What should I study to understand knot theory?

Question

I want to ask for some advice :)

I am aware that it might be a little early to get interested in specific areas/subareas at his stage, but the theory of knots interests me a lot.

I've went through PMA, topology at the level of Munkres and 《Basic topology》 by Armstrong, Dummit&Foote, Linear algebra(quite rigorously), basic set theory, ...etc.

What should I study next in order to start understanding knot theory?

(I've read the Knot book by Adams, by the way. I just wanted to know what I should study to learn some more.)

Thanks in advance! I would really appreciate any word of advice.

Answer 1

I'd say at least a solid knowledge of algebraic topology and some experience in analysis on manifolds.

Answer 2

Serious work with knot theory involves homology and category theory and a lot of authors quickly go there. However, for basic understanding, elementary courses in topology should suffice.

For a more down-to-earth perspective, I suggest (in addition to Adams), also these books:

Alexei Sossinsky - Knots: Mathematics with a twist
Prasolov & Sossinsky - Knots, links, braids and 3-manifolds