Knot theory has recently emerged as a productive field of study in both the physical and mathematical sciences. This book is concerned with the fundamental question of the classification of knots, and more generally the classification of arbitrary (compact) topological objects which can occur in the normal space of physical reality. The author explains his classification algorithmusing the method of normal surfacesin a simple and concise way. The reader is thus shown the relevance of such traditional mathematical objects as the Klein bottle or the hyperbolic plane to this basic classification theory. The Classification of Knots and 3-Dimensional Spaces will be of interest to mathematicians, physicists, and other scientists who want to apply this algorithm to their research in knot theory.
Table of Contents
PART I: Preliminaries
2. What is a knot?
3. How to Compare Two Knots
4. The Theory of Compact Surfaces
5. Piecewise Linear Topology
PART II: The Theory of Normal Surfaces
6. Incompressible Surfaces
7. Normal Surfaces
8. Diophantine Inequalities
9. Fundamental Solutions
10. The "Easy" Case
11. The "Difficult" Case
12. Why is the "Difficult" Case Difficult?
13. What to do in the Difficult Case
PART III: Classifying Homeomorphisms of Surfaces
14. Straightening Homeomorphisms
15. The Conjugacy Problem
16. The Size of a Homeomorphism
17. Small Curves
18. Small Conjugating Homeomorphisms
19. Classifying Mappings of Surfaces
20. The Final Result