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A Cohomology Theory for Planar Trivalent Graphs with Perfect Matchings

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arxiv 1810.07302 v3 pith:TAZY7D6M submitted 2018-10-16 math.GT math.CO

classification math.GTmath.CO
keywords perfectcohomologygraphsmatchingsinvariantsplanarpolynomialpolynomials
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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.

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Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. OpenAlex reports about 3 citations worldwide. Full citation record

  1. New relations for the vertex polynomial

    math.CO 2026-07 conditional novelty 6.0 of 10

    The vertex polynomial satisfies local relations for digon, triangle, quadrilateral, and pentagon faces, extending it to arbitrary-degree graphs.

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