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<inf>g</inf>) where f(t),g(t)∈R[t], where R is an elementary divisor domain and C<inf>g</inf> 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<inf>g</inf>) to that of the matrix F(C<inf>G</inf>), 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<inf>g</inf>) as a resultant. A key tool is the observation that a matrix ring generated by C<inf>g</inf> – the companion ring of g(t) – is isomorphic to the polynomial ring Q<inf>g</inf>=R[t]/<g(t)>. We relate several features of the Smith form of f(C<inf>g</inf>) to the properties of the polynomial g(t) and the equivalence classes [f(t)]∈Q<inf>g</inf>. 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<inf>g</inf>). 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.