Conical SL(3) foams

TL;DR

This paper extends the unoriented SL(3) foam theory by considering more general singular vertices based on planar trivalent graphs, including the dodecahedron, and shows that associated modules are free of rank 60.

Contribution

It introduces a generalization of the SL(3) foam theory to include broader classes of singular vertices with specific graph conditions, resolving certain module rank issues.

Findings

Modules for the dodecahedron graph are free of rank 60.

Generalized singular vertices include more complex planar trivalent graphs.

The extension simplifies the homology theory for these graphs.

Abstract

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.