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A contribution of the trivial connection to the Jones polynomial and Witten's invariant of 3d manifolds, I

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Abstract

We use a path integral formulation of the Chern-Simons quantum field theory in order to give a simple “semi-rigorous” proof of a recently conjectured limitation on the 1/K expansion of the Jones polynomial of a knot and its relation to the Alexander polynomial. A combination of this limitation with the finite version of the Poisson resummation allows us to derive a surgery formula for the contribution of the trivial connection to Witten's invariant of rational homology spheres. The 2-loop part of this formula coincides with Walker's surgery formula for the Casson-Walker invariant. This proves a conjecture that the Casson-Walker invariant is proportional to the 2-loop correction to the trivial connection contribution. 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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Communicated by G. Felder

Work supported in part by the National Science Foundation under Grants No. PHY-92 09978 and 9009850 and by the R. A. Welch Foundation.

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Rozansky, L. A contribution of the trivial connection to the Jones polynomial and Witten's invariant of 3d manifolds, I. Commun.Math. Phys. 175, 275–296 (1996). https://doi.org/10.1007/BF02102409

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