We investigate the Reshetikhin-Turaev invariants associated to SU(2) for the 3-manifolds M 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 to propose a formula for the leading asymptotics of the invariants in the limit of large quantum level. We analyze this expression using the saddle point method. We construct a certain surjection from the set of stationary points for the relevant phase functions onto the space of conjugacy classes of nonabelian SL(2, C)-representations of the fundamental group of M and prove that the values of these phase functions at the relevant stationary points equals the classical Chern-Simons invariants 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 [14]. This leads to a slightly finer asymptotic description of the invariant than predicted by the volume conjecture [24].
We derive explicit formulas for the Reshetikhin–Turaev invariants of all oriented Seifert manifolds associated to an arbitrary complex finite dimensional simple Lie algebra [Formula: see text] in terms of the Seifert invariants and standard data for [Formula: see text]. A main corollary is a determination of the full asymptotic expansions of these invariants for lens spaces in the limit of large quantum level. This result is in agreement with the asymptotic expansion conjecture due to Andersen [1,2].
We calculate the RT-invariants of all oriented Seifert manifolds directly from surgery presentations. We work in the general framework of an arbitrary modular category as in [Tu], and the invariants are expressed in terms of the S -and T -matrices of the modular category. In another direction we derive a rational surgery formula, which states how the RTinvariants behave under rational surgery along framed links in arbitrary closed oriented 3-manifolds with embedded colored ribbon graphs. The surgery formula is used to give another derivation of the RT-invariants of Seifert manifolds with orientable base.
Let X be a topological space, Sin k X the space of singular k-simplices with the compact-open topology, and let c k Xbe the real vector space of all compactly supported signed Borel Measures of bounded total variation on Sin k X. There are linear operators d X c k X 3 c kÀ1 X, so that c à X Y d f gis a chain complex. The homology H " à X is the measure homology of X of Thurston and Gromov. The main results in this paper are that H " à À satis¢es the Eilenberg-Steenrod axioms for a wide class of topological spaces including all metric spaces, and is ordinary homology with real coe¤cients for CW-complexes.
We calculate the RT-invariants of all oriented Seifert manifolds directly from surgery presentations. We work in the general framework of an arbitrary modular category as in [Tu], and the invariants are expressed in terms of the S -and T -matrices of the modular category. In another direction we derive a rational surgery formula, which states how the RTinvariants behave under rational surgery along framed links in arbitrary closed oriented 3-manifolds with embedded colored ribbon graphs. The surgery formula is used to give another derivation of the RT-invariants of Seifert manifolds with orientable base.
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