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?
What Is Knot Theory? Why Is It in Mathematics? [pdf]
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Re: What Is Knot Theory? Why Is It in Mathematics? [pdf]
#12Why 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?
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.
Re: What Is Knot Theory? Why Is It in Mathematics? [pdf]
#13Why 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?
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 :]
Re: What Is Knot Theory? Why Is It in Mathematics? [pdf]
#14Why 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?
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.
Re: What Is Knot Theory? Why Is It in Mathematics? [pdf]
#15One of my favorite mathematical diagrams: http://www-math.mit.edu/~andyp/Figures/FIGURE2.pdf, from Matveev, Fomenko, Algorithmic and computer methods in three-dimensional topology.
Re: What Is Knot Theory? Why Is It in Mathematics? [pdf]
#16In the late 1800's knot theory was quite popular with physicists. Now there is the much bigger string theory: https://www.sciencedaily.com/releases/2016/02/160210170411.h... On a lighter note I could use some knot theory to explain why earphone or computer cables always seem to tie themselves up, despite my best efforts to keep them apart.
Knot theory is still quite popular with physicists of certain sorts, such as condensed matter physicists that study "topological phases" - a popular account that I like can be found here: https://www-thphys.physics.ox.ac.uk/people/SteveSimon/PWsept...
Re: What Is Knot Theory? Why Is It in Mathematics? [pdf]
#17In the late 1800's knot theory was quite popular with physicists. Now there is the much bigger string theory: https://www.sciencedaily.com/releases/2016/02/160210170411.h... On a lighter note I could use some knot theory to explain why earphone or computer cables always seem to tie themselves up, despite my best efforts to keep them apart.
There are fewer desired unknot states to your cable than there are tangled and knotted states to them. There also is some confirmation bias as you are less likely to noticed the desired state versus having to battle through the knots for 5 minutes.
Re: What Is Knot Theory? Why Is It in Mathematics? [pdf]
#18Here's a good video from Numberphile: https://www.youtube.com/watch?v=aqyyhhnGraw
Re: What Is Knot Theory? Why Is It in Mathematics? [pdf]
#19I think a lot of human understanding is basically intuition about topological invariants in various "spaces". If you go around asking famous thinkers what they see when they think they all describe similar kinds of imagery, fuzzy shapes that merge and unmerge in various ways as they probe the subject. One good book I've found on the subject is https://en.wikipedia.org/wiki/Where_Mathematics_Comes_From .
Intuition quickly becomes unreliable when you move to spaces with weird topologies, like non-Hausdorff and non-(pseudo)metrizable spaces. When your intuition stops being useful, you actually need to calculate.
Re: What Is Knot Theory? Why Is It in Mathematics? [pdf]
#20I think a lot of human understanding is basically intuition about topological invariants in various "spaces". If you go around asking famous thinkers what they see when they think they all describe similar kinds of imagery, fuzzy shapes that merge and unmerge in various ways as they probe the subject. One good book I've found on the subject is https://en.wikipedia.org/wiki/Where_Mathematics_Comes_From .
Intuition quickly becomes unreliable when you move to spaces with weird topologies, like non-Hausdorff and non-(pseudo)metrizable spaces. When your intuition stops being useful, you actually need to calculate.
I suspect a few folks that studied p-adic numbers extensively would disagree. And in general topologists and algebraists that study non-euclidean things in general.