# What are these 3-manifolds from surgery?

I know that surgery on the unlink with +0 slope gives $$S^2 \times S^1$$ (where all the links above are embedded in $$S^3$$). I figured (I think) that surgery on the hopf link (with +0 on both) returns $$S^3$$ by performing the surgery on one of the unknots and pulling the second unknot into the new torus (the 'slam dunk' trick) - but here is my first question:

In "On Kirby's Calculus of Links", Fenn and Rourke prove that you can get from any one integral surgery presentation of a 3-manifold to any other presentation through 'Kirby Moves' - where these are performing surgery on +1 unknots (or adding +1 unknots into the link) changing the surgery coefficients and linking of the components correctly. How do I perform these kirby moves to get that the surgery on the hopf link shown above equals $$S^3$$?

(Actually, I think this picture solves my first question:)

I'm still not sure what 3-manifold the final surgery corresponds to - I couldn't simplify it.

• The 3d one is the Euler class $-2$ circle bundle over $RP^2$. See eg chapter 4 of Gompf-Stipsicz which would make good reading for you. This is more of a Mathstackexchange level question; I'd suggest going there with similar such questions. – Danny Ruberman May 1 at 11:33
• Another good resource for these questions is Rolfsen's Knots and Links'' especially Section 9H. (Though Gompf-Stipsicz might be the more canonical choice.) Rolfsen's book should help you with your question, because it promotes the mentality that you can undo twist regions via 1/n Dehn surgery. – Neil Hoffman May 1 at 14:22