  In the unoriented SL(3) foam theory, singular vertices are generic
singularities of two-dimensional complexes. Singular vertices have
neighbourhoods homeomorphic to cones over the one-skeleton of the tetrahedron,
viewed as a trivalent graph on the two-sphere. In this paper we consider foams
with singular vertices with neighbourhoods homeomorphic to cones over more
general planar trivalent graphs. These graphs are subject to suitable
conditions on their Kempe equivalence Tait coloring classes and include the
dodecahedron graph. In this modification of the original homology theory it is
straightforward to show that modules associated to the dodecahedron graph are
free of rank 60, which is still an open problem for the original unoriented
SL(3) foam theory.
