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.