[HN Gopher] Algebraic topology: knots links and braids
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       Algebraic topology: knots links and braids
        
       Author : marysminefnuf
       Score  : 57 points
       Date   : 2026-03-09 14:28 UTC (8 hours ago)
        
 (HTM) web link (aeb.win.tue.nl)
 (TXT) w3m dump (aeb.win.tue.nl)
        
       | marysminefnuf wrote:
       | Who are you degenerates who like without commenting? If you like
       | it tell hn why lol
        
         | random3 wrote:
         | They do have nice pictures
        
         | andrewflnr wrote:
         | 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.
        
           | marysminefnuf wrote:
           | 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.
        
       | jesuslop wrote:
       | 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. Schrodinger 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".
        
       | orlandpm wrote:
       | 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
        
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