# Small knots becoming isotopic after connect sum

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).

• What do you mean when you say “this can’t happen in dimension 2 (say for some geometric reason)”? If I understand your question, this would be about embedded $S^0$’s in $S^2$, which for silly reasons are always isotopic. Oct 16, 2022 at 14:54
• @AndyPutman Oh yes -- I'm sorry I was totally incorrect in my phrasing there. I was thinking of curves on a surface (just as fun low low dimensional analogy). I'll edit - thank you. Oct 16, 2022 at 16:57

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).