  We propose the Volume Conjecture for the relative Reshetikhin-Turaev
invariants of a closed oriented $3$-manifold with a colored framed link inside
it whose asymptotic behavior is related to the volume and the Chern-Simons
invariant of the hyperbolic cone metric on the manifold with singular locus the
link and cone angles determined by the coloring. We prove the conjecture in the
case that the ambient $3$-manifold is obtained by doing an integral surgery
along some components of a fundamental shadow link and the complement of the
link in the ambient manifold is homeomorphic to the fundamental shadow link
complement, for sufficiently small cone angles. Together with Costantino and
Thurston's result that all compact oriented $3$-manifolds with toroidal or
empty boundary can be obtained by doing an integral surgery along some
components of a suitable fundamental shadow link, this provides a possible
approach of solving Chen-Yang's Volume Conjecture for the Reshetikhin-Turaev
invariants of closed oriented hyperbolic $3$-manifolds. We also introduce a
family of topological operations (the change-of-pair operations) that connect
all pairs of a closed oriented $3$-manifold and a framed link inside it that
have homeomorphic complements, which correspond to doing the partial discrete
Fourier transforms to the corresponding relative Reshetikhin-Turaev invariants.
As an application, we find a Poisson Summation Formula for the discrete Fourier
transforms.
