A New look at holes in 4D

Discussion of shapes with curves and holes in various dimensions.

A New look at holes in 4D

Postby wendy » Mon Feb 10, 2014 9:49 am

I turned to a different topological device to look at holes in four dimensions. It works in three dimensions as well, but in 4D, it tells us that the holes are out of this world. Knotted holes indeed!

We look at a hollow sphere with 'N' holes let into it. If we open this figure out by expanding it and laying it on a plane, it cones to a pancake witn N-1 holes let into it. This N-1 is the 'genus' or kind of figure in 3d. A cube, with a hole drilled from face to opposite face, has a genus of 1. Drill a hole on a different axis, you get '3', and the third direction gives 5. This can be seen in part, in that the drill makes 1, 3, or 5 entries and a similar number of exits.

When you take just the vertices and edges, you flatten it out to a schaegel diagram, which is five hollow squares. The sixth square becomes the outside.

Central inversion preserves the general shape. Since the outside is simply 'hole nr 0', we are free to move a point of central inversion to any of the holes on the flattened sphere, and this gives us a different hole.

Links are formed between pairs of holes by a pair of points. In practice, the point-pair is where a vertical circle crosses the plane. We see that you can replace the point-pair with a circle, sphere, .. where a sphere, glome, .. cross the plane.

Four Dimensions

We use a hollow glome, and spread it out onto a chorid (3d) plane. Central inversion still works, in that if we move the point to some different space, it becomes 'hole 0'. But it's no longer circular pancakes on the pan.

A terid glomohedrix (spherated 3-sphere-surface) still gives a sphere with a hole in it. It's basically the spherinder walls flattened out.

A terid glomolatrix (spherated 2-sphere-surface), gives a torus. We see that in order to snare this little thing, we need more than just a point-pair: we need a circle (ie a sphere-equator) to grab it.

A tiger, is then a hollow torus. It has two holes, and we can catch it by running a circle through one of the holes, or a point-pair one in both. The central inversion, taken from a point inside the hollow of the torus, will revert outside into a hollow torus, and the hollow to the outside.

This is a topologically different beast to something like a torus with a little torus-shaped hole, not around the central spindle.

Since we now have two kinds of hole in 4d, one is a point-like form as in 3d, and one is a line-like form. Because these are restricted to the 3d plane, line-holes do here what lines do in 3d: tangle up and knot and link. The holes of the tiger are linked. You can't separate them with an enclosing sphere, so that one is inside, and one is outside.

The clifford-holes, by making a torus-shaped thing follow a clifford-parallel, corresponds to a cylinder full of pipes, given a full-turn before the ends are joined. Every pair of pipes are linked to each other and to the outside.

However, one might suppose that there will come a trifoil hole.

Another possibility is that the hole might be in the shape of a pretzel: that is, it could have arms like a letter B or & or something. One can show by using a point-pair that these can not be separated into two or more simple loops, because that act would allow a circle to pick the figure up.
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