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We introduce a new cohomology theory for planar trivalent graphs with perfectmatchings. The graded Euler characteristic of the cohomology is a one variablepolynomial called the 2-factor polynomial that, if nonzero when evaluated atone, implies that the perfect matching is even and therefore the graph is4-face colorable. We also define several new polynomials invariants of graphswith and without perfect matchings that are invariants of abstract tensorssystems and spin networks defined by Roger Penrose in the 1970s. We show howsome of these polynomials can be ``categorified'' into their own homologytheories.

A Cohomology Theory for Planar Trivalent Graphs with Perfect Matchings