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

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

1994-01-14

arxiv.org/abs/hep-th/9401061v1

Commun.Math.Phys. 175 (1996) 275-296

Physics

High Energy Physics

High Energy Physics - Theory

28 pages

Scientific paper

10.1007/BF02102409

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.

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