Continuant, Chebyshev polynomials, and Riley polynomials
Autor: | Kyeonghee Jo, Hyuk Kim |
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Jazyk: | angličtina |
Rok vydání: | 2022 |
Předmět: |
Mathematics - Geometric Topology
Algebra and Number Theory Computer Science::Information Retrieval Mathematics::History and Overview FOS: Mathematics Astrophysics::Instrumentation and Methods for Astrophysics Computer Science::General Literature 57K14 57K31 Geometric Topology (math.GT) Computer Science::Computation and Language (Computational Linguistics and Natural Language and Speech Processing) |
Popis: | 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. Comment: 24 pages |
Databáze: | OpenAIRE |
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