I'm assuming that by "Hatcher Algebraic topology book" you actually mean Algebraic Topology instead of one of his other books (say, Vector Bundles and K Theory), and that this is Kauffman's Quantum Topology. Do bear in mind that this is a book intended for professional mathematicians, so when you ask what background you need, you're essentially outlining a graduate career! What follows are some quick recommendations on my part for introductory texts to the general field in which the Kauffman text seems to fall.
A working knowledge of basic algebraic topology is certainly necessary to understand the material in this book (really, necessary to understand much of modern mathematics), but it appears to me by no means sufficient. Since the subject appears to be applications of the mathematics of quantum theory to low-dimensional topology, you'll probably also want to study low-dimensional topology and quantum mechanics.
For a quick introduction to ideas in low-dimensional topology, you might consider picking up Thurston's book Three-Dimensional Geometry and Topology (much broader but less polished lecture notes are available online) and Scorpan's Wild World of 4 Manifolds. It looks like you'll also probably want an introduction to knots as well; you might check out Lickorish, An Introduction to Knot Theory, or Rolfsen, Knots and Links. For a deeper guide to four-dimensional theory (very different in flavor from three-dimensional theory!) you might check out 4-Manifolds and Kirby Calculus by Stipsicz and Gompf.
At the same time, you'll want to study the mathematical formalism of quantum field theories. I don't know that I can help you much here, but from time to time I hear buzzwords like "geometric quantization" and " For a classical start, I have been recommended Reed and Simon, Methods in Modern Mathematical Physics.
Mind, this doesn't mean you shouldn't read your book as well! You should never try to learn mathematics linearly (even though we often teach our undergraduates that way), and the first article, at least, looks like a reasonably clean survey.