We study three types of
statistical mechanical models for link invariants (vertex, IRF and spin models) and
some relations between them when they exhibit certain symmetries described by an
Abelian group. In particular we show the equivalence of three kinds of models:
strongly conservative vertex models on an Abelian group X, doubly translation
invariant IRF models on the same group X, and translation invariant spin models on
the direct product X × X. Some examples of constructions of spin models from
vertex models are given (the associated link invariants are the generating function for
the writhe of orientations, the Jones polynomial, and the number of Fox colourings).
Then we introduce a composition of link invariants related to the decomposition of a
link into its components, and we explore the above correspondence between
vertex, IRF and spin models in connection with this operation. As a main
consequence, we show that the link invariant associated with spin models recently
constructed by K. Nomura from Hadamard matrices is a composition of two Jones
polynomials.
