The distortion of a knot in R^3 can be arbitrary large

You need to know: Euclidean space ${\mathbb R}^3$ , Euclidean distance in ${\mathbb R}^3$ , curve in ${\mathbb R}^3$ , closed curves, self-intersecting and non-self-intersecting curves, arclength along the curve.

Background: A knot representative is a closed, non-self-intersecting curve $\gamma$ in ${\mathbb R}^3$ . Two knot representatives $\gamma_1$ and $\gamma_2$ are equivalent if $\gamma_1$ can be transformed smoothly, without intersecting itself, to coincide with $\gamma_2$ . A knot K is the set of all knot representatives $\gamma$ equivalent to a fixed knot representative $\gamma_0$ . The distortion of knot representative $\gamma$ is $\delta(\gamma)=\sup\limits_{p,q \in \gamma}\frac{d_\gamma(p,g)}{d_{{\mathbb R}^3}(p,q)}$ , where $d_\gamma$ denotes the arclength along $\gamma$ and $d_{{\mathbb R}^3}$ denotes the Euclidean distance in ${\mathbb R}^3$ . The distortion of a knot K is $\delta(K) = \inf\limits_{\gamma \in K} \delta(\gamma)$ .

The Theorem: On 10th October 2010, John Pardon submitted to arxiv a paper in which he proved that for any $C>0$ there exists a knot K in ${\mathbb R}^3$ with $\delta(K) > C$ .

Short context: The question whether every knot has a “nice” representative is well-studied for various definitions of “nice”, see here for an example. In 1983, Gromov asked whether every knot K in ${\mathbb R}^3$ have a representative $\gamma$ with $\delta(\gamma) < 100$ . The Theorem shows that this is not the case, with any constant in place of $100$ . In fact, Pardon presented explicit examples of families of knots with distortion going to infinity. For example, let $\gamma_{p,q}$ be the curve in ${\mathbb R}^3$ defined by equations $x=r\cos(p\phi)$ , $y=r\sin(p\phi)$ , $z=-\sin(q\phi)$ , where $r=\cos(q\phi)+2$ and $0\leq \phi <2\pi$ . Set $K_{p,q}$ of curves equivalent to $\gamma_{p,q}$ is called the $(p,q)$ -torus knot. Pardon proved that $\delta(K_{p,q})\geq \frac{1}{160}\min(p,q)$ . The Theorem follows.