Dale Rolfsen is an expert in low-dimensional topology and knot theory, and is co-author of the AMS books Ordering Braids (with Dehornoy, Dynnikov and Wiest), and Ordered Groups and Topology (with Clay). His seminal work Knots and Links helped to popularize the study of low-dimensional topology in the 1970s and continues to be a standard reference today.
Knots and Links, in the AMS Chelsea Series
What made you decide to write the book? Was there a gap in the literature you were trying to fill? Did you use existing notes from teaching?
It was Mike Spivak who encouraged me to write Knots and Links. We had met at the IAS, where we became friends. That was the late 1960’s. Mike had established Publish or Perish, Inc. and encouraged me to write a book and publish with him if the muse ever struck me. A few years later I taught a graduate course on knot theory at UBC, where I was a young professor. It was based partly on notes from a course I had taken as a graduate student myself in Madison, Wisconsin, taught by Joe Martin. I also referred to several books in teaching that course, especially the Crowell and Fox classic, Introduction to Knot Theory. While in Princeton, I also had the pleasure of meeting Ralph Fox and frequently attended his weekly seminar, which further kindled my interest in the subject.
Regarding a gap in the literature… It was more a gap in my knowledge, and a bit my allergy to abstract algebra which was more severe than it is now. I was particularly mystified by the Alexander polynomial, which was treated quite formally and algebraically in all the sources I knew. How to make sense of a process involving putting dots at certain places near the crossing of a picture of a knot, using that to construct a matrix with polynomial entries, delete one column of the matrix and then take the determinant? It was a mystery to me for years. One day I read Milnor’s beautiful paper Infinite Cyclic Coverings and I had a real epiphany when I realized that the Alexander polynomial was actually describing the homology of a certain (infinite cyclic) covering of the knot’s complement. It was this more geometric understanding that I tried to emphasize in the book. In fact it’s fair to say that was one of the reasons I decided to write a book on knots.
How did you decide on the format and style of the book?
It was written in the ‘70s. Well before TeX. The state of the art at the time in preparing mathematical manuscripts was the IBM Selectric typewriter. It had a removable typing ball, so you could change fonts, type Greek letters, etc. Also, I like to draw, and luckily the subject calls for lots of pictures. So the book has illustrations and text interwoven, which is easier to do with cut and paste techniques than it would have been with LaTeX (even if it had existed at the time). I had a lot of fun drawing, cutting and pasting, sometimes using plastic overlays of cross-hatching, dots, etc.
I find I learn new concepts much more deeply by looking at examples and explicit calculations, rather than the formal, axiomatic approach. My experience as an often struggling student motivated me to introduce and motivate concepts by concrete, and hopefully interesting, examples. For both pedagogical and aesthetic reasons, I tried to emphasize the visual and geometric aspects of ideas and convey some of the excitement I felt toward the subject.
What was the writing process like? Did you write every day on a set schedule, or did you have periods of setting it aside?
I started out with a sketchy first draft based on the notes from the course I gave in 1974. But the writing started in earnest when I had a sabbatical in 1975, which I mostly spent on a remote island off the west coast of British Columbia. No phone or electricity but a lot of peace and tranquility. I had a friend on a nearby island where there was electricity, and my rented IBM Selectric was there. So every week or so, I’d take a boat over to visit her and type a couple of pages. I didn’t have a schedule or explicit deadline, though I wanted to finish it before my teaching responsibilities started again. Eventually I heeded Mary-Ellen Rudin’s advice to me, “Don’t try to put everything into that book!”
What did you focus on the most when writing? What was the most challenging aspect? What came easily?
Although no doubt there would be many types of readers of my imagined book, I tried to write to a person a bit like myself as a student – a person who is easily confused, but also keen to see new ideas take shape. Also, it’s actually possible to build a little suspense into a mathematical narrative by hints, examples and questions which are then clarified later by theorems.
Knots are beautiful objects which had inspired some amazing new mathematical ideas – that’s even more true today than when I wrote the book. But to be honest, I don’t really like many aspects of knot theory. My bias is to be more interested in applications of knot theory to understand 3-manifolds, via surgery, branched coverings, etc., than in the program of classifying knots, for example. I was especially excited about surgery when writing the book. Rob Kirby was developing his surgery calculus at the same time. Rob later told me that he was surprised to see that some of his ideas were explored independently in Knots and Links. His work went much deeper, but he considered only integral surgery, whereas the more general rational surgery was developed in the book.
One of my great challenges in writing was that I didn’t have access to a library while on the island, so I did a bad job regarding the history of the subject and attributing names to ideas. I was also, at the time, totally ignorant of the theory of braid groups. Joan Birman’s seminal book Braids, Links and Mapping Class Groups was being written at the same time. But we were unaware of each other’s efforts. Later, Joan and I became, and remain, good friends.
What were the positives and negatives of the experience? Did anything about the experience surprise you?
