 + 0, 0) -- (8 + 0, -1.5) node[below] {0} ;
+  \draw (8 + 4, 0) -- (8 + 5, .75) node[right, rotate = 30] {1} ;
+  \draw (8 + 4, 0) -- (8 + 4, -1.5) node[below] {0} ;
+  \draw (8 + 2, 4) -- (8 + 1, 4.75) node[left, rotate = -30] {2} ;
+  \draw (8 + 2, 4) -- (8 + 3, 4.75) node[right, rotate = 30] {1} ;
+  \node [vertex] at (8 + 0, 0) {} ;
+  \node [vertex] at (8 + 4, 0) {} ;
+  \node [vertex] at (8 + 2, 4) {} ;
+  \node [vertex] at (8 + 2, 1.5) {} ;
+\end{tikzpicture}
+\end{document}
\ No newline at end of file