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 ℤ/22ℤ.