Algebraic topology: knots links and braids | Hacker News Reader

Algebraic topology: knots links and braids

aeb.win.tue.nl

99 pointsmarysminefnuf14 comments

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

I get excited because I went to school with one of Vaughan Jones' children and was (and still am) into math and was blown away when I understood that he was significant.

That's a wonderful set of Knotes. Thank you.

BTW is your book out ? (On algebra and programming)

Thanks very much for sharing these notes. I studied algebraic topology in grad school but somehow avoided knot theory entirely. Reading these has sparked that feeling I had when I first got into topology.

This set of notes is a fabulous document. Thank you so much. My personal interest in this subject extends from machine knitting through category theory (https://www.youtube.com/watch?v=iEaK68VRAng and more particularly https://textiles-lab.github.io/publications/2025-knit-equiva...)

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

They do have nice pictures

A high ratio of upvotes to comments is generally a sign of a high quality HN submission. It means that most people don't feel the need to add anything or argue about it. If we're going to throw around accusations of being a "degenerate", I think it fits better on people who can't stop themselves from commenting even if they have nothing useful to say.

im kidding lol. its just a funny observation. its kinda like if you like it then its good to know the educational value of it for future reference. im always suspicious of high upvote to low comment ratios so i like it when users call it out.

> A high ratio of upvotes to comments is generally a sign of a high quality HN submission.

I think that the parent is trolling, but I don't think what you're saying is true. Low number of comments usually means that no one understands the topic, but they still want to upvote because it sounds interesting or geeky. High number of comments usually means a topic where everyone feels like they can chime in without reading the article, just reacting to the title.

We're not that sophisticated. And you have evidence of this on the front page right now. A story about AI copyright with 388 comments, versus Scott Aaronson's short rant about quantum algorithms with 12 comments.

> High number of comments usually means a topic where everyone feels like they can chime in without reading the article, just reacting to the title.

Would we be in agreement that this describes a low quality submission? If not in a way that's the submitter's fault, then at least indicating a low-quality discussion?

Edit: I guess even in your cynical framing, I think the topic only upvoted because people think it's geeky is still usually better than the one where people are reacting to the title.

Upvoting is for recognising that something is worth sharing to other users.

Commenting is for discussion.

These are orthogonal, mutually exclusive matters.

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