The Knot Book: Introduction to the Mathematical Theory of Knots (1994) [pdf] | Hacker News Reader

The Knot Book: Introduction to the Mathematical Theory of Knots (1994) [pdf]

math.harvard.edu

204 pointslainon26 comments

This is the book that inspired me to create a WebGL knot gallery (https://prideout.net/knotgl/) which I eventually rewrote using Filament (https://prideout.net/knotess/).

1: 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

Nice! How did you get the equations of the knots? I.e. how did you compute the paths followed in 3-space by the 'strings'?

Those are really cool!

I 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-kno...

http://matwbn.icm.edu.pl/ksiazki/bcp/bcp42/bcp4216.pdf

there are also knotted proteins, although they are rarer:

https://en.m.wikipedia.org/wiki/Knotted_protein

I 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.

After 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).

I 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.

Wow - 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.

Additional 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

I will get two entangled moebius strips, right!?

Only one way to find out!

I 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?

A 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.

This 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

Link to Up and Atom: https://www.youtube.com/channel/UCSIvk78tK2TiviLQn4fSHaw/

For 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

The 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 :)

I 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.

Puts me in mind also of The 85 Ways to Tie a Tie [https://en.wikipedia.org/wiki/The_85_Ways_to_Tie_a_Tie].

No joke, I've been wanting an introductory book on this.

Knot jokes, page 276.

Why Knot? https://www.amazon.com/dp/B00IF3UOQ4/

What are the n most useful knots for sailors? Other outdoorspeople? Eagle scouts? All-round handy problem-solver types?

Very cool to see knot theory on HN! I studied it in undergrad and wrote a paper about it as a class project.

I remember my highschool math teacher bought me this book.

Reply on news.ycombinator.com