A refinement of the LMO invariant for 3-manifolds with the first Betti number 1

Abstract

It is known that the LMO invariant of 3-manifolds with positive first Betti numbers is relatively weak and can be determined by “(semi-)classical” invariants such as the cohomology ring, the Alexander polynomial, and the Casson–Walker–Lescop invariant.

In this paper, we formulate a refinement of the LMO invariant for 3-manifolds with the first Betti number 1. It dominates the perturbative SO(3) invariant of such 3-manifolds, which is the power series invariant formulated by the arithmetic perturbative expansion of the quantum SO(3) invariants of such 3-manifolds. As the 2-loop part of the refinement of the LMO invariant, we define the 2-loop polynomial of such 3-manifolds. Further, as the $𝔰{𝔩}_m$ reduction at large $m$ limit of the $\ell$-loop part of the refinement of the LMO invariant for $\ell \le 5$, we formulate an $\ell$-variable polynomial invariant of such 3-manifolds whose Alexander polynomial is constant.