Continuant, Chebyshev polynomials, and Riley polynomials

TL;DR

This paper explores the splitting properties of Riley polynomials for 2-bridge knots, expressing them through generalized Chebyshev polynomials and providing criteria for their irreducibility and unimodality.

Contribution

It introduces a novel expression of Riley polynomials using epsilon-Chebyshev polynomials and derives explicit formulas for splitting polynomials, advancing understanding of their algebraic structure.

Findings

Riley polynomial can be expressed by epsilon-Chebyshev polynomials.

Provides explicit formulas for the splitting polynomial g(u).

Identifies conditions for Riley polynomial irreducibility and unimodality.

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.