The Knot Book by Colin C. Adams

by Colin Conrad Adams

In February 2001, scientists at the Department of Energy's Los Alamos National Laboratory announced that they had recorded a simple knot untying itself. Crafted from a chain of nickel-plated steel balls connected by thin metal rods, the three-crossing knot stretched, wiggled, and bent its way out of its predicament--a neat trick worthy of an inorganic Houdini, but more than that, a critical discovery in how granular and filamentary materials such as strands of DNA and polymers entangle and enfold themselves. A knot seems a simple, everyday thing, at least to anyone who wears laced shoes or uses a corded telephone. In the mathematical discipline known as topology, however, knots are anything but at 16 crossings of a "closed curve in space that does not intersect itself anywhere," a knot can take one of 1,388,705 permutations, and more are possible. All this thrills mathematics professor Colin Adams, whose primer offers an engaging if challenging introduction to the mysterious, often unproven, but, he suggests, ultimately knowable nature of knots of all kinds--whether nontrivial, satellite, torus, cable, or hyperbolic. As perhaps befits its subject, Adams's prose is sometimes, well, tangled ("a knot is amphicheiral if it can be deformed through space to the knot obtained by changing every crossing in the projection of the knot to the opposite crossing"), but his book is great fun for puzzle and magic buffs, and a useful reference for students of knot theory and other aspects of higher mathematics. --Gregory McNamee

Genres: Mathematics, Science, Nonfiction, Textbooks, Physics

Paperback

First published March 1, 1994

Reviews

[Sasha Seliutina · Rating 3 out of 5]

You wish you were as cool as me: a person who spent her whole summer reading a book on KNOTS. 😎
I was lowkey interested in a large chunk of the information presented here; the chapters on knot application to the sciences (chemistry specifically) and the possibility of knots in the color dimension were definitely my favorite parts. I can't say that I understood everything (or most) of what I read, but I still found the book very approachable!! Taking notes on this during summer break kept my brain alive and for THAT, I am thankful!! <3

Favorite Quote (yes, there is a favorite quote):

"That was satisfying. It really makes you appreciate surfaces."
^ Is my brain broken, or is that objectively hilarious?!?!?!?

[Barb · Rating 3 out of 5]

The book provides a general overview that is fairly accessible to a lay person like myself. I had some difficulty with explanations of rational knots. The accompanying illustration seemed different from the text explanation. The explanation of hyperbolic knots requires much more mathematical knowledge than I currently have. Fortunately, the annotated bibliography is very thorough, so I'll try a couple of those books.

I had to return this book to the library before I could finish it. I expect to reread it at some future time.

[Jon · Rating 4 out of 5]

Covers a large number of topics (e.g. analysis, algebra, topology) without assuming much background. Provides lots of interesting things to think about, and will often gloss over the technical details, which I think makes this book work very well actually.

My favorite parts were probably supercoiling, knot and graph polynomials, and the discussions of three-manifolds.

[Philip Rideout · Rating 4 out of 5]

This book is a rarity because it sits exactly at the right level of math for me: it's advanced enough to feel like a real mathematics book with theories and equations and whatnot, but it's still much more engaging than a college textbook. I admit that I just never "got" some of the more advanced stuff like hyperbolic knots, but Jones Polynomials etc are fairly easy to understand and fascinating.

[Malia · Rating 5 out of 5]

The most readable math text about what I find to be the most interesting field in math.

[kye · Rating 4 out of 5]

Only read the first ~5 chapters up to Conway's notation. They were great; I wish I could've read the rest of the book! Great diagrams and explanations. Wished there were solutions to the exercises.

[Mundy Reimer · Rating 5 out of 5]

TL;DR - Excellent beginner's intro to Knot Theory. Assumes no prerequisites besides basic algebra 2, some geometry, and decent reasoning skills.

I self-studied this before officially taking a graduate-level elective class in it and using this as the textbook. The book is taught more like a survey course, where you dabble in sub-topic after sub-topic within knot theory for the first half (where my course ended), and then cover knot theory applications in the second half (applied to various sciences and other math topics). This topic of mathematics ranks as one of my favorites so the book now holds a special place for me.

The 1st chapter already gives you most of the basics from which you can go off on your own and start exploring. It covers composition of knots, the famous Reidemeister moves, Links, Tricolorability, and Knots+Sticks. The 2nd chapter covers how to tabulate or encode knots via the Dowker notation and Conway notation. The 3rd chapter covers other invariants of knots like the unknotting number, bridge number, and crossing number. We then take a geometric and topological turn with Chapter 4, going over how to turn knots into surfaces, boundaries, and genus. Chapter 5 is my favorite as it goes over different types of knots in addition to the awesome topic of Braid theory (I'm definitely a visually-intuitive person who likes drawing and manipulating strings in my imagination, but for those that prefer "crunchiness" in their maths / more algebraic rules or structure, this area is for you). The final chapter in the "theory" half of the book covers the various polynomials or other ways of encoding knots, such as the bracket, Jones, Alexander, and HOMFLY polynomials. The later chapters deal more with applications in biology, chemistry, physics, graph theory, topology, and then finally covering higher dimensional knotting.

The book is also great for self-studying. Adams writes with a nice narrative-like voice, keeping you engaged and motivated, rather than the relatively dry theorem-lemma-proof style of other higher math books. There are lots of pictures, albeit all in black-and-white, but that didn't really prove a detriment. And while there were some areas where problems could be made clearer, in general, exercises were well-structured and balanced between computing things, conceptual explanations, visual / geometric / topological drawings, algebra, and proofs. This wide-coverage and bouncing back between different "ways of thinking" made the material stick better in my mind. There were also mentions of various open problems (with the caveat that it was open at the time of writing), but I still think many of these problems are still open.

Overall, the book covers a nice survey of topics. I highly recommend this books to any beginner in the subject. Though not required at all, I also supplemented my reading of it with An Interactive Introduction to Knot Theory by Johnson & Henrich, Encyclopedia of Knot Theory by Adams (same author), and Quandles: An Introduction to the Algebra of Knots by Elhamdadi & Nelson.

[Isaac van der Meulen · Rating 4 out of 5]

Decent introduction to knot theory. It is somewhere in between a textbook and a popular math book. There are a few theorems, but the tone is fairly conversional. There are a lot of illustrations of knots, however I often find it difficult trying to manipulate the knots in my head. It would have been helpful if there were solutions to the many exercises in the book

[Ivan Karamazov · Rating 3 out of 5]

As the name suggests, it is an elementary (basic) introduction to mathematical knot theory. It is quite accessible to newcomers on the topic and the language is easy, legible and clean. It has hardly any prerequisites apart from basic entry level college mathematics.

[NoWillNoWord · Rating 4 out of 5]

It's really funny book.
It makes me feel I am making mathematics theory by myself.

[Spunkyblah · Rating 5 out of 5]

This book is a wonderful introduction to knot theory. If you're curious about what knot theory is, give this a read. It's not only one of the few math books that you can simply read cover-to-cover, but it's also enjoyable for both mathematicians and non-mathematicians.

[Saman · Rating 5 out of 5]

Knot theory is a mathematical field which studies knots in a three dimensional space. The books is a well written starter to the field. It talks about ideas, history, why it is important and applications. It is well written and does not require any special background in mathematics.

[Bob Woodley · Rating 5 out of 5]

A great introduction to Knot Theory. So accessible and approachable yet the book takes us to far-out mathematical realms such as topological embeddings, Dehn surgery, the Poincare conjecture.

It has a well-annotated bibliography as well. Why don't more writers annotate their bibliographies?