The Fáry-Milnor Theorem, as stated in Kristopher Tapp's Differential Geometry of Curves and Surfaces, states that, for a unit-speed simple closed (Tapp uses the convention that only regular curves are called closed) space curve $\gamma: [a, b] \to \mathbb R^3$ whose curvature function $\kappa$ is nowhere zero, if $\gamma$ is knotted (i.e. if $\gamma$ has no spanning disk), then $\int_a^b\kappa(t) \, dt \geq 4 \pi$. Tapp proves this by showing that the total Gaussian curvature of the portion with nonnegative curvature of a tubular neighborhood around $\gamma$ is at least $8 \pi$. Then, in an exercise, we are asked to show that, in fact, $\int_a^b\kappa(t) \, dt > 4 \pi$.
The only thought I've really had is to show that if the total curvature is exactly $4 \pi$, then $\gamma$ is a plane curve, which would force $\gamma$ to be unknotted by the Jordan curve theorem, but I'm struggling to show this. My attempts have included
- trying to extend Lemma 2.17, which states that a closed curve in $S^2$ whose trace intersects every great circle has arc length at least $2 \pi$ with equality if and only if the curve isa simple parametrization of a great circle (and which Tapp used to give the first proof in the text of Fenchel's theorem), to curves that intersect every great circle more than once (which we know must be true about $\gamma'$ for $\gamma$ to be knotted), and
- splitting $\gamma'$ into two curves, each of which Lemma 2.17 applies to.
However, I haven't really made any progress with the first attempt, and the second doesn't seem very promising since Lemma 2.17 only applies to closed curves, and it doesn't seem we can guarantee that $\gamma'$ can be split into two closed curves (with matching derivatives at the endpoints).
You have to check the equality case in the proof. It is easy to see that equality implies that it must be locally convex plane curve. Since its total curvature is $4{\cdot}\pi$, the curev cannot be simple --- a contradiction.