  We show that if {L_n} is any infinite sequence of links with twist number
tau(L_n) and with cyclotomic Jones polynomials of increasing span, then lim sup
tau(L_n)=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.
