Vertex distortion of lattice knots

Abstract

The vertex distortion of a lattice knot is the supremum of the ratio of the distance between a pair of vertices along the knot and their distance in the [Formula: see text]-norm. Inspired by Gromov, Pardon and Blair–Campisi–Taylor–Tomova, we show that results about the distortion of smooth knots hold for vertex distortion: the vertex distortion of a lattice knot is 1 only if it is the unknot, and there are minimal lattice-stick number knot conformations with arbitrarily high distortion.