a Hopf link $H$ contained in a ball near the point at infinity. As marking data for $(S^{3}, K^{\sharp})$, we take $U_{\mu}$ to be the ball containing $H$, disjoint from $K$, and we take $E_{\mu}$ to have $w_{2}\ne 0$ on the torus which separates the two components of the Hopf link. This marking data is strong. We define $$\Jsharp(K) = J(S^{3}, K^{\sharp};\mu).$$ Given a foam cobordism $\Phi\subset [0,1]\times\R^{3}$ from $K_{1}$ to $K_{2}$, we similarly construct a new foam $\Phi^{\sharp}$ as $\Phi\cup ([0,1]\times H)$, with marking data $\nu=[0,1]\times\mu$. In this way, $\Phi$ gives rise to a linear map $$\Jsharp(\Phi) : \Jsharp(K_{1}) \to \Jsharp(K_{2}).$$ In this way we obtain a functor $\Jsharp$ with values in the category of $\F$-vector spaces, from a category whose objects are webs in $\R^{3}$ and whose morphisms are isotopy classes of foams with boundary in intervals $[a,b]\times \R^{3}$.

Statement of the results {#sec:statements}
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We state now the version of the Theorem \[thm:intro-three-bar\] that we shall prove. Instead of $\R^{3}$, we consider a closed, oriented $3$-manifold $Y$, and three webs $L_{2}$, $L_{1}$, $L_{0}$ in $Y$ which are identical outside a standard ball $B\subset Y$. As in Theorem \[thm:intro-three-bar\], we suppose that, inside the ball $B$, they look 