This year I put some effort into understanding mathematical knots, particularly, the role that dimension plays regarding knots.
First: what is meant by a knot? In mathematical usage, it reflects the notion of a string-like thing that can’t come untied, rather than the more practical idea of something that holds fast. A knot in a piece of string can always be untied, so long as the ends of the string are free, but if the ends of the string are glued together to form a loop, a real knot is there to stay. On the other hand, it is possible to wind a loop of string around so that it looks complicated, yet without cutting the string, the winding can come undone to form a circle of string — that is not regarded as a knot.
So a mathematical knot is defined as the image of a smooth, one-to-one mapping of a circle into three-dimensional space, which can’t be transformed to a circle by continuous movements of the string (movements that don’t break the string) that never allow the string to intersect itself.
Now, some knots can be transformed to other knots by moving their parts around according to those rules, so they’re called “equivalent”. But not all knots are equivalent.
So the first questions for mathematicians is, how to know if two knots are equivalent or not? Another question is, how to classify all the possible non-equivalent knots? Mathematicians began puzzling over such things in the late 1800s, It wasn’t until the 1970s that the classification of all knots was completed.
Now, a knot in three dimensions can’t be represented on a flat surface, without making special notations — usually saying "here, this part of the string goes over this other part of the string." In fact, those notations are used to form a mathematical system to describe and compare knots.
Knots are all about dimension. It turns out that in four dimensions, all knots of the kind described above come untied. There’s just that much extra room for movement in four dimensions. A knot is a configuration of a circle in three-dimensional space.
Is there an analogue for a knot in four dimensions? One might think, well, one-dimensional things like loops and curves can’t form knots, but maybe two-dimensional things can. And a mathematician might think of the Klein bottle, which is a two-dimensional surface somewhat like a torus, twisted in a way that it has no inside and outside. Such a surface can live in four dimensions without intersecting itself — but to represent it in three dimensions, it must intersect itself. This appears analogous to the situation of a knot being projected onto two dimensions. But the two ideas are not analogous. The Klein bottle is intrinsically different from a sphere or a torus, whereas a knot is not intrinsically different from a circle — the difference between knots has to do with how they lie in three dimensions.
I don’t know if there are analogs to knots in four dimensions. I’m trying to find out.
But while I was thinking about these things, it occurred to me that, while a knot can’t be projected one-to-one onto a plane, it might be possible to represent it directly on some other two-dimensional surface. I tried a sphere — but for knots it’s no better than a plane surface. Then I tried a torus — it looked good at first, but I couldn’t get the simplest knot, the "trefoil" knot, to go onto it.
And then I thought of the Klein bottle. And voilà — well, maybe not voilà, because it’s really hard to draw even a Klein bottle in a flat image. But it turns out the trefoil knot can be represented on that surface.
But it gets stranger. The next-most complicated knot can’t be represented on the Klein bottle, but it can be on the torus!
See: Surface-Knots in 4-Space: An Introduction Kamada, Seiichi 2017 Springer Monographs in Mathematics