Infinitely many roots of unity are zeros of some Jones polynomials

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 $(-1{)}^M{\sum }_{k=-M}^M(-1{)}^k{t}^k$. 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.

arXiv comment14 pages, 6 figures