The LMO-invariant of 3-manifolds of rank one and the Alexander polynomial

TL;DR

This paper establishes a deep connection between the LMO-invariant and the Alexander polynomial for rank one 3-manifolds, showing they determine each other, and extends this relationship to knots in rational homology spheres.

Contribution

It proves the equivalence between the LMO-invariant and the Alexander polynomial for rank one 3-manifolds and relates the Alexander polynomial of knots to the Aarhus invariant.

Findings

LMO-invariant determines the Alexander polynomial for rank one 3-manifolds

Alexander polynomial can be derived from the LMO-invariant

Alexander polynomial of knots in rational homology spheres relates to the Aarhus invariant

Abstract

We prove that the LMO-invariant of a 3-manifold of rank one is determined by the Alexander polynomial of the manifold, and conversely, that the Alexander polynomial is determined by the LMO-invariant. Furthermore, we show that the Alexander polynomial of a null-homologous knot in a rational homology 3-sphere can be obtained by composing the weight system of the Alexander polynomial with the Aarhus invariant of knots.