TL;DR - Accessible to a topology newbie, but needs to be supplemented w/ other books!
Easy to follow and good examples, but just not enough of them! ThTL;DR - Accessible to a topology newbie, but needs to be supplemented w/ other books!
Easy to follow and good examples, but just not enough of them! The book is dense in concepts and about a pinky finger's amount of thickness, which makes it light and easy to carry, but also requires you to supplement your studies with other topology books, which in my case were Schaum's Outlines - General Topology by Seymour Lipschutz, and Topology Illustrated by Peter Saveliev. With that said, the explanations and deconstructions of the proofs were pretty well done, just incomplete in certain proof directions, requiring the reader to fill them in as exercises. Also important note for those that understand this, but this book mainly approaches things from the Point-Set Topology perspective, so if you are looking for metric, algebraic, or differential approaches, look elsewhere!...more
Pros: - Nice beginner knot theory book that is much heavier on the expository / dialogue side compared to the math-textbook-y side.
- Each chapter givesPros: - Nice beginner knot theory book that is much heavier on the expository / dialogue side compared to the math-textbook-y side.
- Each chapter gives a relatively self-contained introduction to knot theory from a different perspective and historical/mathematical approach (my favorite was the braid theory section as this has inspired me to think about some potential ideas wrt rewriting systems and encoding "proofs as knots" (although this idea was not mentioned anywhere in the book)).
- The partition function connection to statistical physics models was rather fascinating! (although I wish it could have been explained much better)
Cons: - The way the hardcover book was formatted could be improved. Many of the pictures were on pages whose explanations were on another page. This required constant flipping of the pages to see what the author was talking about. I suspect changing the size of the book could improve this aspect.
- Some chapters, especially in the latter half could have used more "meat" and explanatory dialogue to lead the reader through the mathematical steps. I had to do many rereads and external referencing to look up how the specific formalism / rules of that "knot invariant" worked.
- Many of the examples need to be worked out fully, without skipping steps. I would also like to have seen more worked-out examples in general....more
TL;DR - Excellent beginner's intro to Knot Theory. Assumes no prerequisites besides basic algebra 2, some geometry, and decent reasoning skills.
I selfTL;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.
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