Thanks for sharing this! I've gotten into knot tying in the new year, and keep finding applications and things to do with knots.
Here are some other resources I've used: Ashley Book of Knots [0], Grog's animated knots [1], and youtube.
A lot of useful knots build on basic knots and variations of basic knots. More complex knots used to be daunting to me (trying to memorize how to tie the whole thing). But they become easier when you can recognize the basic knots used in the complex ones. Learning the use cases for the knots also helped.
I think learning and tying knots is cool too, but just a heads up, the mathematical knot theory discussed in the OP is
not related to that. It's a branch of topology that studies the possible shapes and transformations of closed curves in 3-space (without self intersection).
It's not concerned with practical properties of knots tied with rope, like strength, stability, ease of use, etc.
Of course they are related! A knot is a knot. It's just that mathematicians are obsessed with proving things rigorously [1], whereas the sailor just wants his sails to hang right.
The applications mentioned in the book are to chemistry and to molecular biology. (As well as another, mentioned only fairly briefly, elsewhere in pure mathematics, to what's called the classification of 3-manifolds.)
It is absolutely related, though most real world things we call knots are usually in fact closer to tangles or links with the ends of the rope looping back to the other end virtually.
Neck ties are a mathematical knot if you again assume the two ends are connected. See other comment in this thread.
Knot theory here only determines the "shapes" of the knots and which are "allowed", there is no friction, width, etc.
Not only 3-space. An example of a slice knot would be a normal 1-dimensional knot suspended in 3-space which turns out to be the 3-dimensional slice/cross-section of a 2-dimensional sphere suspended and knotted in 4-space.
I've always loved knots and have learned a few cool ones throughout my life, some useful, some just cool. But I've always wanted to learn the discipline more holistically. I wanna "get" knots and be able to think in knots.
What kind of cool uses have you found for your new knot skills?
Any recommendations for learning how to "get" knots other than just looking at knots in books?
I end up using knots quite a lot. I cycle so when I need to transport stuff I reach for the piece of parachute cord I keep in my backpack and tie things to bike rack with harvester's hitch, bowline, round turn and two, clove hitch, as appropriate. Also, instead of using a ratchet tie down ona trailer load, try using knot skills to keep things in place. That way you get a really practical feel for what type of know does what, and then Ashley or other things can take you further in a particular direction
Thanks for the links. I memorised a number of decorative knots as a kid, but it never really occurred to me then that they were formed of smaller parts. That didn’t seem to be necessary at the time. Also, you do have prime knots, which are not decomposable in a certain way.
One of my favourite applications of knot theory is to classify all aesthetic tie knots, using topology to uncover unknown sartorial gems. It is a great piece of research from Cambridge, a short and accessible paper called “Tie knots, random walks and topology” by Thomas M.A. Fink and Yong Mao published in Physica A 276 (2000) 109–121.
Here are some other resources I've used: Ashley Book of Knots [0], Grog's animated knots [1], and youtube.
A lot of useful knots build on basic knots and variations of basic knots. More complex knots used to be daunting to me (trying to memorize how to tie the whole thing). But they become easier when you can recognize the basic knots used in the complex ones. Learning the use cases for the knots also helped.
[0]: https://archive.org/details/TheAshleyBookOfKnots/mode/2up
[1]: https://www.animatedknots.com/