L_{2} = \mathfig{0.079}{figures/Graph-Xbar}, \qquad
        L_{1} = \mathfig{0.079}{figures/Graph-H}, \qquad
        L_{0} = \mathfig{0.079}{figures/Graph-I}.$$ There are standard foam cobordisms between these (see section \[sec:statements\] for a fuller description): $$\cdots \longrightarrow \mathfig{0.079}{figures/Graph-Xbar}
       \longrightarrow \mathfig{0.079}{figures/Graph-H}
          \longrightarrow \mathfig{0.079}{figures/Graph-I}
                \longrightarrow \mathfig{0.079}{figures/Graph-Xbar}
                     \longrightarrow\cdots$$ We then have:

\[thm:intro-three-bar\] The sequence of $\F$-vector spaces obtained by applying $\Jsharp$ to the above sequence of webs and foams is exact: $$\cdots \longrightarrow \Jsharp(L_{2}) \longrightarrow
                 \Jsharp(L_{1}) \longrightarrow
                  \Jsharp(L_{0}) \longrightarrow
                    \Jsharp(L_{2}) \longrightarrow \cdots$$

There is a variant of this exact triangle. Consider three webs differing in the ball as in the following diagrams: $$K_{2} = \mathfig{0.079}{figures/Graph-X}, \qquad
        K_{1} = \mathfig{0.079}{figures/Graph-Res1}, \qquad
        K_{0} = \mathfig{0.079}{figures/Graph-Res0}.$$ Again, there are standard cobordisms between these. In [@KM-unknot], we established an exact triangle in $\Isharp$ relating these three. The corresponding sequence of vector spaces $\Jsharp(K_{i})$ do *not* form an exact triangle. Instead, there is an exact triangle involving $L_{i+2}$, $K_{i+1}$ and $K_{i}$, for each $i$ (with the indices interpreted cyclically modulo $3$). Thus:

\[thm:intro-one-bar\] For each $i=0, 1, 2$, we have an exact sequence of $\F$-vector spaces, $$\cdots \longrightarrow \Jsharp(L_{i+2}) \longrightarrow
                 \Jsharp(K_{i+1}) \longrightarrow
                  \Jsharp(K_{i}) \longrightarrow
                    \Jsharp(L_{i+2}) \longrightarrow \cdots,$$ in which the maps are obtained by applying $\Jsharp$ to standard foam cobordisms.

![\[fig:J-octahedron\] An octahedral diagram whose faces are four 