According to answers to this question every metrics on $S^3$ admits a simple closed geodesic. Given a knot (or link) $K$, it's also quite simple to build a metric on $S^3$ such that $K$ is a geodesic (for some parametrisation).

Is there a Riemannian metric on $S^3$ such that

allits simple closed geodesics are topologically knotted?

every unknotis has intrinsic curvature. $\endgroup$