People have been interested in knots at least since the time of Alexander the Great and his encounter with the Gordian knot. There are famous knot illustrations in the "Book of Kells" and throughout traditional Islamic art. Lord Kelvin believed that atoms were knots in the ether and he encouraged Tait to compile a table of knots in the late 19th century. In recent years the Jones polynomial has stimulated interest in possible relationships between knot theory and physics. This book is concerned with the fundamental question of the classification of knots and, more generally, the classification of arbitrary (compact) topological objects which occur in our normal space of physical reality. Professor Hemion explains his classification algorithm - using the method of normal surfaces - in 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" should be of interest to mathematicians, physicists, and other scientists who want to apply this basic classification algorithm to their research in knot theory.
`an informal account of Haken's classification of sufficiently large 3-manifolds by means of normal surfaces ... appropriate for someone who wants a broad overview of this theorem in 3-dimensional topology'
Martin Scharlemann, Mathematical Reviews, Issue 94g
Part I: Preliminaries
1: What is a knot?
2: How to compare two knots
3: The theory of compact surfaces
4: Piecewise linear topology
Part II: The Theory of Normal Surfaces
5: Incompressible surfaces
6: Normal surfaces
7: Diophantine inequalities
8: Fundamental solutions
9: The "easy" case
10: The "difficult" case
11: Why is the "difficult" case difficult?
12: What to do in the difficult case
Part III: Classifying Homeomorphisms of Surfaces
13: Straightening homeomorphisms
14: The conjugacy problem
15: The size of a homeomorphism
16: Small curves
17: Small conjugating homeomorphisms
18: Classifying mappings of surfaces
19: The final result