Re: 4d-world [Was: Re: I'm back!]

On Wed, 22 Sep 2004, Muke Tever wrote:

> On Tue, 21 Sep 2004 20:46:41 -0700, Apollo Hogan <apollo@...> wrote:
> > On Tue, 21 Sep 2004, H. S. Teoh wrote:
> >
> >> On Tue, Sep 21, 2004 at 03:37:43PM -0700, Apollo Hogan wrote:
> >> [...]
> >>
> >> Interesting, can you actually tie a sphere into a knot in 4D? I
> >> thought you could only do it to a 2D surface in 4D. But maybe I'm
> >> wrong.
> >Well, I meant S^2, the _skin_ or surface of a ball, not the solid ball.
> > (I.e. the unique-up-to-homeomorphism compact two-manifold without boundary
> > and with genus -2, :-)  But I do think that you can tie it into a knot
> > in 4-space.  You can also link together spheres (S^2) inseparably, just as we
> > can link rings inseparably in 3-space.
> >One way (not the only way) to generate knotted spheres in 4-space is to
> > "suspend" ordinary knots in 3-space... The idea is something like the knot
> > becomes the "equator" of the sphere... just take two hemispheres, sew them
> > together along the knot (of course you can't do this in 3-space, because of
> > the twisting of the knot, but there's room in 4-space).  Voila, you've got
> > a knot.  (Warning: I've not thought very much about this, so I'm bound to
> > say something wrong, but the idea is something like this.)
>
> Is that like the one where you take a Möbius strip and close it by sewing a
> panel along the edge?
Yes, same process of "surgery", the difference is that the Mobius strip only
has one edge, of course, so you only need one panel.  For the knot, I guess you
thicken it a bit so it is like a strip of paper tied in a knot (but not twisted
so it still has two edges) and then glue on two panels.

--Apollo