The interlace polynomial: a new graph polynomial

We define a new graph polynomial, the interlace polynomial, for any undirected graph. Also, we show how to count Euler circuits and circuitdecompositions for any directed or undirected Eulerian graph, by a straightforward reduction formula. For 2-in, 2-out directed graphs D, any Euler circuit induces an undirected "interlace" graph H, and there is a close relationship between the number of circuit decompositions of D and the interlace polynomial of H. We explore this relationship, and properties of the interlace polynomial in general.