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Published online by Cambridge University Press: 03 February 2010
We define a polynomial W on graphs with colours on the edges, by generalizing the spanning tree expansion of the Tutte polynomial as far as possible: we give necessary and sufficient conditions on the edge weights for this expansion not to depend on the order used. We give a contraction-deletion formula for W analogous to that for the Tutte polynomial, and show that any coloured graph invariant satisfying such a formula can be obtained from W. In particular, we show that generalizations of the Tutte polynomial obtained from its rank generating function formulation, or from a random cluster model, can be obtained from W. Finally, we find the most general conditions under which W gives rise to a link invariant, and give as examples the one-variable Jones polynomial, and an invariant taking values in ℝ/22ℝ.
Introduction
Basic definitions
Throughout this paper, we shall consider the set of finite multigraphs, with loops allowed. Usually, we shall call an element of a graph, but sometimes we shall write multigraph for emphasis.
The Tutte polynomial, or dichrornate of [34], is an isomorphism-invariant function which arises in many different ways. We shall consider four different definitions of the Tutte polynomial.
The first definition, due to Tutte [34], is based on a spanning tree expansion. We take an order ø on E(G), and, for each spanning tree S of G, use ø and S to classify the edges of G into four types. We then assign a weight to each edge, depending on its type, and multiply these weights to find the weight of S.
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