Asymptotics of the quantum invariants for surgeries on the figure 8 knot

Jørgen Ellegaard Andersen, Søren Kold Hansen

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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 R.Kashaev. This leads to a slightly finer asymptotic description of the invariant than predicted by the volume conjecture due to R.Kashaev, H.Murakami and J.Murakami.
    Original languageEnglish
    JournalJournal of Knot Theory and Its Ramifications
    Issue number4
    Pages (from-to)479-548
    Publication statusPublished - 2006

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