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Knot theory. (English) Zbl 0887.57008

The Carus Mathematical Monographs. 24. Washington, DC: The Mathematical Association of America. xviii, 240 p. (1993).
This book is a good text for an undergraduate course in knot theory. To give a quick overview, here are the chapter titles: 1. A century of knot theory; 2. What is a knot?; 3. Combinatorial techniques; 4. Geometric techniques; 5. Algebraic techniques; 6. Geometry, algebra, and the Alexander polynomial; 7. Numerical invariants; 8. Symmetries of knots; 9. High-dimensional knot theory; 10. New combinatorial techniques. Two appendices contain a table of knots with no more than nine crossings and a table of their Alexander polynomials. There are three main parts to this book. The first part, comprising Chapter 2 through 5, develops the fundamentals of knot theory. Chapter 1 includes a discussion of the recent history of the study of knots. In the process, some of the most interesting problems of knot theory are described. Chapter 2 focuses on the basic material of the subject, the precise definitions of knots and their deformations. It is here that one begins to see how mathematical methods can be applied to the study of knotting. The three main techniques of knot theory appear in the next chapters: Chapter 3 is devoted to combinatorial methods, Chapter 4 presents geometric techniques, and Chapter 5 illustrates algebraic tools. These chapters demonstrate the nature of the techniques and the types of problems to which each apply. The second part presents advanced topics in knot theory. Chapter 6 describes relationships among the methods of the earlier chapters. The sources of these relationships are quite deep and subtle. As a consequence the work is delicate, but the results provide many new insights. In Chapter 7 several properties of knots are presented. The intention here is to describe some of the very natural questions that occur about knots and to illustrate how the methods developed so far can give detailed answers to these questions. Chapter 8 is devoted to the study of symmetry, one of the most beautiful properties of knotting. It is here that the tremendous power of the techniques developed earlier becomes most evident. The third part is independent of the material of the second part. Two modern aspects of the subject are explored in these closing chapters. Chapter 9 provides an introduction to high dimensional knot theory and briefly indicates how the methods of classical knot theory can be applied. Chapter 10 describes new combinatorial methods. These methods greatly extend those of Chapter 3; the study of these combinatorial invariants is one of the most active and fascinating areas of knot theory today.
Reviewer: Y.Akira (Tokyo)

MSC:

57M25 Knots and links in the \(3\)-sphere (MSC2010)
57Q45 Knots and links in high dimensions (PL-topology) (MSC2010)
57-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to manifolds and cell complexes
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