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Introduction to Knot Theory

Richard H. Crowell and Ralph H. Fox
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The Basic Library List Committee recommends this book for acquisition by undergraduate mathematics libraries.

[Reviewed by
Fabio Mainardi
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This book first appeared in 1963 and quickly become quite popular. As its title claims, it is intended as an introduction and is appropriate for undergraduate students with some knowledge of abstract topology. It is completely self-contained. To achieve that, some chapters are dedicated to introduce from scratch certain topics in algebraic topology (e.g. fundamental groups) or algebra (e.g. presentations of groups). This makes the book very accessible. Nevertheless each chapter contains detailed proofs as well as a set of exercises, some of which may be challenging for the beginner.

Probably the best review of this book was written by R. Crowell himself, in his preface to the 1977 edition: “the book could be certainly be rewritten by including more material and also by introducing topics in a more elegant and up-to-date style. […] this book achieves qualities of effectiveness, brevity, elementary character, and unity. These characteristics would be jeopardized, if not lost, in a major revision”.

This search for simplicity has some drawbacks. For instance, the Alexander polynomials are introduced in a way that, while it is perfectly suitable for computations, make them a bit mysterious; where do they come from? In order to motivate their construction, one should ultimately go further and introduce homology groups of covering spaces. One would lose the elementary character, but would gain a deeper understanding.

Summing up, I think this is a great book for the beginner who is looking for a quick overview and who wants to taste the flavour of the theory. Afterwards, the no-more beginner will turn to more advanced and complete textbooks (for example, Lickorish’s Introduction to Knot Theory, Springer, Graduate Texts in Mathematics 175).

Fabio Mainardi earned a PhD in Mathematics at the University of Paris 13. His research interests are mainly Iwasawa theory, p-adic L-functions and the arithmetic of automorphic forms. At present, he works in a "classe préparatoire" in Geneva. He may be reached at



Chapter 1. Knots and Knot Types

Chapter 2. The Fundamental Group

Chapter 3. The Free Groups

Chapter 4. Presentation of Groups

Chapter 5. Calculation of Fundamental Groups

Chapter 6. Presentation of a Knot Group

Chapter 7. The Free Calculus and the Elementary Ideals

Chapter 8. The Knot Polynomials

Chapter 9. Characteristic Properties of the Knot Polynomials

Appendix I. Differentiable Knots are Tame

Appendix II. Categories and groupoids

Appendix III. Proof of the van Kampen theorem

Guide to the Literature