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Let $i:T^{2}\rightarrow S^{1}\times S^{2}$ be an embedding map and $i_{*}:\pi(T^{2})\rightarrow\pi(S^{1}\times S^{2})$ be the homomorphism induced by $i$. If $i(T^{2})$ is separable in $S^{1}\times S^{2}$ and $\operatorname{Im}{ i_{*}}=0$, how can I prove that $$S^{1}\times S^{2}-i(T^{2})\cong K^{c}\cup S^{1}\times D^{2}\sharp S^{1}\times D^{2}$$ or $$S^{1}\times S^{2}-i(T^{2})\cong S^{1}\times D^{2}\cup S^{1}\times D^{2}\sharp K^{c}$$ Here $S^{1}\times D^{2}$ is an open torus and $K^{c}$ is a knot complement in $S^{3}$. Someone can me recommendation some book or article about these type of identification.

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Did you try Dehn surgery? – Ilies Zidane Sep 4 '12 at 15:04

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