  In this paper, we consider the Reshetikhin-Turaev invariants of knots in the
three-sphere obtained from a twisted Drinfeld double of a Hopf algebra, or
equivalently, the relative Drinfeld center of the crossed product
$\text{Rep}(H)\rtimes\text{Aut}(H)$. These are quantum invariants of knots
endowed with a homomorphism of the knot group to $\text{Aut}(H)$. We show that,
at least for knots in the three-sphere, these invariants provide a
non-involutory generalization of the Fox-calculus-twisted Kuperberg invariants
of sutured manifolds introduced previously by the author, which are only
defined for involutory Hopf algebras. In particular, we describe the
$SL(n,\mathbb{C})$-twisted Reidemeister torsion of a knot complement as a
Reshetikhin-Turaev invariant.
