Perturbative 3-manifold invariants by cut-and-paste topology

TL;DR

This paper provides a purely topological definition of perturbative quantum invariants for links and 3-manifolds, aligning closely with Kontsevich's approach, and establishes their fundamental properties related to finite type invariants.

Contribution

It introduces a topological framework for perturbative invariants that matches Kontsevich's formulation and proves their universality as finite type invariants under specific surgeries.

Findings

Invariants are universally finite type with respect to algebraically split surgery.

Invariants are universally finite type with respect to Torelli surgery.

Torelli surgery generalizes blink and clasper surgeries.

Abstract

We give a purely topological definition of the perturbative quantum invariants of links and 3-manifolds associated with Chern-Simons field theory. Our definition is as close as possible to one given by Kontsevich. We will also establish some basic properties of these invariants, in particular that they are universally finite type with respect to algebraically split surgery and with respect to Torelli surgery. Torelli surgery is a mutual generalization of blink surgery of Garoufalidis and Levine and clasper surgery of Habiro.