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Abstract <p>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.</p>

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

  • 期刊论文

  • 发表日期：

    2025年2月1日

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  • A. D. Mednykh,
  • I. A. Mednykh,
  • G. K. Sokolova

摘要

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’ 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.