On links with cyclotomic Jones polynomials

Abstract

We show that if $\left\{L_n\right\}$ is any infinite sequence of links with twist number $\tau \left(L_n\right)$ and with cyclotomic Jones polynomials of increasing span, then $limsup\tau \left(L_n\right)=\infty$. This implies that any infinite sequence of prime alternating links with cyclotomic Jones polynomials must have unbounded hyperbolic volume. The main tool is the multivariable twist–bracket polynomial, which generalizes the Kauffman bracket to link diagrams with open twist sites.