Volume conjecture and quantum hyperbolic invariants: the figure eight knot complement

Overview & Synopsis

We compute the real part of the semi-classical limit of the sequence of quantum hyperbolic invariants (QHI) of the figure-eight knot complement M . We show that it is rigid, in the sense that it does not depend on the choice of holonomy representation of M , and it is either 0 or equal to the hyperbolic volume of M divided by 2π, depending on a parity condition satisfied by logarithms of the holonomy eigenvalues on the canonical longitude, where the logarithms are parameters of the QHI of M . Along the way we also survey some relevant general features of the QHI.