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Higher-Dimensional Knot Theory

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Jonny Evans

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Nov 6, 2003, 8:35:58 AM11/6/03
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Hi!

I've recently become interested in knots, and I was wondering if
anyone could clarify a few points for me...

1. I read on mathworld (http://mathworld.wolfram.com/Knot.html) that
it has been proved that knots cannot exist in dimension greater than
or equal to four. Is this actually saying that 1-manifolds embedded in
R^4 are basically equivalent to the unknot?

2. I read elsewhere that we could generalise the definiton of knot to
an embedding of an n-manifold in an n+2-manifold. Is this dependent at
all on the metric structure of the manifold? e.g. are
"pseudo-Riemannian" knots any different from conventional ones? The
only reason I ask is in relation to Campbell's theorem, where the
number of dimensions needed to embed Riemannian and pseudo-Riemannian
manifolds in locally flat space are different.

3. Is it the case that in for higher-dimensional knots, embedding them
in yet higher dimensional spaces enables us to untie them (in the
sense of question 1)?

I'd be grateful to anyone who could answer these questions!

from Jonny!

Lee Rudolph

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Nov 8, 2003, 6:57:04 AM11/8/03
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larryp...@hotmail.com (Jonny Evans) writes:

>Hi!
>
>I've recently become interested in knots, and I was wondering if
>anyone could clarify a few points for me...
>
>1. I read on mathworld (http://mathworld.wolfram.com/Knot.html) that
>it has been proved that knots cannot exist in dimension greater than
>or equal to four. Is this actually saying that 1-manifolds embedded in
>R^4 are basically equivalent to the unknot?

That's exactly what it's saying (or should be saying, since
that's what's true).

>2. I read elsewhere that we could generalise the definiton of knot to
>an embedding of an n-manifold in an n+2-manifold. Is this dependent at
>all on the metric structure of the manifold? e.g. are
>"pseudo-Riemannian" knots any different from conventional ones? The
>only reason I ask is in relation to Campbell's theorem, where the
>number of dimensions needed to embed Riemannian and pseudo-Riemannian
>manifolds in locally flat space are different.

Generally when seeking to generalize classical knot theory, one
assumes that there *is* at least one "embedding" (of whatever sort)
of X in Y, and only then tries to classify *all* embeddings (perhaps
of that sort, or perhaps of a more--or maybe less--restricted sort)
of X in Y (or at least distinguish from some others). So, for example,
you might give X a particular metric, find a locally flat space Y
in which X embeds, and then dare to call the study of all such
embeddings of (that fixed) X in (that fixed) Y "knot theory";
depending on who you were talking to, you might or might not
get away with it.

>3. Is it the case that in for higher-dimensional knots, embedding them
>in yet higher dimensional spaces enables us to untie them (in the
>sense of question 1)?

Yes (but you might need to increase the dimension of the target space
more than you'd think).

Lee Rudolph

Andrew Ranicki

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Nov 8, 2003, 11:18:04 AM11/8/03
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larryp...@hotmail.com (Jonny Evans) wrote in message news:<33467a68.03110...@posting.google.com>...

> 1. I read on mathworld (http://mathworld.wolfram.com/Knot.html) that
> it has been proved that knots cannot exist in dimension greater than
> or equal to four. Is this actually saying that 1-manifolds embedded in
> R^4 are basically equivalent to the unknot?

Yes. This is true for locally flat topological embeddings
(H.Gluck, "Unknotting S^1 in S^4", Bull. A.M.S. 69 (1963), 91-94).

> 2. I read elsewhere that we could generalise the definiton of knot to
> an embedding of an n-manifold in an n+2-manifold. Is this dependent at
> all on the metric structure of the manifold?

No. There is a purely topological theory of high-dimensional knots.
There is a severely algebraic treatment in my book "High-dimensional
knot theory" (Springer, 1998)

> 3. Is it the case that in for higher-dimensional knots, embedding them
> in yet higher dimensional spaces enables us to untie them (in the
> sense of question 1)?

Yes. It is also possible to topologically untie locally flat n-dimensional
knots in (n+3)-space for any n>1 (J.Stallings, "On topologically
unknotted spheres", Ann. of Maths. 77 (1963), 490-503).

Andrew Ranicki

David Marcus

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Nov 8, 2003, 2:59:15 PM11/8/03
to
Jonny Evans wrote:
> 1. I read on mathworld (http://mathworld.wolfram.com/Knot.html) that
> it has been proved that knots cannot exist in dimension greater than
> or equal to four. Is this actually saying that 1-manifolds embedded in
> R^4 are basically equivalent to the unknot?

Yes. If you want to pass one strand through another, just move the
strand a bit sideways into the fourth dimension. I find it easier to
imagine the fourth dimension as some attribute like color, e.g., red. So
make one strand redder, move it past where the other strand is, then
remove the red color. Voila, the two strands have passed through each
other.

