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Abstract
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, skewcirculant 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 nonzero 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)=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 cyclically presented groups, this provides the homology of all Brieskorn manifolds M(r,s,n) where r,s are coprime.
Original language  English 

Pages (fromto)  119 
Number of pages  19 
Journal  JOURNAL OF ALGEBRA 
Volume  587 
DOIs  
Publication status  Published  1 Dec 2021 
MoE publication type  A1 Journal articlerefereed 
Keywords
 Brieskorn manifold
 Circulant
 Cyclically presented group
 Elementary divisor domain
 Homology
 Smith form
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 1 Active

Noferini_Vanni_AoF_Project: Noferini Vanni Academy Project
Noferini, V., Quintana Ponce, M., Barbarino, G. & Wood, R.
01/09/2020 → 31/08/2024
Project: Academy of Finland: Other research funding