Infinitely many roots of unity are zeros of some Jones polynomials

Let \(N=2n^2-1\) or \(N=n^2+n-1\), for any \(n\ge 2\). Let \(M=\frac{N-1}{2}\). We construct families of prime knots with Jones polynomials \((-1)^M\sum_{k=-M}^{M} (-1)^kt^k\). Such polynomials have Mahler measure equal to \(1\). If \(N\) is prime, these are cyclotomic polynomials \(\Phi_{2N}(t)\),...

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Bibliographic Details
Published in:arXiv.org 2021-02
Main Author: Mroczkowski, Maciej
Format: Article
Language:English
Subjects:
Polynomials
Unity
Online Access:Get full text
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Summary:Let \(N=2n^2-1\) or \(N=n^2+n-1\), for any \(n\ge 2\). Let \(M=\frac{N-1}{2}\). We construct families of prime knots with Jones polynomials \((-1)^M\sum_{k=-M}^{M} (-1)^kt^k\). Such polynomials have Mahler measure equal to \(1\). If \(N\) is prime, these are cyclotomic polynomials \(\Phi_{2N}(t)\), up to some shift in the powers of \(t\). Otherwise, they are products of such polynomials, including \(\Phi_{2N}(t)\). In particular, all roots of unity \(\zeta_{2N}\) occur as roots of Jones polynomials. We also show that some roots of unity cannot be zeros of Jones polynomials.
ISSN:2331-8422