Learning roadmap for Knot theory I want to study Knot theory in the summer break. What are some good resources on this topic? My background is a a first course in Analysis and Linear Algebra, and Hatcher's notes on point-set topology. Also, what should I study next, and from where, to be sufficiently prepared for learning Knot theory? I am interested in learning some Algebraic Topology as well.
 A: I'm a big fan of Justin Roberts excellent set of notes titled "knots knotes" which is available on his web page.
There are other excellent references such as Rolfsen's Knots and Links, Lickorish's Introduction to Knot Theory, and Adam's Knot Book. Related to this field is 3-manifold topology, which is also a very interesting subject to read. Hatcher has some online notes.
I also highly recommend some passing familiarity with homotopies, isotopies, and what have you (basic algebraic topology via Hatcher or Fulton or Bredon etc.) and some differential topology (Milnor or Guellemin-Pollack). If you're completely new to manifolds, Spivak's Calculus on Manifolds is a great short text; Milnor's Differential Topology is even shorter! Below is my take on a few texts.
Roberts - Very approachable, very Modern, easy to read, fun exercises.
Rolfsen - A little dated, but beautiful diagrams and great exercises. Lots of good stuff here.
Lickorish - Seems to be a little more sophisticated than the others. I have no personal experience with this text.
Schulten's Intro to 3 Manifold Topology - Difficult text, but very good.
Have fun reading!
A: The best introduction to the subject I know is The Knot Book: An Elementary Introduction to the Mathematical Theory of Knots by Colin C. Adams (AMS). I recommend it without hesitation. From there you can proceed to more advanced and formal texts.
