Splitting formulae for the Kontsevich-Kuperberg-Thurston invariant of rational homology 3-spheres

TL;DR

This paper explores the properties of the Kontsevich-Kuperberg-Thurston invariant for rational homology 3-spheres, providing explicit formulas that connect it to known invariants like the Walker invariant.

Contribution

It generalizes a sum formula for the Casson-Walker invariant to the Kontsevich-Kuperberg-Thurston invariant, clarifying its behavior under handlebody replacements.

Findings

Explicit formulas for the invariant's behavior under handlebody replacements

Identification of the degree one term of Z with the Walker invariant

Extension of previous sum formulas to a broader class of invariants

Abstract

M. Kontsevich proposed a topological construction for an invariant Z of rational homology 3-spheres using configuration space integrals. G. Kuperberg and D. Thurston proved that Z is a universal real finite type invariant for integral homology spheres in the sense of Ohtsuki, Habiro and Goussarov. We discuss the behaviour of Z under rational homology handlebodies replacements. The explicit formulae that we present generalize a sum formula obtained by the author for the Casson-Walker invariant in 1994. They allow us to identify the degree one term of Z with the Walker invariant for rational homology spheres.