Reshetikhin's Formula for the Jones Polynomial of a Link: Feynman diagrams 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.

33 pages, 11 figures

References in corpus (2)

• A Contribution of the Trivial Connection to Jones Polynomial and Witten's Invariant of 3d Manifolds I
• A Contribution of the Trivial Connection to the Jones Polynomial and Witten's Invariant of 3d Manifolds II

Cited by in corpus (12)

• A Contribution of the Trivial Connection to Jones Polynomial and Witten's Invariant of 3d Manifolds I
• Classical BV theories on manifolds with boundary
• Witten's Invariants of Rational Homology Spheres at Prime Values of $K$ and Trivial Connection Contribution
• Quantum Invariants, Modular Forms, and Lattice Points II
• Lectures on quantization of gauge systems
• A Contribution of the Trivial Connection to the Jones Polynomial and Witten's Invariant of 3d Manifolds II
• Link Invariants from Classical Chern-Simons Theory
• The Topological Theory of the Milnor Invariant $\bar{\mu }(1,2,3)$
• A contribution of a U(1)-reducible connection to quantum invariants of links II: Links in rational homology spheres
• Invariants from classical field theory
• Quantum cyclotomic orders of 3-manifolds
• The universal sl_2 invariant and Milnor invariants