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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 theRiley polynomial. We show that the Riley polynomial can be expressed by`$\epsilon$-Chebyshev polynomials', which is a generalization of Chebyshevpolynomials 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 splittingpolynomial $g(u)$ also as $\epsilon$-Chebyshev polynomials. As applications, wefind a sufficient condition for the irreducibility of the Riley polynomials andshow the unimodal property of the symmetrized Riley polynomial.

Continuant, Chebyshev polynomials, and Riley polynomials