The Mystery of Knots

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

• A Knot Theory Primer:
  • A General Understanding of Topology
  • Knot Theory as a Branch of Topology
  • The Regular Presentations of Knots
  • The Equivalence Moves
  • The Knot Invariants
  • Elements of Group Theory
  • The Fundamental Group
  • The Knot Group
  • The Colorization Invariants
  • The Alexander Polynomial
  • The Theory of Linear Homogeneous Systems
  • Calculating the Alexander Polynomial
  • The “Minor” Alexander Polynomials
  • The Meridian-Longitude Invariants
  • Proving a Knot's Chirality
  • Braid Theory — Skein Invariants
  • Calculating the HOMFLYPT Polynomials
  • Knot Theory After the HOMFLYPT
• The Problem of Knot Tabulation:
  • Basic Concepts of Computer Programming
  • The Dowker Notation
  • Drawing the Knot
  • When is a Notation Drawable?
  • The “Equal Drawability” Moves
  • Multiple Notations for Equivalent Knots
  • Ordering the Dowker Notations
  • Calculating the Notation Invariants
  • A Few Examples
  • The Knot Tabulation Algorithm
  • The Pseudocode
  • The Flowchart
  • Actual Results
• The Table of Knots