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Integral Expressions for the Vassiliev Knot Invariants

Dylan P. Thurston
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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.

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Dylan P. Thurston. 1999-01-26. Integral Expressions for the Vassiliev Knot Invariants. https://arxiv.org/abs/math/9901110

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