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Knots and Links
Dale Rolfsen, University of British Columbia, Vancouver, Canada
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AMS Chelsea Publishing
1976; 439 pp; hardcover
Volume: 346
ISBN-10: 0-8218-3436-3
ISBN-13: 978-0-8218-3436-7
List Price: US$61
Member Price: US$54.90
Order Code: CHEL/346.H
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Rolfsen's beautiful book on knots and links can be read by anyone, from beginner to expert, who wants to learn about knot theory. Beginners find an inviting introduction to the elements of topology, emphasizing the tools needed for understanding knots, the fundamental group and van Kampen's theorem, for example, which are then applied to concrete problems, such as computing knot groups. For experts, Rolfsen explains advanced topics, such as the connections between knot theory and surgery and how they are useful to understanding three-manifolds.

Besides providing a guide to understanding knot theory, the book offers "practical" training. After reading it, you will be able to do many things: compute presentations of knot groups, Alexander polynomials, and other invariants; perform surgery on three-manifolds; and visualize knots and their complements. It is characterized by its hands-on approach and emphasis on a visual, geometric understanding.

Rolfsen offers invaluable insight and strikes a perfect balance between giving technical details and offering informal explanations. The illustrations are superb, and a wealth of examples are included.

Now back in print by the AMS, the book is still a standard reference in knot theory. It is written in a remarkable style that makes it useful for both beginners and researchers. Particularly noteworthy is the table of knots and links at the end. This volume is an excellent introduction to the topic and is suitable as a textbook for a course in knot theory or 3-manifolds.

Other key books of interest on this topic available from the AMS are The Shoelace Book: A Mathematical Guide to the Best (and Worst) Ways to Lace your Shoes and The Knot Book.

Readership

Advanced undergraduates, graduate students, and research mathematicians interested in knot theory and its applications to low-dimensional topology.

Table of Contents

  • Introduction
  • Codimension one and other matters
  • The fundamental group
  • Three-dimensional PL geometry
  • Seifert surfaces
  • Finite cyclic coverings and the torsion invariants
  • Infinite cyclic coverings and the Alexander invariant
  • Matrix invariants
  • 3-manifolds and surgery on links
  • Foliations, branched covers, fibrations and so on
  • A higher-dimensional sampler
  • Covering spaces and some algebra in a nutshell
  • Dehn's lemma and the loop theorem
  • Table of knots and links
  • References
  • Index
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