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

Loading...
Thumbnail Image
Journal Title
Journal ISSN
Volume Title
A1 Alkuperäisartikkeli tieteellisessä aikakauslehdessä
Date
2021-12-01
Major/Subject
Mcode
Degree programme
Language
en
Pages
19
1-19
Series
JOURNAL OF ALGEBRA, Volume 587
Abstract
We study the Smith forms of matrices of the form f(Cg) where f(t),g(t)∈R[t], where R is an elementary divisor domain and Cg 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(Cg) to that of the matrix F(CG), 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(Cg) as a resultant. A key tool is the observation that a matrix ring generated by Cg – the companion ring of g(t) – is isomorphic to the polynomial ring Qg=R[t]/<g(t)>. We relate several features of the Smith form of f(Cg) to the properties of the polynomial g(t) and the equivalence classes [f(t)]∈Qg. 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(Cg). 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.
Description
Funding Information: The authors thank Alessia Cattabriga for insightful discussions on Brieskorn and Dunwoody manifolds, Steve Mackey for enlightening comments on the history of the Smith Theorem, and the referees for providing feedback on the presentation. Vanni Noferini acknowledges support by an Academy of Finland grant (Suomen Akatemian päätös 331240 ). Gerald Williams was supported for part of this project by Leverhulme Trust Research Project Grant RPG-2017-334 . Publisher Copyright: © 2021 The Author(s)
Keywords
Brieskorn manifold, Circulant, Cyclically presented group, Elementary divisor domain, Homology, Smith form
Other note
Citation
Noferini , V & Williams , G 2021 , ' Matrices in companion rings, Smith forms, and the homology of 3-dimensional Brieskorn manifolds ' , JOURNAL OF ALGEBRA , vol. 587 , pp. 1-19 . https://doi.org/10.1016/j.jalgebra.2021.07.018