The Fundamental Theorem of Vassiliev Invariants

TL;DR

This paper examines four different approaches to proving the fundamental theorem of Vassiliev invariants, highlighting unresolved issues and the ongoing lack of complete understanding of the theorem.

Contribution

It provides a comparative analysis of topological, geometrical, algebraic, and physical approaches to the theorem's proof, emphasizing their limitations.

Findings

All approaches have unresolved issues.

The theorem's proof remains incomplete.

Current methods do not fully explain the theorem's foundations.

Abstract

The "fundamental theorem of Vassiliev invariants" says that every weight system can be integrated to a knot invariant. We discuss four different approaches to the proof of this theorem: a topological/combinatorial approach following M. Hutchings, a geometrical approach following Kontsevich, an algebraic approach following Drinfel'd's theory of associators, and a physical approach coming from the Chern-Simons quantum field theory. Each of these approaches is unsatisfactory in one way or another, and hence we argue that we still don't really understand the fundamental theorem of Vassiliev invariants.