Tóm tắt
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.
