  We show that in a prime, closed, oriented 3-manifold M, equivalent knots are
isotopic if and only if the orientation preserving mapping class group is
trivial. In the case of irreducible, closed, oriented $3$-manifolds we show the
more general fact that every orientation preserving homeomorphism which
preserves free homotopy classes of loops is isotopic to the identity. In the
case of $S^1\times S^2$, we give infinitely many examples of knots whose
isotopy classes are changed by the Gluck twist.
