Graphs, foams, tensors, polytopes, and homology

Abstract

In the study of knotted trivalent graphs and their higher dimensional analogue, knotted foams, some of the moves have alternative interpretations. Here three interpretations are given. (1) As the boundaries of chains in a homology theory, (2) as a system of abstract tensor relations, and (3) as a collection of polyhedra that include the permutohedron. The homological interpretation will allow for a solution to the abstract tensor system.