  We introduce a new cohomology theory for planar trivalent graphs with perfect
matchings. The graded Euler characteristic of the cohomology is a one variable
polynomial called the 2-factor polynomial that, if nonzero when evaluated at
one, implies that the perfect matching is even and therefore the graph is
4-face colorable. We also define several new polynomials invariants of graphs
with and without perfect matchings that are invariants of abstract tensors
systems and spin networks defined by Roger Penrose in the 1970s. We show how
some of these polynomials can be ``categorified'' into their own homology
theories.
