I am interested in the following situation: I have two codimension-2 knots $K_1$ and $K_2$ in $S^n$ and they are not isotopic. Furthermore, $K_1$ is not isotopic to the mirror image of $K_2$ and vice versa. Could it be that $K_1$ and $K_2$ become isotopic after connect summing (away from the knots) another $n$-manifold $X$ (so that $K_1$ and $K_2$ are then isotopic inside of $X$)?
(In the case where say $K_2$ is the mirror image of $K_1$, we could connect sum on a nonorientable $X$ and going around a nonorientable loop would do the trick -- I don't know any other tricks.)
I would like to know the answer to this question specifically in low-dimensions (<5). My guess: this can't happen in dimension 2 (here changing the codimension to consider curves on the surface -- say for some geometric reasons), this can't happen in dimension 3 (say by Gordon-Lueke and the fact that the fundamental group of the complement says a lot), this can happen in dimension 4 (since it's a jungle out there).