Continuant, Chebyshev polynomials, and Riley polynomials

Abstract

In the previous paper, we showed that the Riley polynomial R-K(lambda) of each 2-bridge knot K is split into R-K(-u(2)) = +/- g(u)g(-u), for some integral coefficient polynomial g(u) is an element of Z[u]. In this paper, we study this splitting property of the Riley polynomial. We show that the Riley polynomial can be expressed by 'is an element of-Chebyshev polynomials', which is a generalization of Chebyshev polynomials containing the information of is an element of(i)-sequence (is an element of(i) = (-1)([i beta/alpha])) of the 2-bridge knot K = S(alpha, beta), and then we give an explicit formula for the splitting polynomial g(u) also as is an element of-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.