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.



Surgery of $S^3$
$S^3$ is a union of two solid tori, glued along their boundary, where a meridian is glued to the longtitude, and conversely. When you do the surgery, you get a solid torus doubled along the boundary. If you think of a solid torus as $D^2 \times S^1,$ the double is a double of $D^2$ times $S^1.$ The double of $D^2$ is $S^2,$ hence, when the smoke clears, you get $S^2 \times S^1.$


