Continuant, Chebyshev polynomials, and Riley polynomials

Abstract

In the previous paper, we showed that the Riley polynomial ${\mathcal{R}}_K(\lambda )$ of each 2-bridge knot $K$ is split into ${\mathcal{R}}_K(-u^2)=\pm g(u)g(-u)$, for some integral coefficient polynomial $g(u)\in \mathbb{Z}[u]$. In this paper, we study this splitting property of the Riley polynomial. We show that the Riley polynomial can be expressed by `$\epsilon$-Chebyshev polynomials', which is a generalization of Chebyshev polynomials containing the information of ${\epsilon }_i$-sequence $({\epsilon }_i=(-1{)}^{[i\frac{\beta }{\alpha }]})$ of the 2-bridge knot $K=S(\alpha ,\beta )$, and then we give an explicit formula for the splitting polynomial $g(u)$ also as $\epsilon$-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.

24 pages

References in corpus (2)

• Rotundus: triangulations, Chebyshev polynomials, and Pfaffians
• Classification of parabolic generating pairs of Kleinian groups with two parabolic generators