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I am not familiar with topology. We know that in $R^3$, we cannot separate two "rings": two copies of $S^1$, if they are "linked".

I wonder that is there any similar results for two copies of $S^1\times I^k$ embedded in $R^{2k+3}, I:= [-1,1]$? Thanks a lot!

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  • $\begingroup$ Some keywords: "knotted spheres," "knotted tori," "knotted surfaces." $\endgroup$
    – Qiaochu Yuan
    Commented Nov 7, 2014 at 19:21

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A pair of circles can be unlinked in any dimension above three. Thickening the rings doesn't change this. However, it is possible to have linked $n$-spheres inside of $\mathbb{R}^{2n+1}$.

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