Vertex distortion of lattice knots

TL;DR

This paper investigates the vertex distortion of lattice knots, establishing that only the unknot has minimal distortion and demonstrating the existence of complex knots with arbitrarily high distortion.

Contribution

It extends classical results on smooth knot distortion to the lattice knot setting, revealing new properties of lattice knot conformations.

Findings

Vertex distortion equals 1 only for the unknot.

Existence of lattice knots with arbitrarily high vertex distortion.

Minimal lattice-stick number knots can have high distortion.

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 l1-norm. We show analogous results to those of Gromov, Pardon and Blair-Campisi-Taylor-Tomova 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 that there are minimal lattice-stick number knot conformations with arbitrarily high distortion.