Matrices in companion rings, Smith forms, and the homology of 3-dimensional Brieskorn manifolds

We study the Smith forms of matrices of the form f(C_g) where f(t),g(t)∈R[t], where R is an elementary divisor domain and C_g is the companion matrix of the (monic) polynomial g(t). Prominent examples of such matrices are circulant matrices, skew-circulant matrices, and triangular Toeplitz matrices. In particular, we reduce the calculation of the Smith form of the matrix f(C_g) to that of the matrix F(C_G), where F,G are quotients of f(t),g(t) by some common divisor. This allows us to express the last non-zero determinantal divisor of f(C_g) as a resultant. A key tool is the observation that a matrix ring generated by C_g – the companion ring of g(t) – is isomorphic to the polynomial ring Q_g=R[t]/<g(t)>. We relate several features of the Smith form of f(C_g) to the properties of the polynomial g(t) and the equivalence classes [f(t)]∈Q_g. As an application we let f(t) be the Alexander polynomial of a torus knot and g(t)=tⁿ−1, and calculate the Smith form of the circulant matrix f(C_g). By appealing to results concerning cyclic branched covers of knots and cyclically presented groups, this provides the homology of all Brieskorn manifolds M(r,s,n) where r,s are coprime.