Knot intensity distribution: a local measure of entanglement

TL;DR

This paper introduces the knot intensity distribution, a local measure to identify and analyze regions of entanglement in curves, aiding in understanding knot tightness and topological changes.

Contribution

The paper presents a novel local quantifier called the knot intensity distribution, which helps locate entanglement regions and assess knot tightness.

Findings

Distribution computed for ideal knots.

Effective in locating entanglement regions.

Identifies regions leading to topological changes.

Abstract

The problem of finding robust and effective methods for locating entanglement in embedded curves is relevant to both applications and theoretical investigations. Rather than focusing on an exact determination, we introduce the knot intensity distribution, a local quantifier for the contribution of a curve's region to global entanglement. The integral of the distribution yields a measure of tightness for knots. We compute the distribution for ideal knots, and study its behaviour on prime and composite random knots. Intensity distributions provide an effective method to locate entanglement. In particular, they identify regions in knots that accommodate passages leading to topological changes.