تقرير
Continuant, Chebyshev polynomials, and Riley polynomials
| العنوان: | Continuant, Chebyshev polynomials, and Riley polynomials |
|---|---|
| المؤلفون: | Jo, Kyeonghee, Kim, Hyuk |
| المصدر: | Journal of Knot Theory and Its Ramifications, Vol. 31, No. 1 (2022) 2150078 |
| سنة النشر: | 2022 |
| المجموعة: | Mathematics |
| مصطلحات موضوعية: | Mathematics - Geometric Topology, 57K14, 57K31 |
| الوصف: | 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 |
| نوع الوثيقة: | Working Paper |
| DOI: | 10.1142/S0218216521500784 |
| URL الوصول: | http://arxiv.org/abs/2201.03922 |
| رقم الأكسشن: | edsarx.2201.03922 |
| قاعدة البيانات: | arXiv |
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