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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We study spin models for invariants of links as defined by Jones [22]. We consider the two algebras generated by the weight matrices of such models under ordinary or Hadamard product and establish an isomorphism between them. When these algebras coincide they form the Bose-Mesner algebra of a formally self-dual association scheme, We study the special case of strongly regular graphs, which is associated to a particularly interesting link invariant, the Kauffman polynomial [27]. This leads to a classification of spin models for the Kauffman polynomial in terms of formally self-dual strongly regular graphs with strongly regular subconstituents [7]. In particular we obtain a new model based on the Higman-Sims graph [171. 24 FRAN(~OIS JAEGER generalizations of the dichromatic polynomial [38], called spin models (see, for instance, [3], [15], [22]).A state of a graph is an assignment to each vertex of one spin, or color, chosen in a certain finite set. Then each edge receives a weight depending only on its sign and on the unordered pair of colors of its ends (this weight function defines the model), and the weight of a state is the product of weights of the edges. The corresponding graph invariant is the partition function of the model, i.e. the sum of weights of states. Spin models which define link invariants can be characterized by a few equations to be satisfied by the weight functions [22], and the Kauffman polynomial will be obtained, provided a single additional equation is satisfied.The above equations can be easily written in matrix form, using the usual matrix product and the Hadamard (i.e. componentwise) product. In Section 2, after a presentation of the general framework, we associate to any spin model which satisfies these equations a pair of algebras (one for each type of product) and we establish an isomorphism between them. An interesting case is when both algebras coincide with the Bose-Mesner algebra of some association scheme (see, for instance, [1], [5]). Then the above isomorphism acts as a formal duality (a combinatorial abstraction of the discrete Fourier transform). Moreover, the equation which defines the Kauffman polynomial leads to this situation, with the additional constraint that the Bose-Mesner algebra has dimension at most 3. In other words, formally self-dual strongly regular graphs are the natural setting for the study of spin models for the Kauffman polynomial.We present such a study in Section 3. It turns out that a primitive formally self-dual strongly regular graph will yield a spin model if and only if each of its subconstituents is strongly regular (a property already studied by Cameron et al. [7]). This yields a classification of the spin models for specializations of the Kauffman polynomial which unifies the description of previously known models and also exhibits a new one based on the Higman-Sims graph [17].We conclude in Section 4 with a few questions and perspectives of future research. LINK INVARIANTS AND SPIN MODELS Link Diagrams and Link InvariantsA link is a finite col...

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