Universal invariants, the Conway polynomial and the Casson-Walker-Lescop invariant

TL;DR

This paper presents a diagrammatic surgery formula for the Casson-Walker-Lescop invariant, linking it to the Conway polynomial and Kontsevich integral coefficients in 3-manifold topology.

Contribution

It introduces a new combinatorial approach to the Casson-Walker-Lescop invariant using the LMO invariant and explicit relations with the Conway polynomial.

Findings

Derived a surgery formula for the Casson-Walker-Lescop invariant

Connected Conway polynomial coefficients to Kontsevich integral coefficients

Provided a diagrammatic, combinatorial perspective on 3-manifold invariants

Abstract

We give a general surgery formula for the Casson-Walker-Lescop invariant of closed 3-manifolds, seen as the leading term of the LMO invariant, in a purely diagrammatic and combinatorial way. This provides a new viewpoint on a formula established by C. Lescop for her extension of the Walker invariant. A central ingredient in our proof is an explicit identification of the coefficients of the Conway polynomial as combinations of coefficients in the Kontsevich integral. This latter result relies on general \lq factorization formulas\rq\, for the Kontsevich integral coefficients.