I've looked around, but I haven't found anything in particular on Google or here, so I figure I'd ask. What are some solid prerequisites to be able to tackle Kirby Calculus?
I have a solid foundation in undergraduate analysis, working through graduate algebra now, and I have taken some courses in Statistics, along with a very brief introduction to topology. Thanks in advance!
On
Some experience with Poincare duality, (co)homology, the fundamental group, and vector bundles is probably necessary to get started (these should all be covered either in first year courses on differential or algebraic topology at most universities).
Some experience with knot theory, Morse functions, characteristic classes, and projective algebraic geometry (especially regarding blowing up the projective plane) can be very useful.
The first 3 chapters of Hatcher and a good portion of a book like Gulliemin-Pollack would probably be enough to give a book like Gompf-Stipsicz a try.
The Kirby calculus is a way of coding how to generate $3$-manifolds from links and determine whether or not manifolds thus generated are equivalent. (There's also a $4$-manifold version with handlebody decompositions that's a bit more complicated. In extremely broad terms that shouldn't be taken seriously, the topology of dimensions $1$ and $2$ is trivial, dimensions $\geq 5$ are dominated by the cobordism theorems, $3$ is very different and geometric, and $4$ is just weird.) It's a bit odd to consider it before a general course in geometric or algebraic topology, since it's designed to answer more technical questions (e.g., constructing exotic $4$-manifolds) than arise at the level of, say, Hatcher's book on algebraic topology or Lee's on smooth manifolds. That having been said, if you're willing to ignore the motivation, you could probably get into the subject with a crash course of reading those two books.
In general, my response to questions about prerequisites for a subject is just to dive in and see how far you get. The worst case scenario is that you get stuck reading a book or paper, and you won't be able to understand everything the first time through; that doesn't mean that the time is wasted or that you can't get anything out of it. Especially at the undergrad or grad level, there's a huge amount of math out there to learn, and you shouldn't be dissuaded from something you're interested in.