Description
Book SynopsisIn this book the authors develop the theory of knotted surfaces in analogy with the classical case of knotted curves in 3-dimensional space. In the first chapter knotted surface diagrams are defined and exemplified; these are generic surfaces in 3-space with crossing information given. The diagrams are further enhanced to give alternative descriptions. A knotted surface can be described as a movie, as a kind of labeled planar graph, or as a sequence of words in which successive words are related by grammatical changes. In the second chapter, the theory of Reidemeister moves is developed in the various contexts. The authors show how to unknot intricate examples using these moves. The third chapter reviews the braid theory of knotted surfaces. Examples of the Alexander isotopy are given, and the braid movie moves are presented. In the fourth chapter, properties of the projections of knotted surfaces are studied. Oriented surfaces in 4-space are shown to have planar projections with
Trade ReviewThe authors must be congratulated on their heroic endeavors to bring known and unknown results into one book. Who should buy this book? Certainly all topologists with geometric leanings should do so, and their students too. -- Bulletin of the London Mathematical Society
Table of Contents
- Diagrams of knotted surfaces
- Moving knotted surfaces
- Braid theory in dimension four
- Combinatorics of knotted surface diagrams
- The fundamental group and the Seifert algorithm
- Algebraic structures related to knotted surface diagrams
- Bibliography
- Index
