Vassiliev Invariants in the Context of Chern-Simons Gauge Theory

TL;DR

This paper reviews recent progress in understanding Vassiliev invariants through perturbative Chern-Simons gauge theory, emphasizing the potential for combinatorial formulas and illustrating explicit expressions in terms of Gauss diagrams.

Contribution

It highlights the approach of using perturbative Chern-Simons theory to derive combinatorial formulas for Vassiliev invariants, including explicit results for order four invariants.

Findings

Progress in deriving combinatorial formulas for Vassiliev invariants

Explicit expressions for order four primitive invariants in Gauss diagrams

Support for the perturbative Chern-Simons approach as promising

Abstract

We summarize the progress made during the last few years on the study of Vassiliev invariants from the point of view of perturbative Chern-Simons gauge theory. We argue that this approach is the most promising one to obtain a combinatorial universal formula for Vassiliev invariants. The combinatorial expressions for the two primitive Vassiliev invariants of order four, recently obtained in this context, are reviewed and rewritten in terms of Gauss diagrams.