Let us start with a $S^1$ 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 (1) cut out a thin tube $X$ around $S^1$, and then (2) modify this $X$ under some transformation $\eta$ (like 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 $X$.
EDIT: Following Ryan's 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 $M^d$.