I have been troubled by this seemingly simple question recently. How do we easily visualize the statement:

Surgery of $S^3$ over a trivial unknot gives $S^1 \times S^2$?

All I can think of for the first step is to remove a point from $S^3$ so we can work on $R^3$, but then I have no idea how to go on.

As a further generalization, what is the result if we do surgery of $S^3$ over $n$ unlinked unknot?

I am an algebraist so I have no clue how people usually visualize these objects in more than 3 dimension... Any hints will be appreciated.