Quantum cyclotomic orders of 3-manifolds (Q1591431)

The \(SO(3)\) quantum invariant \(\tau_p(M)\) of a \(3\)-manifold \(M\) is a combination of the Jones' polynomial for links, evaluated at a convenient root of unity. From its construction, this invariant lives in a cyclotomic field \(\mathbb{Q}(\zeta)\), with \(\zeta^p=1\). If \(p\) is a prime, then it was shown by \textit{H. Murakami} [Math. Proc. Camb. Philos. Soc. 117, No. 2, 237-249 (1995; Zbl 0854.57016)] and by \textit{G. Masbaum} and \textit{J. D. Roberts} [ibid. 121, No. 3, 443-454 (1997; Zbl 0882.57010)] that the invariant \(\tau_p(M)\) belongs to the ring of integers \(\mathbb{Z}[\zeta]\). The authors study number theoretic properties of this invariant. In the ring \(\mathbb{Z}[\zeta]\) there exists a unique prime ideal containing \(p\), and it is a prime ideal. The quantum \(p\)-order of the \(3\)-manifold invariant is the valuation associated with this ideal. The main results in this paper give relations between this order and classical invariants. In particular, it is shown that the \(p\)-order is bounded below by a linear function of the mod \(p\) first Betti number. A sharper bound is obtained for a special class of \(3\)-manifolds, and examples are computed.