from the $L_{i}$, which we write as $\Sigma(L_{i+1},
K_{i})$, $\Sigma(K_{i+1}, L_{i})$, and $\Sigma(L_{i+1}, L_{i})$. These are all obtained from $\Sigma(K_{i+1},K_{i})$ by adding one or two disks. Pictures of $\Sigma(L_{1}, K_{0})$ and $\Sigma(L_{1}, L_{0})$ are given in Figure \[fig:cobordism\]. The latter foam has a single tetrahedral point. In the picture of the cobordism from $L_{1}$ to $L_{0}$, we have labeled as $\delta_{1}$ and $\delta_{0}$ the edges of the webs $L_{1}$ and $L_{0}$ which are contained in the interior of the ball. These edges appear on the boundary of disks $\Delta^{+}_{1}$ and $\Delta^{-}_{0}$ in the foam $\Sigma(L_{1}, L_{0})$. The tetrahedral point is the unique intersection point $\Delta^{+}_{1}\cap\Delta^{-}_{0}$.

![\[fig:cobordism\] The cobordisms from $L_{1}$ to $K_{0}$ (left) and from $L_{1}$ to $L_{0}$ (right).](figures/cobordism)

The standard cobordisms give maps such as $$J(I\times Y, \Sigma(L_{i+1}, L_{i}) ; \nu)
    : J(Y, L_{i+1}; \mu) \to  J(Y, L_{i}; \mu)$$ where the marking data $\nu$ is the product $I\times \mu$. Thus we have a sequence of maps with period $3$, $$\label{eq:J-exact-triangle}
        \dots \to J(Y, L_{2}; \mu) \to  J(Y, L_{1}; \mu) \to  J(Y,
        L_{0}; \mu) 
   \to  J(Y, L_{2}; \mu) \to  \cdots$$

\[thm:exact-triangle\] The above sequence is exact.

As a special case, we can consider webs in $\R^{3}$ and apply the functor 