Reshetikhin's Formula for the Jones Polynomial of a Link: Feynman diagrams and Milnor's Linking Numbers

TL;DR

This paper proves a formula for the Jones polynomial of a link using Feynman diagrams, connecting it to Milnor's linking numbers through integral representations and stationary phase analysis.

Contribution

It provides a Feynman diagram-based proof of Reshetikhin's formula for the Jones polynomial, linking it to Milnor's linking numbers and flat connections.

Findings

Expresses the Jones polynomial as an integral over coadjoint orbits.

Relates the stationary phase approximation to flat connections in the link complement.

Conjectures a relation between the phase's dominant part and Milnor's linking numbers.

Abstract

We use Feynman diagrams to prove a formula for the Jones polynomial of a link derived recently by N.~Reshetikhin. This formula presents the colored Jones polynomial as an integral over the coadjoint orbits corresponding to the representations assigned to the link components. The large $k$ limit of the integral can be calculated with the help of the stationary phase approximation. The Feynman rules allow us to express the phase in terms of integrals over the manifold and the link components. Its stationary points correspond to flat connections in the link complement. We conjecture a relation between the dominant part of the phase and Milnor's linking numbers. We check it explicitly for the triple and quartic numbers by comparing their expression through the Massey product with Feynman diagram integrals.