Integral Expressions for the Vassiliev Knot Invariants

TL;DR

This thesis provides an elementary topological approach to integral expressions for Vassiliev knot invariants, introduces new constructions of configuration space compactification, and relates integrals to diagram counting, producing invariants in Q.

Contribution

It offers a new topological perspective on integral knot invariants, including a novel construction of configuration space compactification and diagram-based invariants.

Findings

Integral integrals converge and produce knot invariants.

New construction of configuration space compactification.

Integrals can be interpreted as counting tinkertoy diagrams.

Abstract

It has been folklore for several years in the knot theory community that certain integrals on configuration space, originally motivated by perturbation theory for the Chern-Simons field theory, converge and yield knot invariants. This was proposed independently by Gaudagnini, Martellini, and Mintchev and Bar-Natan. The analytic difficulties involved in proving convergence and invariance were reportedly worked out by Bar-Natan, Kontsevich, and Axelrod and Singer. But I know of no elementary exposition of this fact. ... This thesis is an attempt to remedy this lack. I adopt an almost exclusively topological point of view, rarely mentioning Chern-Simons theory. There are also a few new results in this thesis. These include a new construction of the functorial compactification of configuration space (Section 3.2) as well as some variations on the integrals. For a suitable choice of this variation, the integral reduces to counting tinkertoy diagrams (Section 4.5). In particular, the invariants constructed take values in Q.