In Mikhail Gromov's "Filling Riemannian Manifolds" he defines the distortion of a knot $K$ embedded in $S^3$ as $$\delta (K) := \inf_{\gamma \in K} \sup_{x,y \in \gamma} \frac{d_{\gamma}(x,y)}{||x-y||} \geq 1. $$
It is clear to me why this is well defined, since this is independent of the representation of the knot K you choose. The theorem I'm having trouble proving (and couldn't find a proof of) is as follows -
$$\delta(K) = \frac{\pi}{2} $$ if and only if $K$ is the unknot.
I've successfully proven one direction. If $K$ is the unknot then the furthest apart two points can be is $\pi r$ where r represents the radius of the embedding. In the bottom, you get something like $$ \sqrt{2r^2 - 2r\cos(2 \pi(x-y)) }. $$ By inspection, this is minimized for $r =1 $, that is the standard embedding of the unknot. Further elementary reasoning shows that $ 2 - 2(-1) = 4 \Rightarrow \sqrt{4} = 2. $ Hence $$\delta(K) = \frac{\pi}{2}$$ whenever $K$ is the unknot.
The other direction has me stumped. I know that Sullivan and Denne showed that $$ \delta(K) \geq \frac{5\pi}{3}$$ whenever $K$ is a knotted curve. Moreover, Pardon has some bounds for $p,q$ Torus knots in the form of $$ \delta{K} \geq \frac{1}{160}\min(p,q). $$ However, Gromov was able to prove this fact independent of the more modern results, I was hoping someone could help me prove this fact.
I think I got to the bottom of this - it's a little shaky, but it's what I got so far.
To prove the converse, set $ \delta(K) = \frac{\pi}{2}.$ Without loss of generality, suppose $\gamma([0,1]) = K$ satisfies $$ |\gamma| := \int_{0}^{1} |\gamma '(t)| \mathrm{d} t = 2\pi $$ for all $x,y \in K$ then $$ ||x-y||_{\mathbb{R}^3} \leq 2. $$ Thus $$ f(x) := \frac{x-y}{2} $$ is non-increasing whenever $y \in K$ such that $$\pi = d_{\gamma}(x,y) = \sup_{x' \in K} d_{\gamma}(x,z). $$ Note that such a $y \in K$ exists since $K$ is compact and $$ d_{\gamma} (x,y) := \int_{t_0}^{t_1} |\gamma'(t)| \mathrm{d} t $$ where $\gamma(t_0) = x$ and $\gamma(t_1) = y$ and $\gamma$ is a continuous mapping of a compact set. So, $d_{\gamma}(x, -)$ is a continuous function over the compact set $K$.
Therefore, $$2 = d_{\mathbb{R}^3}(f(x),f(y)) \leq d_{\mathbb{R}^3}(f(x),f(z)) $$ for any $z \in K.$ That is, our map $f$ is distance non-increasing; a basic fact in geometry implies that $f$ is an isometry from $K$ to the unit circle. It goes without saying that the unit circle is an embedding of the unkot.