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.
The Knot Book: Introduction to the Mathematical Theory of Knots (1994) [pdf]
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Re: The Knot Book: Introduction to the Mathematical Theory of Knots (1994) [pdf]
#22A 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.
Re: The Knot Book: Introduction to the Mathematical Theory of Knots (1994) [pdf]
#23This 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.)
Re: The Knot Book: Introduction to the Mathematical Theory of Knots (1994) [pdf]
#24Earlier quoted context omitted.
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?
Re: The Knot Book: Introduction to the Mathematical Theory of Knots (1994) [pdf]
#25This 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/ ).
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'?
Re: The Knot Book: Introduction to the Mathematical Theory of Knots (1994) [pdf]
#26Re: The Knot Book: Introduction to the Mathematical Theory of Knots (1994) [pdf]
#27What are the n most useful knots for sailors? Other outdoorspeople? Eagle scouts? All-round handy problem-solver types?
Re: The Knot Book: Introduction to the Mathematical Theory of Knots (1994) [pdf]
#28For 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
Re: The Knot Book: Introduction to the Mathematical Theory of Knots (1994) [pdf]
#29For 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 :)