Vassiliev invariants for flows via Chern–Simons perturbation theory

Abstract

The perturbative expansion of Chern–Simons gauge theory leads to invariants of knots and links, the so-called finite type invariants or Vassiliev invariants. It has been proved that at any order in perturbation theory the superposition of certain amplitudes is an invariant of that order. Bott–Taubes integrals on configuration spaces are introduced in the present context to write Feynman diagrams at a given order in perturbation theory in a geometrical and topological framework. One of the consequences of this formalism is that the resulting amplitudes are rewritten in cohomological terms in configuration spaces. This cohomological structure can be used to translate Bott–Taubes integrals into Chern–Simons perturbative amplitudes and vice versa. In this paper, this program is performed up to third order in the coupling constant. This expands some work previously worked out by Thurston. Finally we take advantage of these results to incorporate in the formalism a smooth and divergenceless vector field on the [Formula: see text]-manifold. The Bott–Taubes integrals obtained are used for constructing higher-order average asymptotic Vassiliev invariants extending the work of Komendarczyk and Volić.