# Metrics on the 3-sphere with knotted geodesics

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 all its simple closed geodesics are topologically knotted?

• Perhaps relevant is: arxiv.org/abs/math-ph/9906021 where Etnyre and Grist establish existence of analytic contact flows on the three-sphere whose solution curves realize every knot type. Geodesic flow on the three-sphere however is a contact flow on a two-sphere bundle OVER the three-sphere so is a different matter ... somehow feels easier to me. (? but is it?) – Richard Montgomery Jul 28 '15 at 23:43
• This feels a strange conjecture, because contrapositively you are asking whether, in some metric, every unknot is has intrinsic curvature. – Jesse C. McKeown Mar 24 '16 at 18:35
• Then I go ahead and calculate Euler-Lagrange equations for the action functional $\int \langle \nabla_X X, \nabla_X X\rangle dt$, and indeed there are Riemann curvature terms as well as $\nabla_X^3 X$ terms... – Jesse C. McKeown Mar 28 '16 at 2:22