a similar (and equivalent) statement about the two composites from $L'_{2}$ to $K_{2}$.

Fuller versions of these results are stated in section \[sec:statements\], where we also broaden the scope of the theorems a little by discussing webs embedded in arbitrary oriented $3$-manifolds, rather than in $\R^{3}$.

##### Acknowledgement.

The authors are very grateful for the support of the Radcliffe Institute for Advanced Study, which provided them with the opportunity to pursue this project together as Fellows of the Institute during the academic year 2013–2014.

Review of $\SO(3)$ instanton homology {#sec:recap}
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We briefly recall some of the constructions which are described more fully in [@KM-jsharp]. If $Z$ is an $n$-dimensional orbifold, and $z\in Z$, then we write $H_{z}$ for the local stabilizer group at $z$. All our orbifolds will be orientable, so $H_{z}$ is a subgroup of $\SO(n)$ acting effectively on $\R^{n}$.

#### Bifolds.

For $m\le n$, let $H_{m}\subset \SO(m)$ be the elementary abelian $2$-group of order $2^{m-1}$ consisting of diagonal matrices of determinant $+1$ whose diagonal entries are $\pm
1$. Regard $H_{m}$ also as a subgroup of $\SO(n)$ for $m\le n$. We call $Z$ an $n$-dimensional *bifold* if its local stabilizer groups $H_{z}\subset \SO(n)$ are conjugate to $H_{m}$ for 