The Witten–Reshetikhin–Turaev (WRT) invariants extend the Jones polynomials of links in
to invariants of links in
–manifolds. Similarly, the authors
constructed two
–manifold
invariants
and
which extend the Akutsu–Deguchi–Ohtsuki (ADO) invariant of links in
colored by complex numbers to links in arbitrary manifolds. All these
invariants are based on the representation theory of the quantum group
, where the definition
of the invariants
and
uses a nonstandard
category of
–modules
which is not semisimple. In this paper we study the second invariant,
, and consider
its relationship with the WRT invariants. In particular, we show that the ADO invariant of
a knot in
is a meromorphic function of its color, and we provide a strong relation between its residues
and the colored Jones polynomials of the knot. Then we conjecture a similar relation
between
and a WRT invariant. We prove this conjecture when the
–manifold
is not a rational homology
sphere, and when
is a rational homology sphere obtained by surgery on a knot in
or a
connected sum of such manifolds.