Conical $SL(3)$ foams

Abstract

In the unoriented $SL(3)$ foam theory, singular vertices are generic singularities of two-dimensional complexes. Singular vertices have neighborhoods 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 neighborhoods 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.