  We use SO(3) gauge theory to define a functor from a category of unoriented
webs and foams to the category of finite-dimensional vector spaces over the
field of two elements. We prove a non-vanishing theorem for this SO(3)
instanton homology of webs, using Gabai's sutured manifold theory. It is hoped
that the non-vanishing theorem may support a program to provide a new proof of
the four-color theorem.
