A Polynomial Invariant of Graphs On Orientable Surfaces

Our aim in this paper is to construct a polynomial invariant of cyclic graphs, that is, graphs with cyclic orders at the vertices, or, equivalently, of 2-cell embeddings of graphs into closed orientable surfaces. We shall call this invariant the cyclic graph polynomial, and denote it by the letter C . The cyclic graph polynomial is a three-variable polynomial which generalizes the Tutte polynomial in an essential way. In the next section we de®ne cyclic graphs from two different viewpoints, and give some background on cyclic graphs and the Tutte polynomial. In § 3 we discuss one-vertex cyclic graphs, thought of as chord diagrams, introducing an algebraic notion of the rank of a chord diagram needed to de®ne the cyclic graph polynomial. In § 4 we de®ne the cyclic graph polynomial in terms of recurrence relations and a `boundary condition' on one-vertex cyclic graphs, and state our main result ± that these relations have a (unique) solution. This is proved in § 5. In § 6 we state and prove a universal property of the cyclic graph polynomial. In § 7 we give an alternative description of the rank of a chord diagram, as the genus of the surface naturally associated to the diagram. Finally, in § 8 we give some further properties of the cyclic graph polynomial, showing that it has a spanning tree expansion, and, more importantly, that it depends on the orders (or embedding) in an essential way.