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Knots and Links

Peter Cromwell
Cambridge University Press
Publication Date: 
Number of Pages: 
[Reviewed by
Ioana Mihaila
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When I first came across Knots and Links , I knew nothing of the subject, but I was very interested in learning about it. The clear structure and good graphics made me think that this would be a great introduction to knots. Sixty pages later I was more frustrated than enlightened.

This is by no means saying that the book is bad; quite the opposite, in fact. Since my first attempt I had the opportunity to participate in a knot theory workshop ran by Colin Adams. I read Adams’ book on knot theory and worked my way through several knot theory papers. Only then I came to appreciate Cromwell’s book.

Knots and Links is a new monograph on knot theory, and as such it contains more information than other classical books on the subject (such as Adams or Livingston), and it has updated tables on both knots and links. However, it is not a “first book” in knot theory. The treatment is in some sense more pedantic — see for example the definition of a knot, on page 25 — but at the same time it skips details in proofs.

The fact that knot theory is a relatively new field in which notations are not completely standardized, adds to a beginner’s difficulties. For example, reading through the chapter on rational tangles with a colleague, we discovered that the direction in which the Conway notation is used differs in various books and papers. Cromwell’s book put us in the right direction for investigating the connections between the continued fractions expression of a rational tangle, the use of determinants, and the achirality of a rational knot. However, finding the proofs of the results was quite a battle and we had to appeal to many sources.

In conclusion, Knots and Links is a great book for those who have the background and patience to dig a little deeper.

Ioana Mihaila ( is Assistant Professor of Mathematics at Cal Poly Pomona. Her research area is analysis, and she is also interested in mathematics competitions.

Preface. 1. Introduction; 2. A topologist's toolkit; 3. Link diagrams; 4. Constructions and decompositions of links; 5. Spanning surfaces and genus; 6. Matrix invariants; 7. The Alexander-Conway polynomial; 8. Rational tangles; 9. More polynomials; 10. Closed braids and arc presentations; Appendix A. Knot diagrams; Appendix B. Numerical invariants; Appendix C. Properties; Appendix D. Polynomials; Appendix E. Polygon coordinates; Appendix F. Family properties; Bibliography; Index.