Perturbative invariants of 3–manifolds with the first Betti number 1

Abstract

It is known that perturbative invariants of rational homology 3–spheres can be constructed by using arithmetic perturbative expansion of quantum invariants of them. However, we could not make arithmetic perturbative expansion of quantum invariants for 3–manifolds with positive Betti numbers by the same method.

In this paper, we explain how to make arithmetic perturbative expansion of quantum $SO\left(3\right)$ invariants of 3–manifolds with the first Betti number $1$. Further, motivated by this expansion, we construct perturbative invariants of such 3–manifolds. We show some properties of the perturbative invariants, which imply that their coefficients are independent invariants.