Communications in Mathematical Physics

, Volume 175, Issue 2, pp 275–296 | Cite as

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

  • L. Rozansky
Article

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.

Keywords

Neural Network Manifold Quantum Field Theory Quantum Computing Integral Formulation 

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Copyright information

© Springer-Verlag 1996

Authors and Affiliations

  • L. Rozansky
    • 1
  1. 1.Physics DepartmentUniversity of MiamiCoral GablesUSA

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