--
David Marcus

Ryan Budney

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Nov 8, 2003, 5:40:53 PM11/8/03
to

> 1. I read on mathworld (http://mathworld.wolfram.com/Knot.html) that
> it has been proved that knots cannot exist in dimension greater than
> or equal to four. Is this actually saying that 1-manifolds embedded in
> R^4 are basically equivalent to the unknot?

> 3. Is it the case that in for higher-dimensional knots, embedding them


> in yet higher dimensional spaces enables us to untie them (in the
> sense of question 1)?

Yes. In general, one can not "knot" an n-manifold in R^m provided m>2n+1.
Precisely: there is only one embedding up to isotopy. I believe the first
person who gave arguments to this effect was Hassler Whitney. This was
more-or-less implicit in his proof of the weak embedding theorem.

The rough outline of the proof is this: if f,g:N-->R^m are two embeddings,
they are homotopic. Let F:IxN-->R^m be the straight-line homotopy from
f to g, F(t,x)=(1-t)f(x)+tg(x)

Define G(t,x)=(F(t,x),t). This is a function from IxN to R^{m+1}.

By the weak embedding theorem, G is epsilon-close to an embedding G'.
Similarly, there is a dense collection of unit direction vectors v in the
m-sphere so that if pr_v is projection onto the orthogonal complement of
v, then pr_v(G') is a 1-parameter family of embeddings. So choose v so
that it is close to the vector (0,0,...,0,1). This gives you pr_v(G') a
1-parameter family of embeddings such that pr_v(G'(0,x)) is epsilon-close
to f(x) and pr_v(G'(1,x)) is epsilon-close to g(x) (for all x in N).

Of course, I've skipped lots of steps here, primarily, the proof of the
weak embedding theorm. This can be found in Hirsch's "differential
topology" of Guillemin and Pollack's "differential topology".

> 2. I read elsewhere that we could generalise the definiton of knot to
> an embedding of an n-manifold in an n+2-manifold. Is this dependent at
> all on the metric structure of the manifold? e.g. are
> "pseudo-Riemannian" knots any different from conventional ones? The
> only reason I ask is in relation to Campbell's theorem, where the
> number of dimensions needed to embed Riemannian and pseudo-Riemannian
> manifolds in locally flat space are different.

There are knotted codimension-1 manifolds, too: compact surfaces in R^3.
It turns out that all embedded 2-spheres in R^3 bound 3-balls, so they
are all isotopic ie: no knotted 2-spheres in R^3. Similarly, all
embedded tori S^1xS^1 in S^3 bound a solid torus D^2xS^1 so studying
embeddings of tori in R^3 or S^3 is essentially the theory of classical
knots in R^3. Once you get to higher genus surfaces things become more
complicated.

For example: There is a connected-sum decomposition of embeddings of
compact surfaces in S^3. Defn: an embedded surface N in S^3 is a
connected-sum if you can write S^3 a union of two 3-balls B_1, B_2 with
(B_1 intersect B_2) a 2-sphere which intersects the surface N in a single
closed curve C. The connected-sum is non-trivial provided the closed
curve C cuts the surface N into two components, neither one a disc. An
embedded surface is prime if it does not admit a non-trivial connected-sum
decomposition. With a little work, you can construct embeddings of genus 2
surfaces in S^3 which are prime. Perhaps someone here knows a good
reference? I've never seen a reference myself, these things just pop-up
occasionally.

In dimension 4 there is an even more basic problem. Given an embedded S^3
in S^4, does it bound a 4-ball in S^4? are there exotic smoothly-embedded
S^3's in S^4? This is the Schoenflies problem in dimension 4. In all
other dimensions the answer is known (see for example the Kirby problem
list http://www.math.berkeley.edu/~kirby/problems.ps.gz)

-ryan

Thomas Mautsch

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Nov 10, 2003, 1:18:47 PM11/10/03
to
In <MPG.1a17151ad...@news.rcn.com> schrieb David Marcus:
> Jonny Evans wrote:
[ ... ]
>> Is this actually saying that 1-manifolds embedded in
>> R^4 are basically equivalent to the unknot?
>
> Yes. If you want to pass one strand through another, just move the
> strand a bit sideways into the fourth dimension. I find it easier to
> imagine the fourth dimension as some attribute like color, e.g., red. So
> make one strand redder, move it past where the other strand is, then
> remove the red color. Voila, the two strands have passed through each
> other.

Great, but this proof only works under the assumption
that the knot is contained in a 3-dimensional subspace of R^4.

Isn't it at least as hard to show
that any knot can be brought into such a position
as showing the triviality by the other mentioned, more technical methods?
--
Thomas Mautsch

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