exact triangles and four commutative triangles.](figures/J-octahedron)

The four exact triangles contained in the two theorems above can be arranged as four of the triangular faces in an octahedral diagram, a particular realization of the diagram for the octahedral axiom for a triangulated category. In the diagram, Figure \[fig:J-octahedron\], the top vertex $L'_{2}$ has a crossing of a different sign from the picture of $L_{2}$. The exact triangles in this octahedron are $$\begin{aligned}
            \cdots &\longrightarrow \Jsharp(K_{2}) \longrightarrow
                 \Jsharp(K_{1}) \longrightarrow
                  \Jsharp(L_{0}) \longrightarrow
                    \Jsharp(K_{2}) \longrightarrow \cdots \\
            \cdots &\longrightarrow \Jsharp(K_{2}) \longrightarrow
                 \Jsharp(L_{1}) \longrightarrow
                  \Jsharp(K_{0}) \longrightarrow
                    \Jsharp(K_{2}) \longrightarrow \cdots \\
            \cdots &\longrightarrow \Jsharp(K_{0}) \longrightarrow
                 \Jsharp(K_{1}) \longrightarrow
                  \Jsharp(L'_{2}) \longrightarrow
                    \Jsharp(K_{0}) \longrightarrow \cdots \\
            \cdots &\longrightarrow \Jsharp(L_{0}) \longrightarrow
                 \Jsharp(L_{1}) \longrightarrow
                  \Jsharp(L'_{2}) \longrightarrow
                    \Jsharp(L_{0}) \longrightarrow \cdots .
\end{aligned}$$ The last two are duals of exact triangles in Theorems \[thm:intro-one-bar\] and \[thm:intro-three-bar\] respectively. The other four faces of octahedron become commutative diagrams of $\F$-vector spaces on applying $\Jsharp$. For example, the triangle $$\xymatrix{  \Jsharp(K_{1})  & & \Jsharp(K_{0}) \ar[dl] \ar[ll]\\
        & \Jsharp(K_{2}) \ar[ul]&}$$ is a commutative diagram. Finally, the two different composites from $K_{2}$ to $L'_{2}$, $$\begin{aligned}
    \mathfig{0.079}{figures/Graph-X} & \longrightarrow \mathfig{0.079}{figures/Graph-H}
    \longrightarrow \mathfig{0.079}{figures/Graph-Xbardag} \\
  \mathfig{0.079}{figures/Graph-X} & \longrightarrow \mathfig{0.079}{figures/Graph-Res1}
    \longrightarrow \mathfig{0.079}{figures/Graph-Xbardag},
\end{aligned}$$ give the same map $\Jsharp(K_{2}) \to \Jsharp(L'_{2})$, with 