I've seen this article https://www.math.cuhk.edu.hk/~ztwu/JonesCosmetic.pdf on the Jones Polynomial and Cosmetic Surgery and I've looked at the Wikipedia entry on Dehn surgery as well. My background in algebraic topology is at the level of the second section of Munkres' Topology 2ed. However, in looking at the above mentioned paper, it still seems rather alien and the references cited aren't much better. It does seem like a very interesting area of topology to study though and I am interested enough to ask: Where do I start? What knowledge should I build up to understand this paper?
I am aware of graduate courses offered in topology as well as simple google searches for various keywords in the paper itself but, the former shoots FAR below what is done in that paper and Google search results tend to yield related stuff from publications and conferences that are just as inaccessible. I would be grateful for any feedback on this.
You will need to read something on modern knot invariants, say, "Lectures on Heegaard Floer Homology" by Ozsvath and Szabo or/and "Heegaard Floer Homologies: lecture notes" by Robert Lipschitz. However, I would suggest, first do some reading on basic knot theory (my favorite is Rolfsen's book "Knots and Links" but there are many other options). Also, make sure you took a course in abstract algebra (groups, fields, rings) and more advanced algebraic topology, covering homology/cohomology.