[wmsiler]

You are right that interesting knot theory does exist in higher dimensions. It is appropriately called higher dimensional knot theory. It considers spheres of dimension m embedded in n-dimensional space. When m = 1, you get a 1-dimensional sphere, which is a circle. There are restrictions on which m and n yield interesting math. Intuitively, if the dimension of the sphere is too small compared to the ambient space (i.e. m is much smaller than n), then there will be so much wiggle room, that any knot can be "untied" without crossing itself (i.e. every knot is the trivial unknot). If the dimension of m is too big compared to n, then there is not enough room to twist things around, and so again nothing can get knotted.

It's been a while since I've studied this, but I believe it's the case that the only time you get nontrivial knots is when n = m + 2. The most well known case, of course, is when m = 1 and n = 3. But for every value of m >= 1, there are nontrivial knots in dimension m + 2. I believe it is indeed a rich area of research.