SO(3) invariants of Seifert manifolds and their algebraic integrality

Abstract

For Seifert manifold $M=X({p_1}/_{\f{q_1}},{p_2}/_{\f{q_2}}, ...,{p_n}/_ {\f{q_n}}), \tau^{'}_r(M)$ is calculated for all $r$ odd $\geq 3$. If $r$ is coprime to at least $n-2$ of $p_k$ (e.g. when $M$ is the Poincare homology sphere), it is proved that $(\sqrt {\dfrac{4}{r}}\sin \dfrac{\pi}{r})^{\nu}\tau^{'}_r(M)$ is an algebraic integer in the r-th cyclotomic field, where $\nu$ is the first Betti number of $M$. For the torus bundle obtained from trefoil knot with framing 0, i.e. $X_{tref}(0)=X(-2/_{\f{1}},3/_{\f{1}},6/_{\f{1}}), \tau^{'}_r$ is obtained in a simple form if $3\mid\llap /r$, which shows in some sense that it is impossible to generalize Ohtsuki's invariant to 3-manifolds being not rational homology spheres.