Knots and Links

Rolfsen’s Knots and Links is a gem and a classic. Every mathematics library should own a copy and every mathematician should read at least some of it. The writing is clear and engaging, while the choice of examples is genius. Each example leads the reader to understand techniques that apply much more generally. They have just the right level of difficulty — not deceptively simple, but also not overwhelmingly difficult. The examples are assisted by numerous figures. Most pages have multiple figures, all hand-drawn and beautiful.

I am like many geometric topologists in that Knots and Links was the book I grew up with. My reading course (with Daryl Cooper at University of California, Santa Barbara) solidified my desire to study 3-manifolds. Looking back through the book now, several years post-Ph.D., I have fond memories of each chapter. I remember the moments spent drawing and re-drawing the figures and struggling to turn my mental images into actual proofs. Unfortunately, I came to the text just a semester too late and wasn’t able to procure one of the Publish or Perish editions of the book. I had to borrow Daryl’s copy rather than buy my own since it had just gone out-of-print. Fortunately, not long afterwards the AMS included it in their Chelsea series, and I was able to get my own copy.

Every chapter of Knots and Links is connected to major themes in low-dimensional topology, but the focus is on the relationship between classical knots and 3-manifolds. Higher dimensional knots are prevalent and often appear as a contrast to the classical setting. Similarly, the horned sphere and Antoine’s necklace are explained well, but function primarily as motivation for introducing tameness hypotheses. The book is particularly strong on the basics of simple closed curves in surfaces, cyclic coverings of knots, and Dehn surgery on knots. Arguments often make use of simple, but profound, results from group theory or homology theory. Despite this, I have successfully used portions of the book with undergraduates having only limited exposure to algebraic topology. Rolfsen also frequently refers to Rourke and Sanderson’s Introduction to Piecewise-Linear Topology. Fortunately, that book is also now back in print. The PL-prerequisites won’t, however, hamper Rolfsen’s readers. A more recent book, also called Knots and Links but this time by Cromwell, has a similar spirit to Rolfsen’s book but is more oriented toward knot theory, rather than low-dimensional topology more broadly, and requires less background than Rolfsen’s book. Part of the joy of reading Rolfsen’s text, however, is the joy of the scenic detour!

Since its original publication in 1976 (the year I was born!), significant new topics in knot theory and low-dimensional topology have emerged: increased attention to knot diagrams; the development of new homology theories for knots and 3-manifolds; the boom in geometric group theory; and, above all, the re-introduction of geometry to the study of 3-manifolds. Even so, Rolfsen’s book continues to be a beautiful introduction to some beautiful ideas.

In conclusion, I can’t resist encouraging you to surf on over to the Book Ends interview with Dale Rolfsen.

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Scott Taylor is an associate professor at Colby College. Rolfsen’s Knots and Links is on his desk almost as often as it’s on his bookshelf.