The mystery of knots: Computer programming for knot tabulation

✍ Scribed by Aneziris C.N.

• Publisher: WS
• Year: 1999
• Tongue: English
• Leaves: 409
• Series: Series on Knots and Everything, Volume 20
• Category: Library

✦ Synopsis

One of the most significant unsolved problems in mathematics is the complete classification of knots. The main purpose of this text is to introduce the reader to the use of computer programming to obtain the table of knots. The author seeks to present this problem as clearly and methodically as possible, starting from the very basics. Mathematical ideas and concepts are extensively discussed, and no advance background is required.

✦ Table of Contents

Cover......Page 1
Title page......Page 4
Acknowledgment......Page 6
Preface......Page 8
CONTENTS......Page 10
INTRODUCTION......Page 12
GLOSSARY......Page 17
A KNOT THEORY PRIMER......Page 22
1. A General Understanding of Topology......Page 24
2. Knot Theory as a Branch of Topology......Page 30
3. The Regular Presentations of Knots......Page 36
4. The Equivalence Moves......Page 40
5. The Knot Invariants......Page 46
6. Elements of Group Theory......Page 54
7. The Fundamental Group......Page 60
8. The Knot Group......Page 64
9. The Colorization Invariants......Page 72
10. The Alexander Polynomial......Page 82
11. The Theory of Linear Homogeneous Systems......Page 86
12. Calculating the Alexander Polynomial......Page 94
13. The "minor" Alexander Polynomials......Page 102
14. The Meridian-Longitude Invariants......Page 108
15. Proving a Knot's Chirality......Page 116
16. Braid Theory - Skein Invariants......Page 124
17. Calculating the HOMFLYPT Polynomials......Page 136
18. Knot Theory after the HOMFLYPT......Page 146
THE PROBLEM OF KNOT TABULATION......Page 152
1. Basic Concepts of Computer Programming......Page 154
2. The Dowker Notation......Page 162
3. Drawing the Knot......Page 166
4. When is a Notation Drawable?......Page 172
5. The "Equal Drawability" Moves......Page 178
6. Multiple Notations for Equivalent Knots......Page 182
7. Ordering the Dowker Notations......Page 188
8. Calculating the Notation Invariants......Page 192
9. A Few Examples......Page 202
10. The Knot Tabulation Algorithm......Page 212
11. The Pseudocode......Page 222
12. The Flowchart......Page 236
13. Actual Results......Page 246
THE TABLE OF KNOTS......Page 252
REFERENCES......Page 394
INDEX......Page 400