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

TL;DR

This paper uses Chern-Simons theory to connect the trivial connection's contribution to Witten's invariant with the Casson-Walker invariant, proving a conjecture and deriving a surgery formula.

Contribution

It proves that the Casson-Walker invariant is a 2-loop correction to the trivial connection contribution in Witten's invariant and derives a surgery formula for this correction.

Findings

Casson-Walker invariant is a 2-loop correction to Witten's invariant.

Derived a surgery formula for trivial connection contributions.

Calculated contributions for Seifert manifolds.

Abstract

We use the Chern-Simons quantum field theory in order to prove a recently conjectured limitation on the 1/K expansion of the Jones polynomial of a knot and its relation to the Alexander polynomial. This limitation allows us to derive a surgery formula for the loop corrections to the contribution of the trivial connection to Witten's invariant. The 2-loop part of this formula coincides with Walker's surgery formula for Casson-Walker invariant. This proves a conjecture that Casson-Walker invariant is a 2-loop correction to the trivial connection contribution to Witten's invariant of a rational homology sphere. A contribution of the trivial connection to Witten's invariant of a manifold with nontrivial rational homology is calculated for the case of Seifert manifolds.