On the distortion of knots on embedded surfaces

TL;DR

This paper establishes a new lower bound for the distortion of certain knots, specifically torus knots, answering a question posed by Gromov in 1983.

Contribution

It provides the first nontrivial lower bound for the distortion of torus knots, advancing understanding of knot geometry on embedded surfaces.

Findings

Distortion of torus knots exceeds (1/160) times the minimum of p and q.

Answers a long-standing open question by Gromov from 1983.

Provides a quantitative measure of knot complexity on surfaces.

Abstract

Our main result is a nontrivial lower bound for the distortion of some specific knots. In particular, we show that the distortion of the torus knot $T_{p,q}$ satisfies $\delta(T_{p,q})>\frac 1{160}\min(p,q)$. This answers a 1983 question of Gromov.