The ideas are quite easy to understand, although mathematicians like to have terminology which is well-defined and tend to call things like a pair of linked tori something that sounds complicated (since you can embed all these knots on them), pictures exhibit these semantics. It's just a lot of fancy words, necessarily so because it's very much like needing a computer language, with well-defined data types and operations. Please Register or Log in to view the hidden image! Call the torus in the left image T, and the torus linked to it on the right T*. T has the knot K(3,2) on it (embedded, as mathematicians say), and T* has K(2,3) on it. Except for the minor detail that in the RH image, both knots are lifted slightly away from the nice smooth representation of two linked tori. But topology doesn't care, both the knots are embedded . . . An exercise I just went through, involves figuring out how to reduce the number of loops, on say T. See if you can convince yourself the only way to reduce the three blue loops wrapped around T (i.e. those "on a cylinder") is by cutting the blue line somewhere, then shrinking the line and splicing it back together. For instance, to remove the leftmost blue loop (so K(3,2) becomes K(2,2)), cut the line above this loop, unwrap the loop and join it back to the cut; you need to lift somewhere, the cut part over an uncut part of the line to stay planar. Ed. Sorry, that's a bit confused; it's easiest to cut through two lines (which are near each other) in two places and rejoin it a certain way. See if you can figure it out.
Video ended too early. Looks like turning inside out is another property - like tying a knot - that can only be done in 3 dimensions. (Can a 4-D hypersphere be turned inside out? If my conjecture is correct, no. Knots in 4 dimensions fall apart.)
Since we're on it, the eversion of a sphere and the impossibility of eversion of other surfaces with an inside and an outside, are based on the notion of a rotation of something by 180°. We have to find the something, we assert it exists and that we can section a sphere "the right way", such that if we allow a surface to intersect itself and "pass through" while remaining continuous everywhere, we can rotate the surface 180° everywhere. We're allowed to cheat by cutting the sphere apart, as long as we glue it back together.
More on the trefoil. Please Register or Log in to view the hidden image! I'm not really that clued up about this, but it's a Heegaard diagram which has the knot embedded partly inside the torus (the blue part). In the first diagram the trefoil knot is inside the sphere (i.e. embedded in \( \mathbb R^{3} \)). The perspective shows the three crossings and the green parts are called handles or handlebodies . . . I can follow what happens between the first two diagrams, but the last one still has me thinking. ed. you should be able to get from the bottom diagram to the top right one by cutting the torus somewhere so its topologically a cylinder as in the upper diagram.
Sorry those diagrams are actually part of a way to derive a planar diagram--the Heegaard diagram, of a p,q torus knot. The points on the sphere at the ends of the green bits are free to move around, as they are on the torus. The inside and outside parts follow along of course.
This is how we do it. Please Register or Log in to view the hidden image! You move the points around the torus so the blue dashed line unwraps from the hole and wraps round the torus. Which is to say, you take K(0,1) to K(1,0).
Ok, so the business of deriving a planar diagram which, in essence, traces the movement of a pair of points on a torus, by embedding part of the trefoil knot in its interior with the remainder free to wrap around it--is equivalent to the business of pulling part of the trefoil away from the surface of the torus, and pushing some of it into the interior (of the 'solid' torus). Which you can do without cutting the knot, which would mean you don't have a trefoil knot, you have an open link (except, a link isn't open, it's closed). To get the Heegaard diagram you need an additional bit of structure, a way to trace the movement of the two points. This is acheived by constructing a boundary between them, a circle going around the hole and not around the torus. Then you deform this boundary. In the above diagram this boundary isn't included and the blue line wraps around the hole, but you could change this so it wraps around the torus (but stays in the interior), then wrap it around the hole by moving the points of intersection of the knot. So the idea is that an initial boundary between these points (a meridian) gets distorted by the rearrangement. This distortion then, is a way to get the trefoil knot, and other knots, to 'write information' about how curled up it is, on the surface of a torus (which will always have a dual torus if you can rotate it about one of two centres).
