Every non-trivial knot group is fully residually perfect

Given a class script upper P P P of groups we say that a group G is fully residually script upper P P P if for any finite subset F of G , there exists an epimorphism from G to a group in script upper P P P which is injective on F . It is known that any non-trivial knot group is fully residually finite, but not residually free. For hyperbolic knots, its knot group is fully residually closed hyperbolic 3–manifold group, and such a group is fully residually simple. In this paper, we show that every non-trivial knot group is fully residually perfect, closed 3–manifold group.