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

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Companion Matrix for Composition of Polynomials and Its Application to Knot Theory. / Mednykh, A. D.; Mednykh, I. A.; Sokolova, G. K.

в: Doklady Mathematics, Том 111, № 1, 7, 02.2025, стр. 36-43.

Результаты исследований: Научные публикации в периодических изданиях › статья › Рецензирование

BibTeX

@article{16d337f934204a70ba05e0ed3cbc3a13,

title = "Companion Matrix for Composition of Polynomials and Its Application to Knot Theory",

abstract = "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{\textquoteright} 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.",

keywords = "Smith normal form, branched covering, companion matrix, homology group, knot, НОРМАЛЬНАЯ ФОРМА СМИТА, СОПРОВОЖДАЮЩАЯ МАТРИЦА, УЗЕЛ, ГРУППА ГОМОЛОГИЙ, РАЗВЕТЛЕННОЕ НАКРЫТИЕ",

author = "Mednykh, {A. D.} and Mednykh, {I. A.} and Sokolova, {G. K.}",

note = "This work was performed as part of the state assignment at the Sobolev Institute of Mathematics of the Russian Academy of Sciences, project no. FWNF-2022-0005.",

year = "2025",

month = feb,

doi = "10.1134/S106456242460266X",

language = "English",

volume = "111",

pages = "36--43",

journal = "Doklady Mathematics",

issn = "1064-5624",

publisher = "Maik Nauka-Interperiodica Publishing",

number = "1",

}

RIS

TY - JOUR

T1 - Companion Matrix for Composition of Polynomials and Its Application to Knot Theory

AU - Mednykh, A. D.

AU - Mednykh, I. A.

AU - Sokolova, G. K.

N1 - This work was performed as part of the state assignment at the Sobolev Institute of Mathematics of the Russian Academy of Sciences, project no. FWNF-2022-0005.

PY - 2025/2

Y1 - 2025/2

N2 - 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.

AB - 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.

KW - Smith normal form

KW - branched covering

KW - companion matrix

KW - homology group

KW - knot

KW - НОРМАЛЬНАЯ ФОРМА СМИТА

KW - СОПРОВОЖДАЮЩАЯ МАТРИЦА

KW - УЗЕЛ

KW - ГРУППА ГОМОЛОГИЙ

KW - РАЗВЕТЛЕННОЕ НАКРЫТИЕ

UR - https://www.scopus.com/pages/publications/105019388562

UR - https://www.elibrary.ru/item.asp?id=83051288

UR - https://www.elibrary.ru/item.asp?id=80559508

UR - https://www.mendeley.com/catalogue/08b07fad-276e-3c0b-9b12-94dbe7d5d0d3/

U2 - 10.1134/S106456242460266X

DO - 10.1134/S106456242460266X

M3 - Article

VL - 111

SP - 36

EP - 43

JO - Doklady Mathematics

JF - Doklady Mathematics

SN - 1064-5624

IS - 1

M1 - 7

ER -

ID: 71373089