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Let $L$ be a $0$-framed trivial knot in $S^3 \subset B^4$. Take $B^3 \subset B^4$ such that $B^3$ splits $B^4$ into two and $\partial B^3$ intersects $L$ only two points.

Take a neighborhood $U$ of $L$ in $S^3$, which is $S^1 \times B^2$. We attach a 2-handle $B^2 \times B^2$ to $B^4$ along the identification $U=S^1 \times B^2=\partial B^2 \times B^2$.

We condider the union of the resulting 4 manifold with narrow regular heigborhood $B^3 \times [\epsilon, \epsilon]$ of $B^3$ in $B^4$.

I'd like to prove that this union can be identified with a cylinder over a torus $S^1\times B^2 \times [\epsilon, \epsilon]$.

Could you show me a proof? Also if you have a geometrical explanation please explain it.

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Whew. That got me lost. But I'm not into those things so it's not a remark about the quality of the question, just me being lost in space while reading your thing. =P –  Patrick Da Silva Feb 29 '12 at 9:38

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