\input{figures/exterior_paths}
\begin{figure}[htbp]
\centering
\begin{tikzpicture}[line width=.75pt,line cap=round,line join=round]
  \begin{scope}[xshift=-5.15cm,scale=.62]
    \drawDodecahedralExterior
    \draw (portNW)--(v0)--(portNE);
    \draw (portSW)--(v1)--(portSE);
    \draw[line width=1.45pt] (v0)--(v1);
    \fill (v0) circle[radius=1.8pt];
    \fill (v1) circle[radius=1.8pt];
    \node[font=\small,inner sep=1pt,anchor=east] at (1.05,.68) {$0$};
    \node[font=\small,inner sep=1pt,anchor=east] at (1.05,-.68) {$1$};
    \node[font=\normalsize] at (.35,-4.40) {$D=L_0$};
  \end{scope}
  \begin{scope}[scale=.62]
    \drawDodecahedralExterior
    \node[font=\small,inner sep=1pt,anchor=south west] at (-.39,1.54) {$9$};
    \node[font=\small,inner sep=1pt,anchor=south west] at (2.29,1.70) {$10$};
    \node[font=\small,inner sep=1pt,anchor=north west] at (2.29,-1.70) {$2$};
    \node[font=\small,inner sep=1pt,anchor=north west] at (-.39,-1.54) {$11$};
    \node[font=\normalsize] at (.35,-4.40) {$E$};
  \end{scope}
  \begin{scope}[xshift=5.15cm,scale=.62]
    \drawDodecahedralExterior
    \draw (portNW)--(1.00277702,0.18193783);
    \draw (1.16889297,-0.18193783)--(portSE);
    \draw (portNE)--(portSW);
    \node[font=\normalsize] at (.35,-4.40) {$K_2$};
  \end{scope}
\end{tikzpicture}
\caption{The dodecahedron $D$, the common exterior $E$, and $K_2$,
shown with the same planar exterior. Delete the portion inside the dashed
circle, including the bold edge $01$, to obtain $E$. Its four ends lead
to the labelled vertices $9,10,2,11$, in northwest, northeast, southeast
and southwest order. Reinsert two crossing arcs to obtain $K_2$; the
northwest--southeast arc passes underneath. The same four ends are used
in Figure~\ref{fig:octahedron}.}
\label{fig:exterior-webs}
\label{fig:dodecahedron-exterior}
\end{figure}