Abstract
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.
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Mroczkowski, M. Infinitely many roots of unity are zeros of some Jones polynomials. Geom Dedicata 216, 43 (2022). https://doi.org/10.1007/s10711-022-00708-4
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DOI: https://doi.org/10.1007/s10711-022-00708-4