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The Danish National Research Foundation:
Network in Mathematical Physics and Stochastics



Funded by The Danish National Research Foundation

MPS-RR 2003-28
October 2003




Asymptotics of the Quantum Invariants for Surgeries on the Figure 8 Knot

by: Jørgen Ellegaard Andersen, Søren Kold Hansen

Abstract

We investigate the Reshetikhin--Turaev invariants associated to $frsl_{2}(C)$ for the $3$--manifolds obtained by doing any rational surgery along the figure $8$ knot. In particular, we express these invariants in terms of certain complex double contour integrals. These integral formulae allow us propose a formula for the leading asymptotic of the invariants in the limit of large quantum level. We analyze this expression using the saddle point method. We prove that the stationary points for the relevant phase function are in one to one correspondence with flat $SL(2,C)$--connections on the $3$--manifold and that the value of the phase function at the relevant critical points equals the classical Chern-Simons invariant of the corresponding flat $SU(2)$--connections. Our findings are in agreement with the asymptotic expansion conjecture. Moreover, we calculate the leading asymptotics of the colored Jones polynomial of the figure $8$ knot following Kashaev [Kash]. This leads to a slightly finer asymptotic description of the invariant than predicted by the volume conjecture [MM].

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This paper has now been published in Journal of Knot Theory and its Ramifications (to appear).