Continuant, Chebyshev polynomials, and Riley polynomials

Abstract

In the previous paper, we showed that the Riley polynomial [Formula: see text] of each 2-bridge knot [Formula: see text] is split into [Formula: see text], for some integral coefficient polynomial [Formula: see text]. In this paper, we study this splitting property of the Riley polynomial. We show that the Riley polynomial can be expressed by ‘[Formula: see text]-Chebyshev polynomials’, which is a generalization of Chebyshev polynomials containing the information of [Formula: see text]-sequence [Formula: see text] of the 2-bridge knot [Formula: see text], and then we give an explicit formula for the splitting polynomial [Formula: see text] also as [Formula: see text]-Chebyshev polynomials. As applications, we find a sufficient condition for the irreducibility of the Riley polynomials and show the unimodal property of the symmetrized Riley polynomial.