Distortion and tree-width of knots after Pardon, Guth-Gromov and work of the speaker with some colleagues.

Given a number $m$ there are knots so that no matter the embedding of the knot there exists a pair of points $x,y$ such that $d_K(x,y)/|x-y|>m$. Here $|x-y$| is the distance for a bird and $d_K(x,y)$ is the distance for an ant walking on the knot. In the proof the tree-width will be a key invariant of independent interest.