Sorry, technical fault. Meanwhile, here is where I got the images: https://mathematics.ceu.edu/sites/mathematics.ceu.hu/files/attachment/basicpage/29/yuthesis.pdf If you can wade through it, it explains what the significance of the Heegaard diagram is; basically since you can start with the trivial arc and move the points wherever you like, you can make any knot (single knotted link), on the 2-torus. Hence you can 'record' any knot in a diagram which is a trace of the deformations moving the points generates in an otherwise smooth meridian of the torus. The right diagram in the middle row on p13, is like a start state in which you can have a loop (arc) between the two points going around the hole, or not.
Sorry again, that should be a line of longitude, which is what the Greenwich meridian is on our nearly spherical planet. On the 2-torus a meridian goes around the torus according to some authors. So the author, Yu, describes a line of longitude which is deformed by the movement of the two points (what happens if you don't move one and move the other, though?), which is the red circle in the diagrams.
Knots, or knotting a length of string, have probably been used for a long time as a way to tally goods, animals etc. That there is nowadays a lot of interest in knot theory, at the mathematical (and of course logical) level, is because it seems that there are strong connections between braid groups (a knot is a closed braid) and quantum symmetries. Diagram monoids and other planar representations of knots could be seen as just a development of the original uses of knots; I suppose the navies of the world developed an early knot classification "algebra", and that it was heavily biased toward usefulness, how hard it was to make etc. Not that anything so far is meant to be saying the numbers exist because we can tie knots (but who knows?).
The Ashley Book of Knots This is the definitive book on knots. Here are approximately 3900 different kinds, from simple hitches to “Marlinspike Seamanship.” Mr. Ashley has included almost everything there is to know about them. Precisely named and classified (some new ones for the first time officially), they can be easily found in the big index. He tells when they appeared, something about their history, and what they are good for.[/quote]
It's also occured to me that asking the question "Where do numbers come from?" might not be well framed. The question assumes there's a place numbers "come from", but that could be my misunderstanding. Maybe also it's like asking "Where do knots come from?", or "Where do straight lines come from?", and as meaningful. We can construct knots and straight (or very nearly straight) lines. When we assert the knots and straight lines, or other constructions with multiple "sides" all straight say, are representations of "mathematical, abstract" objects, what are we really doing? What do we do when we make a closed loop of wire (maybe the handle we construct for the loop has multiple twists), then arrange for a soap film to form inside it? The next obvious step is to generate "free" bubbles (i.e. 2-spheres with a liquid boundary), by blowing air at the surface of the film. If you catch a free bubble in the loop and pop half of it, you have a film like you had initially. It seems that knots and minimal surfaces have a closely connected algebra (or if you like, an arithmetic). p.s. I would post some more nice piccys, but am away from my familiar home PC. Also note how, in any example I imply a stepwise, or algorithmic abstraction exists (i.e. "there is a program P").
I would hesitate to call it a program because that would imply a programmer, but IMO, the "program" is an emergent logical aspect of universal evolution along with the unfolding (explication, becoming) of universal potentials into physical reality. This is why I like Tegmark's perspective. The universal values and functions, the program of how the inherent potentials of the geometry and physics of the universe become expressed, has been identified and symbolically translated into a human scientific discipline, which we have named "mathematics", which is able to explain in exquisite detail how universal potentials become expressed in reality. Which leads to the inevitable conclusion that the universe behaves in a way consistent with what we have observed and symbolized as "mathematical values and functions" and "logical patterns" (algorithmic abstractions). As Tegmark observes, there is no difference in molecular content of a dead beetle or a live beetle. The only difference is the pattern in which the molecules are arranged! One pattern is a static (entropic) pattern (dead), the other is a dynamic (sustaining) pattern (alive).
I think the program, or an algorithm, is something humans are good at finding. At least that's the theme I'm disbursing, we're good at "problem-solving". But we do it locally, we build computers that are necessarily in some container, and I don't see that we'll try to build computers anywhere else than on earth for the foreseeable future. Why though, build computers in the first place? Why is it useful to us, currently, to be able to "hand over", as it were, a large class of problems to machines? Dumb question.
Efficiency? The universe is full of data. Computers are good at processing data....Please Register or Log in to view the hidden image! And especially good at storing large numbers of data.....Please Register or Log in to view the hidden image!
