Infinitely many roots of unity are zeros of some Jones polynomials

TL;DR

The paper constructs families of prime knots with Jones polynomials that have roots of unity as zeros, demonstrating infinitely many roots of unity are roots of some Jones polynomials, and explores their algebraic properties.

Contribution

It explicitly constructs Jones polynomials with roots at roots of unity, linking knot theory with cyclotomic polynomials and Mahler measure analysis.

Findings

All roots of unity $\

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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.