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The Knot Book: Introduction to the Mathematical Theory of Knots (1994) [pdf]
- case_of_snakes 8y agoNo joke, I've been wanting an introductory book on this.
- ianai 8y agoVery cool to see knot theory on HN! I studied it in undergrad and wrote a paper about it as a class project.
- dannykwells 8y agoI worked from this book extensively during my undergraduate thesis, and it was an absolute joy to read and learn from. Compared to Lickorish ("An Introduction To Knot Theory"), the explanations were easy even if you hadn't had 3 semesters of graduate abstract algebra. If you want a fun application of where knot theory can be used "in the real world" there are some interesting applications to DNA untangling and the function of DNA Topoisomerase - e.g.: https://sinews.siam.org/Details-Page/untangling-dna-with-knot-theory https://sinews.siam.org/Details-Page/untangling-dna-with-kno... http://matwbn.icm.edu.pl/ksiazki/bcp/bcp42/bcp4216.pdf http://matwbn.icm.edu.pl/ksiazki/bcp/bcp42/bcp4216.pdf
- lumberjack 8y agoI found this book a bit too informal. A very easy introduction to Knot Theory, that is still mathematically rigorous is Cromwell's, one of the few books written explicitly for undergraduates. After you are about 1/3 of the way through, you can start using Lickorish. Combined they make the best introduction, by far.
- dannykwells 8y agoAfter 15 years of doing math, I've decided that, for myself, the best introduction isn't the one that's "rigorous" or "in-depth", it's the one the leaves you wanting to learn more. For me, that was Colin's book. I wish more topics in math had entry-level books that explicitly helped contextualize why certain questions were being asked, rather than defining undergraduate simply by what material is covered/how things are proved (of course, I ended up as an applied mathematician, so it could just be me).
- jkingsbery 8y agoI had a class with Adams while at Williams (multivariable Calculus, never got to take his Knot Theory class). He was a great teacher, and excellent about teaching students why they should care.
- gilleain 8y agothere are also knotted proteins, although they are rarer: https://en.m.wikipedia.org/wiki/Knotted_protein https://en.m.wikipedia.org/wiki/Knotted_protein
- pbk1 8y agoWow - I took introductory topology with Colin just a few years ago. He wove in a few examples from OP's book and I had several friends who took his knot course - excellent resource for anyone interested and he is an amazing person/mathematician. Readers here might appreciate one of my favorite homework "problems" from his topology course- it's a simple but counter-intuitive mathematical result that's easy to replicate even for young children. Rip a sheet of paper (8.5x11 will do) in two long strips. Take the strips and tape them so that one is a ring and the other is a Moebius strip. Here's where the magic happens: make a guess about what happens when you cut each object long-ways, then cut both objects with a pair of scissors. Won't give any spoilers but the result may surprise you :) EDIT: also forgot to mention the knot book and his topology book are great at highlighting open/outstanding problems that precocious undergrads could tackle. I definitely wish more authors of math texts went out of their way to point out avenues for exploration like this.
- posterboy 8y agoI will get two entangled moebius strips, right!?
- pbk1 8y agoOnly one way to find out!
- dilippkumar 8y agoI am not a skilled person. I attempted this a bunch of times and all I have are weird strips of paper. Can someone please share pics if you got something other than weirdly shaped strips of paper?
- nabla9 8y agoAdditional trick. Make a Moebius strip that is thick enoug that you can cut it long-ways into three same width strips. But before you cut them completely, twist the middle strip again. a-------------d b-----b\/c----c c-----c/\b----b d-------------a
- _emacsomancer_ 8y agoPuts me in mind also of The 85 Ways to Tie a Tie [https://en.wikipedia.org/wiki/The_85_Ways_to_Tie_a_Tie https://en.wikipedia.org/wiki/The_85_Ways_to_Tie_a_Tie].
- geraltofrivia 8y agoThis is a really nice introductory video (imo) on the topic by Up and Atom as a guest on Tom Scott's channel. Link to Video: https://youtu.be/-eVd2Ugk9BU https://youtu.be/-eVd2Ugk9BU Link to Up and Atom: https://www.youtube.com/channel/UCSIvk78tK2TiviLQn4fSHaw/ https://www.youtube.com/channel/UCSIvk78tK2TiviLQn4fSHaw/
- deleted 8y ago[deleted]
- cphoover 8y agoI remember my highschool math teacher bought me this book.
- prideout 8y agoThis is the book that inspired me to create a WebGL knot gallery (https://prideout.net/knotgl/ https://prideout.net/knotgl/) which I eventually rewrote using Filament (https://prideout.net/knotess/ https://prideout.net/knotess/).
- cogburnd02 8y ago1: That's pretty cool. 2: Would be nice if I could pause/slow down the rotation. Can't get the Borromean rings to look like the Ballantine logo. (three interlocking perfect circles.) http://mathworld.wolfram.com/BorromeanRings.html http://mathworld.wolfram.com/BorromeanRings.html
- dacohenii 8y agoThose are really cool!
- Ono-Sendai 8y agoNice! How did you get the equations of the knots? I.e. how did you compute the paths followed in 3-space by the 'strings'?
- z2 8y agoI got this book as a part of some math prize in high school. It's supposed to give the reader a sense of appreciation of how knots can be modeled, and in turn can even model other patterns. But when I learned sailing, I realized that trying to read this book as a 14-year old contributed to an irrational fear when learning to tie real knots.
- mauvehaus 8y agoA non-mathematical, but much more practical resource, is the Ashley Book of Knots (fondly known as "ABOK"). Clifford Ashley was a sailor who collected knots, and an accomplished painter and writer. If you find yourself in New Bedford, MA, you can see some of his work in the whaling museum (which I highly recommend). I took a knot theory class as an undergrad, and I don't remember which book we used. It ended up being a pretty superficial introduction to the subject, which is both disappointing, and probably also how I got my first "A" in a math class since 11th grade (which, I would argue, was wholly undeserved) Several key takeaways: 1. The figure 8 knot is the only 4 crossing knot. If you climb, and use it as your tie-in, you can check that you've tied it correctly by checking that you have 5 pairs of strands in the knot. 2. The figure 8 knot is amphichiral. There appear to be two variants (like the left and right-handed trefoil knot), but they are transformable into each other via the "pretzel" configuration, which seems to be the canonical representation in math. 3. If you coil rope with only overhand or underhand loops and pull it out, you put a lot of twist into it. If you alternate overhand and underhand loops, it pulls out untwisted. This is most easily seen with ribbon, which has two distinct sides.
- te 8y agoWhat are the n most useful knots for sailors? Other outdoorspeople? Eagle scouts? All-round handy problem-solver types?
- shagie 8y agoFor knot fans - there's also a code golf problem on Stack Exchange that only has one solution so far - Knot or Not? https://codegolf.stackexchange.com/q/30292 https://codegolf.stackexchange.com/q/30292
- earthicus 8y agoThe spectacular knot homology theories such as Khovanov Homology and Heegaard Floer Homology can detect the unknot 'on the blackboard' as well. Wow!!! Actually it can detect the genus of the knot: that's an amazing theorem! I wish I understood how it worked :)