On Distortion and Thickness of Knots

TL;DR

This paper introduces new notions of thickness for knots based on Gromov's distortion and generalizations of existing measures, providing inequalities and properties that advance understanding of the minimal length needed to tie specific knots.

Contribution

It defines new thickness measures for knots, establishes inequalities between them, and analyzes their continuity properties, contributing to the geometric understanding of knot thickness.

Findings

Established an inequality between the new thickness measures.

Proved that distortion-based thickness is upper semi-continuous.

Suggested the existence of shortest curves of fixed thickness in each knot class.

Abstract

What length of rope (of given diameter) is required to tie a particular knot? To answer this question, we define some new notions of thickness for a space curve, one based on Gromov's distortion, and another generalizing the thickness of Litherland, Simon et al. We prove a basic inequality between these thickness measures, and show that the distortion thickness is upper semi-continuous in the C^0 topology, suggesting that shortest curves of thickness 1 should exist in each knot class.