COMPANION MATRIX FOR SUPERPOSITION OF POLYNOMIALS AND ITS APPLICATION TO KNOT THEORY

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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.

About the authors

A. D Mednykh

Sobolev Institute of Mathematics; Novosibirsk State University

Email: smedn@mail.ru
Novosibirsk, Russia

I. A Mednykh

Sobolev Institute of Mathematics; Novosibirsk State University

Email: ilyamednykh@mail.ru
Novosibirsk, Russia

G. K Sokolova

Sobolev Institute of Mathematics; Novosibirsk State University; Novosibirsk State Technical University

Email: g.sokolova@g.nsu.ru
Novosibirsk, Russia

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