4 ms·

Why is it just in 3D? 1D things making knots in 3D seems like it would have immediate analogs for m-dimensional things in n dimensions. Is that not the case? Is

by imh 10y ago

Why is it just in 3D? 1D things making knots in 3D seems like it would have immediate analogs for m-dimensional things in n dimensions. Is that not the case? Is it not as rich an area of study or something?

detaro 10y ago
AFAIK, the term "knot" is only used for 1D things, and those are not very interesting in higher dimensions because you can always unravel them. The more general field is called (Geometric) Topology, and includes all your m-dimensional objects in n-dimensional spaces.
aznstriker24 10y ago
The following is hand-wavey, but only because I am not equipped to give an actual explanation :] Essentially, knots (i.e. 1D things embedded into some other space without crossing itself) are trivial in dimensions lower than three because we don't have enough space to make it interesting. A 1D thing that doesn't cross itself in the plane must be a circle, warped in some way, but it can be unwarped without ripping the plane. On the other hand, knots in dimension higher than three are trivial because we have too much space to work with. I don't know the details here, but this is what I've been told. It's kinda miraculous that knots in three dimensions have such interesting and rich structure. Hopefully an expert will come by and give a more detailed explanation, but in the meantime, hope this helps :]
wmsiler 10y ago
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.

by imh 10y ago

detaro 10y ago

AFAIK, the term "knot" is only used for 1D things, and those are not very interesting in higher dimensions because you can always unravel them. The more general field is called (Geometric) Topology, and includes all your m-dimensional objects in n-dimensional spaces.

aznstriker24 10y ago

The following is hand-wavey, but only because I am not equipped to give an actual explanation :] Essentially, knots (i.e. 1D things embedded into some other space without crossing itself) are trivial in dimensions lower than three because we don't have enough space to make it interesting. A 1D thing that doesn't cross itself in the plane must be a circle, warped in some way, but it can be unwarped without ripping the plane. On the other hand, knots in dimension higher than three are trivial because we have too much space to work with. I don't know the details here, but this is what I've been told. It's kinda miraculous that knots in three dimensions have such interesting and rich structure. Hopefully an expert will come by and give a more detailed explanation, but in the meantime, hope this helps :]

wmsiler 10y ago

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.