Vertex distortion detects the unknot

TL;DR

This paper proves that trivial vertex distortion characterizes the unknot, confirming a conjecture and providing bounds, a calculator, and new insights into knot complexity.

Contribution

It establishes that trivial vertex distortion implies a knot is the unknot, confirming a key conjecture and advancing understanding of knot invariants.

Findings

Vertex distortion of the unknot is trivial.

Trivial vertex distortion characterizes the unknot.

A vertex distortion calculator is introduced.

Abstract

The first two authors introduced vertex distortion and showed that the vertex distortion of the unknot is trivial. It was conjectured that the vertex distortion of a knot is trivial if and only if the knot is trivial. We will use Denne-Sullivan's bound on Gromov distortion to bound the vertex distortion of nontrivial lattice knots. We will then conclude that trivial vertex distortion implies the unknot, proving the conjecture. Additionally, the first conjecture in vertex distortion's debut article is proven and a vertex distortion calculator is given.