Multiscale Jones Polynomial and Persistent Jones Polynomial for Knot Data Analysis

TL;DR

This paper introduces multiscale and persistent Jones polynomial models to analyze the local entanglement of curves in 3-space, providing robust tools for practical applications in science and engineering.

Contribution

It proposes localized models based on the Jones polynomial, extending classical knot theory to include local structural information and analyzing their stability against perturbations.

Findings

Models are stable under small perturbations.

Multiscale and persistent Jones polynomials are robust to curve variations.

Enhanced analysis of curve entanglement in real-world data.

Abstract

Many structures in science, engineering, and art can be viewed as curves in 3-space. The entanglement of these curves plays a crucial role in determining the functionality and physical properties of materials. Many concepts in knot theory provide theoretical tools to explore the complexity and entanglement of curves in 3-space. However, classical knot theory primarily focuses on global topological properties and lacks the consideration of local structural information, which is critical in practical applications. In this work, two localized models based on the Jones polynomial, namely the multiscale Jones polynomial and the persistent Jones polynomial, are proposed. The stability of these models, especially the insensitivity of the multiscale and persistent Jones polynomial models to small perturbations in curve collections, is analyzed, thus ensuring their robustness for real-world applications.