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Hi all.

I'm not even sure that this is a theorem, but a while ago I heard a topologist friend of mine (whom I haven't been able to reach) saying that given a continuous embedding $f:M^{n}\to N^{r}$ ($M$ and $N$ being topological manifolds), a sufficient condition for $f(M)$ to be (possibly) knotted in $N$ is that $r=2n$. Does anyone know if this is even true? Also, if it is true then the following question seems natural: will the embedding still be knotted if $r>2n$. Thanks a lot.

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Could you make your question more precise, like what do you want knotted to mean for topological embeddings? – Ryan Budney May 1 '12 at 19:43
``Codimension 2'' means $r=n+2$. – Lee Mosher May 1 '12 at 19:43
The logical meaning of "a sufficient condition for STATEMENT to be possibly true" is rather unclear. – Lee Mosher May 1 '12 at 19:45
I guess the only example I have in mind is where $M$ is a torus. By knotted I mean that the image of the torus in $N$ won't be isotopic to a 'standard' embedded torus by a transformation of the ambient space, much like in the case of $n=1$ and $r=3$. – Óscar Guajardo May 1 '12 at 19:47
@Lee: I concur and apologize. What I meant is the following: given the embedding $f$, if $r<2n$ then the embedded manifold will be isotopic to a 'standardly embedded' one (once again, I guess this only makes sense for $M=T^n$ and $N=S^r$) by a transformation of the ambient space; if $r=2n$ then it won't be. I found this: and I'm reading it right now; I'll let you know if there's something interesting there. Thanks. – Óscar Guajardo May 1 '12 at 19:50
up vote 4 down vote accepted

This question is badly posed. If you have a not simply connected manifold (of any dimension $> 3$) then two connected closed curves are isotopic iff they are homotopic, hence you have as many isotopy classes of curves as the conjugacy classes in the fundamental group. Also, following your comment, in codimension two (which means $r = n + 2$) there are a lot of "knotted" submanifolds: classical knots in $S^3$, 2-knots in $S^4$ and in general knotted spheres of dimension $r-2$ in $S^r$. In any manifold there are knotted codimension two submanifolds (i.e. pairs of non-isotopic embeddings of the same manifold in the ambient space). And this satisfies $r < 2 (r - 2)$ if $r > 4$.

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