Hopf Algebras and the Penrose Polynomial

Let λ be a positive integer and let G be a plane graph. LetP (G, λ) be the Penrose polynomial of G. We will present an interpretation ofP (G, − λ) in terms of colourings of G. In order to prove our main theorem we construct a Hopf algebra A of graphs and a homomorphism of Hopf algebras Ψ fromA onto a Hopf algebra of polynomials in one indeterminate. If G is a plane graph, thenΨ (G) coincides with the Penrose polynomial of G.