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

[摘要] We study the Smith forms of matrices of the form f(C-g) where f(t), g(t) is an element of R[t], where Ris an elementary divisor domain and Cgis 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, Gare 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)] is an element of Q(g). As an application we let f(t) be the Alexander polynomial of a torus knot and g(t) = t(n) - 1, and calculate the Smith form of the circulant matrix f(C-g). By appealing to results concerning cyclic branched covers of knots and

[发布日期] 2021-12-01

[关键词] Smith form;Elementary divisor domain;Circulant;Cyclically presented group