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

Adrien Casejuane, Jean-Baptiste Meilhan

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.

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

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.

Paper Structure

This paper contains 23 sections, 31 theorems, 96 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Conventions and Notation
  4. Conway polynomial and the $U_n$ invariant
  5. The Casson-Walker-Lescop invariant
  6. Universal invariants
  7. Chord diagrams and Jacobi diagrams
  8. The Kontsevich integral
  9. The degree $\le 1$ part of the LMO invariant
  10. Coefficients of the Kontsevich integral
  11. Operations on Jacobi diagrams
  12. Preliminaries
  13. Inflations and Infections
  14. Essential diagrams
  15. Factorization results
  16. ...and 8 more sections

Key Result

Theorem 1

Let $L$ be a framed oriented $n$-component link in $S^3$, and let $\mathbb{L}$ denote its linking matrix. Let $S^3_L$ be the result of surgery on $S^3$ along $L$. The Casson-Walker-Lescop invariant $\lambda_L(S^3_L)$ is given by where

Figures (2)

  • Figure 1.1: Two examples of elements in $\mathcal{E}^-(6)$ (left) and $\mathcal{P}(7)$ (right).
  • Figure 3.1: Smoothing chord diagrams on a circle.

Theorems & Definitions (82)

  • Theorem 1: Thm. \ref{['Thm general']}
  • Remark 1.1
  • Theorem 2: Thm. \ref{['c_n']}
  • Definition 2.1
  • Theorem 2.2: Casson
  • Definition 2.3
  • Definition 2.4
  • Remark 2.5
  • Definition 2.6
  • Definition 2.7
  • ...and 72 more