Vassiliev invariants I: Braid groups and rational homotopy theory

TL;DR

This paper explores Vassiliev invariants through the lens of braid groups and rational homotopy theory, connecting Chen's iterated integrals, Malcev completion, and the universal Kontsevich-Vassiliev invariant.

Contribution

It provides a detailed account of Vassiliev invariants, linking braid group completions with universal invariants and extending these concepts to the entire braid group.

Findings

Identification of the pure braid group's injection with the universal Kontsevich-Vassiliev invariant

Extension of the invariant to the whole braid group

Description of the multiplication law for the full braid group

Abstract

We get a detailed account of Vassiliev type invariants starting with Chen's theory of iterated integrals and Malcev's completion of discrete groups. The canonical injection of the group of pure braids into its completion is identified with the universal Kontsevich-Vassiliev invariant.Further we discuss the extension of this morphism to the whole braid group and the multiplication law for the last one.