  We find an explicit formula of the twisted Reidemeister torsion of the
fundamental shadow link complements twisted by the adjoint action of the
holonomy representation of the (possibly incomplete) hyperbolic structures, and
of the double of hyperbolic polyhedral $3$-manifolds twisted by the adjoint
action of the holonomy representation of the hyperbolic polyhedral metrics,
which turn out to be a product of the determinant of the Gram matrix functions
evaluated respectively at the logarithmic holonomies of the meridians and at
the edge lengths. As a consequence, we obtain an explicit formula of the
twisted Reidemeister torsion of closed hyperbolic $3$-manifolds obtained by
doing a hyperbolic Dehn-surgery along a fundamental shadow link complement, and
of the double of a geometrically triangulated hyperbolic $3$-manifold with
totally geodesic boundary, respectively in terms of the boundary logarithmic
holonomies and of the edge lengths. We notice that by \cite{CT} most closed
oriented hyperbolic $3$-manifolds can be obtained from a suitable fundamental
shadow link complement by doing a hyperbolic Dehn-surgery. These formulas play
an essential role in the study of the asymptotic expansion of certain quantum
invariants in \cite{WY}.
