Equimodular curves

This paper is motivated by a problem that arises in the study of partition functions of Potts models, including as a special case chromatic polynomials. When the underlying graphs have the form of ‘bracelets’, the chromatic polynomials can be expressed in terms of the eigenvalues of a matrix. In this situation a theorem of Beraha, Kahane and Weiss asserts that the zeros of the polynomials approach the curves on which the matrix has two eigenvalues with equal modulus. It is shown here that (in general) these ‘equimodular’ curves comprise a number of segments, the end-points of which are the roots (possibly coincident) of a polynomial equation. The equation represents the vanishing of a discriminant, and the segments are in bijective correspondence with the double roots of another polynomial equation, which is significantly simpler than the discriminant equation. Singularities of the segments can occur, corresponding to the vanishing of a Jacobian. In addition, it is proved by algebraic means that the equimodular curves for a reducible matrix are closed curves. The question of dominance is investigated, and a method of constructing the dominant equimodular curves for a reducible matrix is suggested. These results are illustrated by explicit calculations in a specific case.