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 Rep ( H ) â Aut ( H ) \text {Rep}(H)\rtimes \text {Aut}(H) . These are quantum invariants of knots endowed with a homomorphism of the knot group to Aut ( H ) \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 S L ( n , C ) SL(n,\mathbb {C}) -twisted Reidemeister torsion of a knot complement as a Reshetikhin-Turaev invariant.