Math Overflow seems to have a dearth of low dimensional topology, but this seems like an interesting question. Let $L$ be a $2$-component link in $S^3$. Suppose that there is a framed arc joining the two components such that surgery along that arc yields an unknot. (In other words, fatten the arc to a long thin rectangle where two of its edges are subarcs of the link components. Delete these arcs and replace them with the other two edges of the rectangle.) I wonder how restrictive this condition is. For example, can one prove that there are links which don't have this property? I don't really have a good feel for what is possible.

One way to construct an example is to start with an unknotted surface with one boundary component. Now attach a band to the boundary of this surface that travels through all of the surfaces holes. This is now a 2-component link. Snipping the band can be realized by surgery along a transverse arc, and yields the boundary of the original surface, which is an unknot.