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Development of a Python-Based Software for Calculating the Jones Polynomial: Insights into the Behavior of Polymers and Biopolymers

Caleb Musfeldt

Abstract

This thesis details a Python-based software designed to calculate the Jones polynomial, a vital mathematical tool from Knot Theory used for characterizing the topological and geometrical complexity of curves in \( \mathbb{R}^3 \), which is essential in understanding physical systems of filaments, including the behavior of polymers and biopolymers. The Jones polynomial serves as a topological invariant capable of distinguishing between different knot structures. This capability is fundamental to characterizing the architecture of molecular chains, such as proteins and DNA. Traditional computational methods for deriving the Jones polynomial have been limited by closure-schemes and high execution costs, which can be impractical for complex structures like those that appear in real life. This software implements methods that significantly reduce calculation times, allowing for more efficient and practical applications in the study of biological polymers. It utilizes a divide-and-conquer approach combined with parallel computing and applies recursive Reidemeister moves to optimize the computation, transitioning from an exponential to a near-linear runtime for specific configurations. This thesis provides an overview of the software's functions, detailed performance evaluations using protein structures as test cases, and a discussion of the implications for future research and potential algorithmic improvements.

Development of a Python-Based Software for Calculating the Jones Polynomial: Insights into the Behavior of Polymers and Biopolymers

Abstract

This thesis details a Python-based software designed to calculate the Jones polynomial, a vital mathematical tool from Knot Theory used for characterizing the topological and geometrical complexity of curves in , which is essential in understanding physical systems of filaments, including the behavior of polymers and biopolymers. The Jones polynomial serves as a topological invariant capable of distinguishing between different knot structures. This capability is fundamental to characterizing the architecture of molecular chains, such as proteins and DNA. Traditional computational methods for deriving the Jones polynomial have been limited by closure-schemes and high execution costs, which can be impractical for complex structures like those that appear in real life. This software implements methods that significantly reduce calculation times, allowing for more efficient and practical applications in the study of biological polymers. It utilizes a divide-and-conquer approach combined with parallel computing and applies recursive Reidemeister moves to optimize the computation, transitioning from an exponential to a near-linear runtime for specific configurations. This thesis provides an overview of the software's functions, detailed performance evaluations using protein structures as test cases, and a discussion of the implications for future research and potential algorithmic improvements.

Paper Structure

This paper contains 21 sections, 5 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Knot Theory
  3. Reidemeister Moves
  4. Jones Polynomial
  5. Bracket Polynomial
  6. Writhe
  7. The Normalized Bracket Polynomial
  8. The Jones Polynomial of Open Curves in $\mathbb{R}^3$
  9. Existing Methods and Software
  10. Software Overview
  11. $jones\_knot.py$
  12. $JONES.py$
  13. $functions.py$
  14. Implementations of More Efficient Methods
  15. Splitting and Gluing
  16. ...and 6 more sections

Figures (5)

  • Figure 1: Examples of mathematical knots and links. The knots and links we focus on live in 3 dimensions. It is easier, however, to use their 2 dimensional diagrams with over/under information at the crossing to study the 3D knots and links. Image from BestvinaND
  • Figure 2: Reidemeister moves are local moves on diagrams. By applying them multiple times we can go from any knot diagram to any other diagram of the same knot. Image from Baez2018
  • Figure 3: Structures used for testing the code in real systems. The structures were obtained from the protein 2jws.pdb.
  • Figure 4: Execution times for structures of increasing length for the different methods.
  • Figure :