The positive was that I learned a lot of the subject as I went along. Many of the examples and calculations had to be done from scratch. That was really fun. As I said, my library access was very 
limited. That was often frustrating, but forced me to work mostly independently of the literature. And of course there was no internet or even email back then.
How did you choose a publisher? What was important to you when you made the choice?
I chose Publish or Perish, Inc. for several reasons. Mike Spivak was my friend, a writer I admired, and he was very encouraging about the book. He promised to keep the price reasonable, an important issue with me, since students (and libraries) had to pay, even then, outrageous prices for textbooks, monographs and journals. Also I liked the name – it was so cheeky! I thought of my book as being a bit quirky, too. It seemed a good match.
The book went through several printings with P or P over a couple of decades, but at one point Mike told me that his publishing company was having financial difficulties and he couldn’t afford another printing run after his existing stock ran out. He suggested that I find another publisher, as the book was still selling well. I decided to go with the AMS, rather than a commercial publishing house. Another choice I considered was MSP, which is a non-profit mathematical publisher run by mathematicians. However, they wanted it redone in LaTeX, which was more work than I was prepared to engage in! The AMS edition uses the original format, reproduced photographically, with a few corrections put in. It was a challenge making those corrections – I prepared them in TeX using a font that looks like the dirty old IBM Selectric and then pasted them in. I consider it a great honor to be included in the AMS Chelsea series.
Was your writing influenced by other books? Which ones?
Two great influences on my writing (and thinking) were Edwin Abbot’s Flatland and John Milnor’s Topology from the Differential Viewpoint. I read Flatland as a kid, and that’s what got me thinking about higher dimensions and eventually into studying topology. Also, its mixture of mathematical ideas with humor, and even satire, inspired me to understand mathematics as an adventure that can really be fun. Spivak’s famous calculus book was also a guiding light. I admired the clarity of his writing, and was glad to have such a fine writer as my publisher. My ideal would be to write like Milnor, who manages to get right to the heart of the ideas he’s discussing. As I said, his paper on infinite cyclic coverings was a huge influence. Also his work with Fox, Singularities of 2-spheres in 4-space and cobordism of knots, was particularly appealing. It also hearkened back to my Flatland experience that sometimes an extra dimension clarifies things. For example, adding knots by connected sum is only a semigroup operation in 3-space, but if one considers cobordism classes of knots – a four-dimensional point of view – they form a group, and indeed a very complicated and interesting group which is still an active topic of research.
Did you find ways to get feedback while writing your book or was it a solitary effort?
It was mostly a very solitary effort, though of course I got feedback from my students when I taught the course. However, I did ask various people to look at it after the first draft was written, including Cameron Gordon and Andrew Casson, and of course Mike Spivak. They made some very useful suggestions. Also, my grad student, Jim Bailey, was very helpful in helping me edit the book. Crucially, Jim prepared the knot tables at the end of the book, a huge job. The source of those tables was Conway’s paper An enumeration of knots and links, and some of their algebraic properties which was published in 1970. Jim translated Conway’s notation to pencil drawings of the knots and I hired Ali Roth, his girlfriend at the time, to make the nice pen and ink drawings that appear in the tables. I think I paid Ali two dollars per knot! As I said, tabulation is not my strong suit, so I’m grateful to Jim for taking care of that. I always feel a bit guilty when I hear people refer to the “Rolfsen tables.” They’re not really mine. More accurately they should be called the Conway tables as interpreted by Bailey and drawn by Roth.
Did you have a special place where you liked to write? How did you stay motivated and focused?
My special place was a little cabin on the island. I had a little table right next to a wood stove on one side an
d a window on the other. Perhaps I stayed focused because there wasn’t much else to do there in the winter, other than the eternal problem of collecting firewood dry enough to burn. I was not well-prepared! Being in an isolated place without much distraction helped me concentrate and devote daydreams to issues concerning the book, rather than other responsibilities.
Did time pressure or other responsibilities help or hurt your writing?
Let me turn that around. I think that my writing hurt my other responsibilities. As I recall, after the book came out I found it difficult to get down to research on new things for a couple of years. Perhaps I was just a bit exhausted.
What kind of feedback did you get after the book came out?
I’ve gotten a lot of positive feedback from people who used the book when they were students, and still do now and then. That’s really gratifying to me, as I was specifically writing with the student audience in mind. But I’ve also gotten some negative feedback. For example, a book review in the AMS Bulletin criticized the scanty “kudology” in the book. That criticism is completely valid, my scholarship was pretty low-level.
Other feedback was the discovery by Kenneth Perko that two of the knots in the table, which were listed as distinct, are really the same knot. They’re now known as the “Perko pair.” Ken had been a student of Fox and later practiced law in New York for many years. I met him only recently, and we have a sort of correspondence. Recently Perko told me that I could have saved two bucks if only I had known! There were also some errors in some of the link polynomials in the P or P edition, pointed out to me by Nathan Dunfield, who checked them all by computer. They’re fixed in the AMS edition.
What advice would you give to new authors?
Have fun with your writing!