This article is cited in 1 paper

MATHEMATICS

Companion matrix for superposition of polynomials and its application to knot theory

A. D. Mednykh^ab, I. A. Mednykh^ab, G. K. Sokolova<sup>abc

a</sup> Sobolev Institute of Mathematics, Siberian Branch of the Russian Academy of Sciences, Novosibirsk
^b Novosibirsk State University
^c Novosibirsk State Technical University

Abstract: The note provides a new formula for the companion matrix of the superposition of two polynomials over a commutative ring. The results obtained are used to provide a constructive proof of Plans' theorem for two-bridge knots, which states that the first homology group of an odd-sheeted 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-sheeted coverings factored by the reduced homology group of a two-sheeted covering. The structure of the above mentioned Abelian groups is described through Chebyshev polynomials of the second and fourth kind.

Keywords: Smith normal form, companion matrix, knot, homology group, branched covering.

UDC: 517.53+512.714+515.162

Presented: A. T. Fomenko
Received: 05.12.2024
Revised: 14.02.2025
Accepted: 17.02.2025

DOI: 10.31857/S2686954325010096

 English version: Doklady Mathematics, 2025, 111:1, 36–43

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