The Mystery of Knots

Our main focus is the discussion of the problem of tabulation of knots, and to demonstrate how through developing appropriate algorithms to be run as computer programs, one may obtain concrete results. The mathematical background needed to understand the discussion is quite limited. Advanced mathematical knowledge from. Computer Programming, Topology, Group Theory and other areas, is presented as simply as possible, just before it is being used. The only knowledge expected from the reader, is a very elementary understanding of Euclidean Geometry, an intuitive feeling of the idea of continuity, and most important a good understanding of Mathematical Logic and Reasoning.

The material is organized as follows. Following an introduction that includes a historical overview and the statement of the main problem, there is a very extensive presentation of the relevant aspects of Knot Theory in the first part. This presentation is not a synopsis of Knot Theory, since we omit important aspects that are not relevant to the discussion. Ideally, after the end of the first part, one is equipped with all die Knot Theory knowledge that is needed to tackle the problem of knot tabulation.

The second part is exclusively concentrated to the development of the appropriate algorithm, and especially to how the numerous obstacles and hindrances are overcome. Towards the end, the reader may find what in computer slang is called a "pseudocode", followed by a brief summary of the results obtained after we ran the computer program. Finally, there is a third part that contains an extensive table of the tabulated knots, including a proof that they all differ from each other.

Since Knot Theory takes place in the familiar three-dimensional space, we present a number of figures, whose purpose is to encourage the reader to form a visual picture of the concepts involved. It is our belief that such a visualization leads to a better understanding than mere statements of formal. definitions and theorems. In addition, we provide a number of exercises to motivate the reader to be an active participant in the discussion. While some of these exercises are quite involved, and it is not necessary to solve them in order to understand the subsequent theory, readers are encouraged to spend time attempting them, and reading and understanding the solutions provided.

CONTENTS

• PART ONE : A Knot Theory Primer
  1. A General Understanding of Topology
  2. Knot Theory as a Branch of Topology
  3. The Regular Presentations of Knots
  4. The Equivalence Moves
  5. The Knot Invariants
  6. Elements of Group Theory
  7. The Fundamental Group
  8. The Knot Group
  9. The Colorization Invariants
  10. The Alexander Polynomial
  11. The Theory of Linear Homogeneous Systems
  12. Calculating the Alexander Polynomial
  13. The "minor" Alexander Polynomials
  14. The Meridian-Longitude Invariants
  15. Proving a Knot's Chirality
  16. Braid Theory - Skein Invariants
  17. Calculating the HOMFLYPT Polynomials
  18. Knot Theory after the HOMFLYPT
• PART TWO : The Problem of Knot Tabulation
  1. Basic Concepts of Computer Programming
  2. The Dowker Notation
  3. Drawing the Knot
  4. When is a Notation Drawable?
  5. The "Equal Drawability" Moves
  6. Multiple Notations for Equivalent Knots
  7. Ordering the Dowker Notations
  8. Calculating the Notation Invariants
  9. A Few Examples
  10. The Knot Tabulation Algorithm
  11. The Pseudocode
  12. The Flowchart
  13. Actual Results
• PART THREE : The Table of Knots
  
• REFERENCES
  
• INDEX