Perturbative invariant of Seifert fibered rational homology spheres associated with trivial first cohomology class

In this paper, we shall give a general formula for the quantum SU(2)-invariant of Seifert fibered rational homology spheres over a 2-sphere associated with the trivial first cohomology class modulo two, based on the linear skein theory. We shall derive the Casson–Walker invariant from the formula. Also, we shall give a formula for a second coefficient of the perturbative invariant associated with the trivial first cohomology class modulo two, which is a topological invariant. We shall give a relation between the second coefficient and the Ohtsuki invariant. We shall show that the second coefficient for Seifert fibered integral homology spheres belongs to 3Z.