Linked or not
S. B. Gray
separate. In 4D, what if anything defines whether two spheres are
similarly linked? Is this situation even defined? I know that in 4D
there are no knots, so maybe there are no links either.
Steve Gray
Lee Rudolph
"S. B. Gray" <ste...@ROADRUNNER.COM> writes:
>Circles in 3D can be either linked (topologically inseparable) or
>separate.
Though what you say is even true of geometric circles (in
standard Euclidean 3-space), I'm going to assume you mean
by "circle" is an arbitrary (maybe smooth) simple closed
curves (since you bring up knots later). Then I'm going
to go further and assume that below you mean by "sphere"
a topologically (probably smoothly) embedded 2-sphere
in Euclidean 4-space (a "simple closed surface", as it
were). On that assumption:
>In 4D, what if anything defines whether two spheres are
>similarly linked? Is this situation even defined?
It is indeed defined, and there are various algebraic
and/or geometric invariants that with luck can be
calculated to show that, in a given case, two 2-spheres
in Euclidean 4-space are "topologically inseparable"
in various senses (most strongly, perhaps, in the
sense that there is no 4-dimensional ball embedded
in 4-space in such a way that it contains all of
one of the 2-spheres and none of the other).
>I know that in 4D
>there are no knots, so maybe there are no links either.
You may *believe* you "know" that, but it is false
(so if you are one of those people who insists only
truths can correctly be said to be "known", your
belief is false). There are no *one-dimensional*
(smooth) knots in Euclidean 4-space, but there are
plenty of "knotted" 2-spheres in Euclidean 4-space.
See if you can find a copy of Dale Rolfsen's book
_Knots and Links_. Good discussions of everything
you've asked about (and much more) are contained
therein.
Lee Rudolph
Martin Saturka
> _Knots and Links_.
It is a nice book (reachable e.g. via Google books). An example on the
unsplittable links is at page 94, 2-sphere and spun trefoil (something
knotted by itself before its n-sphere embedding yielding rotation).
See e.g.
http://mathworld.wolfram.com/Andrews-CurtisLink.html
http://en.wikipedia.org/wiki/Knot_theory#Higher_dimensions
Another possible (and may be the original) question is about (two) n-
spheres as spheres, not their many embeddings.
Then one can easily see that +-1 change of space dimension requests
+-1 dimension change of one of two n-spheres.
* Two 0-spheres (one 0-sphere is a pair of points with a fixed
distance) can be linked at 1D space, but they are free at 2D space.
* One circle (1-sphere) and one 0-sphere (two points with a fixed
distance) can be linked at 2D space, but they are free at 3D space.
* Two circles (1-spheres) can be linked at 3D space, they intersect
(or are in/out-side) at 2D space, and they are free at 4D space.
* Similarly one (2-)sphere and one 0-sphere can be linked at 3D space.
* One circle (1-spheres) and one (2-)sphere can be linked at 4D space,
two (2-)spheres can be linked at 5D space, etc.
Sometimes, I use colors for the other dimensions to imagine them, e.g.
white for 0 value and red vs. green for negative, positive values.
With it, you can see how the n-spheres try to unlink and so.