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

Jørgen Ellegaard Andersen, Søren Kold Hansen

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Abstract

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].
Original languageEnglish
JournalJournal of Knot Theory and Its Ramifications
Volume15
Issue number4
Pages (from-to)479-548
Number of pages70
ISSN0218-2165
DOIs
Publication statusPublished - 2006
Externally publishedYes

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