ASYMPTOTICS OF THE QUANTUM INVARIANTS FOR SURGERIES ON THE FIGURE 8 KNOT

By J. Andersen, S. K. Hansen. 2005

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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].

Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/456efb611bd4598f66d04adb5971a6cdb3dbe53d

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Citations in the corpus

Exact results for perturbative Chern-Simons theory with complex gauge group. (2009, background): “10) were proved in [1] and the rationality of the numbers sn defined by (4.”
Quantum modular forms (2010, background): “89 × 10(45) Explicitly, J0(ζN ) is given by the the asymptotic formula [1]”
On the asymptotic expansion of the Kashaev invariant of the $5_2$ knot (2016, background): “The volume conjecture has been rigorously proved for some particular knots and links such as torus knots [19] (see also [5](1)), the figure-eight knot (by Ekholm, see also [1](2)), 1A detailed asymptotic expansion of the colored Jones polynomial for torus knots is given in [5].”