\input{figures/web_macros}
\begin{figure}[htbp]
\centering
\begin{tikzpicture}[>=Stealth,line width=.65pt,
 every node/.style={font=\small},
 vertex/.style={circle,minimum size=1.38cm,inner sep=0pt},
 arrow/.style={->,shorten >=3pt,shorten <=3pt}]
\node[vertex] (F) at (0,3.4) {};
\node[vertex] (M) at (-3.25,.8) {};
\node[vertex] (C) at (3.25,.8) {};
\node[vertex] (B) at (-3.25,-2.6) {};
\node[vertex] (A) at (3.25,-2.6) {};
\node[vertex] (W) at (0,-5.2) {};
% Draw the outer arrows first.  White underlays at crossings distinguish
% intersecting arrows from the trivalent vertices in the local web pictures.
\draw[arrow] (F)--node[above left]{$\zeta$}(M);
\draw[arrow] (C)--node[above right]{$\eta$}(F);
\draw[arrow] (B)--node[left]{$t$}(M);
\draw[arrow] (C)--node[right]{$s$}(A);
\draw[arrow] (A)--node[below right]{$a$}(W);
\draw[arrow] (W)--node[below left]{$b$}(B);
\draw[arrow] (M)--node[above]{$\xi$}(C);
\draw[arrow] (A)--node[below]{$\gamma$}(B);
\draw[arrow,preaction={draw=white,line width=3.5pt}]
  (B)--node[pos=.66,left,fill=white,inner sep=1pt]{$\lambda$}(F);
\draw[arrow,preaction={draw=white,line width=3.5pt}]
  (F)--node[pos=.66,right,fill=white,inner sep=1pt]{$\kappa$}(A);
\draw[arrow,preaction={draw=white,line width=3.5pt}]
  (M)--node[pos=.64,left,fill=white,inner sep=1pt]{$q$}(W);
\draw[arrow,preaction={draw=white,line width=3.5pt}]
  (W)--node[pos=.64,right,fill=white,inner sep=1pt]{$r$}(C);
\foreach \p in {F,M,C,B,A,W}
  \filldraw[fill=white,draw=black!65,densely dotted] (\p) circle (.69);
\begin{scope}[shift={(F)},scale=.69]\webLtwoprime\end{scope}
\begin{scope}[shift={(M)},scale=.69]\webLzero\end{scope}
\begin{scope}[shift={(C)},scale=.69]\webLone\end{scope}
\begin{scope}[shift={(B)},scale=.69]\webKone\end{scope}
\begin{scope}[shift={(A)},scale=.69]\webKzero\end{scope}
\begin{scope}[shift={(W)},scale=.69]\webKtwo\end{scope}
\node[above,align=center] at (0,4.18) {$L'_2$\\$F=\J(L'_2)$};
\node[left,align=center] at (-4.02,.8) {$L_0=D$\\$M=\J(D)$};
\node[right,align=center] at (4.02,.8) {$L_1$\\$C=\J(L_1)$};
\node[left,align=center] at (-4.02,-2.6) {$K_1$\\$B=\J(K_1)$};
\node[right,align=center] at (4.02,-2.6) {$K_0$\\$A=\J(K_0)$};
\node[below,align=center] at (0,-5.98) {$K_2$\\$W=\J(K_2)$};
\end{tikzpicture}
\caption{The octahedron of \cite[Figure~1]{KM-tri}, with the local webs
and the homology groups used here shown together. Each dotted circle is
filled into the common exterior $E$ of Figure~\ref{fig:exterior-webs},
without rotating the four ends. The four exact triangles are those in
\eqref{alg:triangles}; the other faces express composition identities.
The crossing in both $K_2$ and $L'_2$ has its northwest--southeast strand
underneath. Intersections of arrows are not vertices.}
\label{fig:octahedron}
\end{figure}