What Is Knot Theory? Why Is It in Mathematics? [pdf]

That is awesome!

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.

Here's a paper on the topic:

"Spontaneous knotting of an agitated string" [1]

From the abstract: "We performed experiments in which a string was tumbled inside a box and found that complex knots often form within seconds. We used mathematical knot theory to analyze the knots."

[1] http://www.ncbi.nlm.nih.gov/pubmed/17911269

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...

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.

Loops tangle creating more loops easierly and do not undo themselves typically but rather get tighter upon pulling

that's knot funny

I think it comes down to the difficulty of finding such things. The goal is usually to find "enough" invariants that all objects can be distinguished, but we settle for just whatever invariants we can happen to find. Finding an invariant is pretty easy: it's straight-forward to show that something is invariant under a certain transformation since you just apply a transformation (in this case, usually a homotopy or similar) and see what it can do to the invariant. For knot theory, you can often check just what it does under the Reidemeister moves, giving you just a couple things to check.

How would you go about showing that something is a dinvariant? You would need to take two objects that are distinct and show that their dinvariants are distinct. I.e. use the fact that a homotopy between the two knots does not exist. This is much more difficult, though it does get done. In particular, it's done whenever we have a complete classification of all objects of a certain type (say orientable surfaces, classified by the single invariant (and also dinvariant) their genus). Though generally here we would be actually approaching this from an invariant perspective but just showing that once you have enough invariants, they collectively become a dinvariant.

Invariants are useful because they allow you to distinguish equivalent knots, which is otherwise really hard to do. If two knots have different invariants, then they are surely different.

Dually, you would want the following: if two knots have identical dinvariants, then they are surely the same. Since "if the knots are equivalent, the assigned values might not be equal", dinvariants cannot accomplish this function, and that makes them mostly useless.

(full disclosure: lawyer who did his undergraduate degree in math, not a topologist)

That is extremely difficult but that is, indeed, the goal. The main business of algebraic topology is doing exactly that.

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.

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 :]

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.

Paper folding is something just as fundamental yet it wasn't until 1893 that humans started mathematically analyzing it.