We know that $S^p$ and $S^q$ can be linked in $S^d$ if $p+q<d$. Let us consider the simplest case where both $S^p$ and $S^q$ are un-knotted spheres.
I am looking for a surgery to unlink $S^p$ and $S^q$ in $S^d$.
Let us start with an example,
(0) The simples process can be done to unlink a Hopf link of a $S^1$ and a $S^1$ linked in $S^3$. In that case, we cut out a $D^2 \times S^1$ solid torus bounding one of the $S^1$ unknot. And we glue the $D^2 \times S^1$ with the remaining $D^2 \times S^1$ to form a $S^2 \times S^1$, Therefore, we successfully unlink two unknots $S^1$ and $S^1$ in the new 3-manifold $S^2 \times S^1$.
(I) Next, say $S^1$ and a $S^2$ are linked in $S^4$. Can I unlink the $S^1$ and $S^2$ by doing some surgery (with certain constraints described below, and let us say both $S^1$ and $S^2$ maintain unknotted)?
Can I do the surgery to first like the above case, say (1) cut out a thin tube $X$ around $S^1$, and then (2) modify this $X$ under some transformation $\eta$ (like any diffeomorphism), and then (3) re-glue it back to make it form a new manifold $M_a$ such that $M_a$ has no boundary: $\partial M_a =0$, and such that $S^1$ and a $S^2$ are eventually unlinked in $M_a$? How to do that? What is $X$, $\eta$ and $M_a$?
Here $X \cup (S^4-X)=S^4$, and $X \cup_{\eta} (S^4-X)=M_a$. Here $\eta$ means possible transformation when gluing two manifolds together. The $X$ may be $D^3 \times S^1$ or $D^2 \times S^1 \times S^1$, or whatever contains the $S^1$ in the interior of $X$.
(II) How about more general cases? To unlink $S^p$ and $S^q$ in $S^d$ by a similar surgery?
p.s. knowing the answer for (I) is good enough to count an answer. p.s. this is a generalized version of a question, rooted in the post from ME with no feedback.
EDIT: Following Kevin Walker suggestion, let me tentatively define,
$\bullet$ $S^p$ and $S^q$ are linked in $M^d$, if and only if their filled ${p+1}$-ball $D^{p+1}$ and ${q+1}$-ball $D^{q+1}$ (bounded by the surface of $S^p$ and $S^q$ respectively) always intersect with each other in $M^d$.
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$\bullet$ $S^p$ and $S^q$ are not linked in $M^d$, if and only if there exists some filled ${p+1}$-ball $D^{p+1}$ and ${q+1}$-ball $D^{q+1}$ (bounded by the surface of $S^p$ and $S^q$ respectively) such that $D^{p+1}$ and $D^{q+1}$ do not intersect with each other in $M^d$.