TITLE: Fáry-Milnor theorem for positively curved metrics on $S^3$?
QUESTION [6 upvotes]: I'm interested in generalizations the following well-known theorem of Fáry and Milnor.
Theorem. (Fáry-Milnor) If a simple closed curve $\gamma \subset \mathbb{R}^3$ is knotted, then the total curvature of $\gamma$ is greater than $4\pi$.
I have found some sources that prove essentially the same result when $\mathbb{R}^3$ is equipped with a non-positively curved metric (e.g. the usual hyperbolic metric), maybe satisfying some extra hypotheses (for example, see [1,2]). I'm interested in the opposite situation.
Question. Is there a version of the Fáry-Milnor theorem for positively curved metrics on $S^3$?
It seems that one could derive some inequality from the usual one using contraction maps to and from Euclidean space and estimating the resulting effect on the total curvature along $\gamma$. However, this would depend strongly on the metrics/contraction maps involved. I would really prefer something metric independent if possible.
[1] F. Brickell and C. C. Hsiung. The total absolute curvature of closed curves in Riemannian manifolds.
[2] C. Schmitz. The Theorem of Fáry and Milnor for Hadamard Manifolds

REPLY [3 votes]: I expand a little on a comment made below Ian Agol's answer, including an improvement proposed by him.
Claim: every knot can be realized on the round sphere $\mathbb{S}^3$ with arbitrarily small (but positive) curvature.
Indeed, a knot can be presented as the closure of a braid; you an realize the closed braid inside a small tubular neighborhood of a closed geodesic, with the braid $C^2$-close to this core geodesic. This ensures the total curvature is as close to zero as you wish.
This is especially easy to represent oneself with torus knots, but actually works for every link, as mentionned by Ian Agol.