Companion Matrix for Composition of Polynomials and Its Application to Knot Theory

Companion Matrix for Composition of Polynomials and Its Application to Knot Theory Full article

Journal

Doklady Mathematics
ISSN: 1064-5624 , E-ISSN: 1531-8362

Output data

Year: 2025, Volume: 111, Number: 1, Pages: 36–43 Pages count : 8 DOI: 10.1134/S106456242460266X

Funding (1)

Sobolev Institute of Mathematics

FWNF-2022-0005

A new formula is given for the companion matrix of the composition of two polynomials over a commutative ring. The results obtained are used to provide a constructive proof of Plans’ theorem for 2-bridge knots, which states that the first homology group of an odd-fold cyclic covering of a three-dimensional sphere branched over a given knot is the direct sum of two copies of some Abelian group. A similar result is also true for the homology of even-fold coverings factored by the reduced homology group of two-fold coverings. The structure of the above-mentioned Abelian groups is described through Chebyshev polynomials of the second and fourth kinds.

Mednykh A.D. , Mednykh I.A. , Sokolova G.K.
Companion Matrix for Composition of Polynomials and Its Application to Knot Theory
Doklady Mathematics. 2025. V.111. N1. P.36–43. DOI: 10.1134/S106456242460266X WOS Scopus

Медных А.Д. , Медных И.А. , Соколова Г.К.
Сопровождающая матрица суперпозиции полиномов и ее применение к теории узлов
Доклады Академии наук. Серия: Математика, информатика, процессы управления. 2025. Т.521. №1. С.72–80. DOI: 10.31857/S2686954325010096 РИНЦ

Submitted:	Dec 5, 2024
Accepted:	Feb 17, 2025
Published print:	Oct 17, 2025
Published online:	Oct 17, 2025

Web of science:	WOS:001595668900009
Scopus:	2-s2.0-105019388562

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