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)
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.