I understand that any orientable $3$-manifold can be obtained by doing Dehn surgery on $S^3$ along a set of circles sitting in it; but why can we further assume the slop to be $0$, i.e. we can obtain any closed orientable $3$-manifold from $3$-sphere by several torus switch?
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There is a detailed explanation of Dehn surgery (and the answer of your question) in Dehn Surgery by Siddhartha Gadgil.