  The distortion of a curve is the supremum, taken over distinct pairs of
points of the curve, of the ratio of arclength to spatial distance between the
points. Gromov asked in 1981 whether a curve in every knot type can be
constructed with distortion less than a universal constant C. Answering
Gromov's question seems to require the construction of lower bounds on the
distortion of knots in terms of some topological invariant. We attempt to make
such bounds easier to construct by showing that pairs of points with high
distortion are very common on curves of minimum length in the set of curves in
a given knot type with distortion bounded above and distortion thickness
bounded below.

Geometric Topology