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Knot Theory and the Jones Polynomial - Bachelor's Thesis

This repository contains all the work I did for my bachelor's thesis titled "Knot Theory and the Jones Polynomial". The thesis explores the mathematical field of knot theory, focusing on the Jones polynomial, a significant knot invariant used to distinguish different types of knots.

The repository includes:

  • Thesis documentation and research.
  • Detailed explanations of key concepts in knot theory.
  • Mathematical derivations related to the Jones polynomial.
  • Code (if applicable) for calculations and visualizations related to knots.

This work is a comprehensive study into the structure and properties of knots, combining theoretical analysis with practical applications.

Invariants Studied

In my thesis, I explored various algebraic invariants used in knot theory to classify and distinguish knots. The key invariants studied are:

  1. Crossing Number

    • The crossing number is the minimum number of crossings in a knot diagram. It is an essential topological invariant, as different knots may have different crossing numbers.

  2. Unknotting Number

    • The unknotting number represents the fewest number of crossing changes required to transform a knot into the trivial knot. It is a crucial invariant but is difficult to calculate in practice.

  3. Genus of a Knot

    • The genus measures the minimal genus (the number of handles) of a surface that bounds the knot. It is used to study knot complexity.

  4. Jones Polynomial

    • The Jones polynomial is a knot invariant that assigns a Laurent polynomial to a knot or link. It was a focal point of the thesis, as this invariant can distinguish different knots and has applications in various mathematical and physical theories.
    • The Jones polynomial was constructed using the Kauffman polynomial and remains invariant under Reidemeister moves, which is central to proving knot equivalence.

Applications of Knot Theory

The thesis also covered several modern applications of knot theory, including:

  1. DNA Knotting

    • Knot theory is used to study the knotting and unknotting of DNA, which can occur during processes such as replication and recombination. This application helps understand biological mechanisms and molecular behavior.

  2. Quantum Computing

    • Knot theory, specifically the Jones polynomial, has applications in the field of quantum computing. The theory of topological quantum computation uses knot-like structures (braiding of anyons) to encode and process information, offering potential solutions to error-resistant quantum systems.

  3. Chirality in Chemistry

    • Knot theory is applied in determining whether molecules are chiral, meaning they cannot be superimposed on their mirror images, which is important in studying the behavior of certain molecules in chemical reactions.

Thesis Supervisor

The thesis was supervised by Doctor Joan Porti Piqué, a professor at the Autonomous University of Barcelona (UAB). His research focuses on geometry, topology, and their applications to fields like hyperbolic geometry and 3-manifolds. His extensive background in these areas provided essential guidance for the thesis.

Bibliography

The bibliography of the thesis includes notable references in knot theory and the study of the Jones polynomial. Key works by mathematicians such as Vaughan Jones, who discovered the Jones polynomial, and Louis Kauffman, are central to the research. The thesis also cites research that links knot theory to modern scientific fields like quantum mechanics and topology in materials science.