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

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.