Let $L$ be a framed link in $S^3$ with $m$ components and let $U$ be a closed regular neighborhood of $L$ in $S^3$.
Let $B^4$ be a closed 4-ball bounded by $S^3$ so that $U \subset S^3$. Gluing $m$ copies of the 2-handle $B^2\times B^2$ to $B^4$ along the identifications of components of regular neighborhood and $\partial B^2 \times B^2$, we get a compact connected 4-manifold denoted by$W_L$.
On the other hand, we can glue $m$ copies of $B^2\times \partial B^2$ to $S^3 \setminus Int(U)$ along the boundary. Let us call the resulting 3-manifold $M$.
I think $M$ is equal (or homeomorphic) to $\partial W_L$. How can I prove it? Could you show me a proof or give me some references.
Also is there any way to visualize this?