We consider a sign-determined Reidemeister torsion with multivariables for a
hyperbolic three-dimensional manifold with cusps. Using a cut and paste
argument, we prove that this Reidemeister torsion is a polynomial invariant
when provided with appropriate conditions on the topology of the manifold and
SL(2, C)-representations of its fundamental group. Under such assumptions, it
is proved that this polynomial invariant is reciprocal like the usual Alexander
polynomial.