Knot Theory and higher dimensions

[Galap]

Hi,

I was thinking about Knot Theory for a while and started thinking about higher dimensionalities. Could the knots we know so well (knots in 3d space) be undone if allowed to be manipulated through a fourth spatial dimension? Could they be made topologically equivalent to the unknot? And if one knot can be 'unknotted' in 4 dimensions, can any? The 2d to 3d analogy doesn't work this time because you can't have knots in 2d.

Personally, it seems to me (without any calculation) that they could, seeing as in a higher dimension, you can move objects past each other wihtout them touching, even though doing so would be impossible in the lower dimension (i.e. you could take a ball out of a box without opening it), so it seems that they could be undone.

The interesting thing about this is that if it is true, it shows that the knots arent really knots at all, just different twistings of the unknot, but rather a manifestation/demonstration of the limitations of manipulation in certain dimensionalities. It also shows that if they are the same in 4d, that different things happen if you lower it to 3d in different ways. Is this similar to how 11d M theory breaks into different string theories in 10d?

 

[Hurkyl]

You are correct -- in four dimensions, there is only one way to knot a string: the unknot. However, I believe there are interesting ways to knot a surface...

Often, there is an odd phenomenon in topology that:

• Things are simple in very few dimensions, because there isn't enough room to be complex
• Things are simple in very many dimensions, because there is a lot of room to maneuver

and so only between the extremes do things become complex.