MPS-RR 2003-28
October 2003
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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