We show that determining the Jones polynomial of an alternating link is #P-hard. This is a special case of a wide range of results on the general intractability of the evaluation of the Tutte polynomial T(M; x, y) of a matroid M except for a few listed special points and curves of the (x, t/)-plane. In particular the problem of evaluating the Tutte polynomial of a graph at a point in the (x, y) -plane is #P-hard except when (x-l)(y-l) = l or when (x,y) equals (

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This article proves the conjecture of Thomas that, for every graph G; there is an integer k such that every graph with no minor isomorphic to G has a 2-coloring of either its vertices or its edges where each color induces a graph of tree-width at most k: Some generalizations are also proved. r

We show that, for every integer n greater than two, there is a number N such that every 3-connected binary matroid with at least N elements has a minor that is isomorphic to the cycle matroid of K 3, n , its dual, the cycle matroid of the wheel with n spokes, or the vector matroid of the binary matrix (I n | J n &I n ), where J n is the n_n matrix of all ones.