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).



Small knots becoming isotopic after connect sum
I believe can happen in dimension $4$, and probably in all higher dimensions. Take two inequivalent knots $K, K'$ with the same exterior (Cappell-Shaneson; Gordon). Then $K'$ is obtained from $K$ by a "Gluck twist", in other words trivialize the normal bundle of $K$ (there's only one way to do this), remove it, and glue back in using the diffeomorphism $f: S^1 \times S^2$ given by $f(\theta,z) = (\theta,$ rotation of $z$ by $\theta$).


There's a general principle that connect summing with $\mathbb{C}P^2$ undoes Gluck twists, and I think that this *should* prove that $K$ and $K'$ are now isotopic in $S^4 \# \mathbb{C}P^2$. I'd have to think about the proof (or maybe someone could point to a reference).


