I am a knot theorist (PhD student) and always tickled when the subject pops up on HN. Curious what the author had in mind including this in his algebraic topology notes. Even the Jones polynomial which is both algebra and topology is not usually called "algebraic topology", but rather "quantum topology".

If anyone is interested in a conversational intro to the subject with lots of pictures, I suggest these semi-famous "Knots Knotes" (amazing title)

https://mathweb.ucsd.edu/~justin/Roberts-Knotes-Jan2015.pdf

To comment something then, the symmetric group (bijections) is generated by permutations of two elements, in the braid group you can braid the left whisker on top of the right one, or below. If you permute twice, you do nothing, but you can twist hair one, two, three, ... any times. that is the contrast in the generators and relations of both groups. If you go to the wiki page of topological quantum computer, the photo expresses a unitary representation of a braid group element. Schrödinger evolution in discrete time is given by unitary matrices (one of the Stone theorems). Now look at the pic, and imagine threads from input 1 to output 1, input 2 to output 2, etc, in adition to the colored threads of the pic. With the extra threads as gluing spec (topological identification) you get a single colorful closed curve (or several ones). The chapter is going to talk about how this is a link/knot. So you get an algebraic understanding from a structural object as the symmetric group is, opening what these closed curves are. People in loop quantum gravity could have had their fingers on that kind of page. There is an accesible description of the Jones polynomial. If you bookmark this, next time the pros juggle the name before you you have a place to go to avoid showing "that face".

Horned sphere is a recursive beauty. Nice thing to refresh my memory on path-connected vs simply connected propety.

Who are you degenerates who like without commenting? If you like it tell hn why lol