Inventing an algebraic knot theory for eight year olds (III)

The reason I don't think it's so obvious is because, while the input knots may be finite structures, the transformation between the two might be unbounded in size. E.g., one can repeatedly "blow up" a knot presentation by adding in superfluous expansions of identities like L=LL; it might be the case that certain combinations of these trivial identities combine to make some nontrivial manipulations of a knot.

This is essentially the reason why there is no algorithm, which when given a finite group presentation and two input words, can decide whether the two words represent the same group element.

New concepts are easy for kids to grasp. But they often fall short in their body of knowledge.

Operators that you're expected to know, and jargon you're expected to understand are a bit challenging for kids. I'm sure he's not talking to the kids about functional and tensorial composition and expecting them to understand. Those are loaded terms. Even to me the term "tensor" takes me a bit of a struggle in my mind because it evokes the idea of a string pulled tight, not so much a mathematical relationship.

But the article isn't written for the 8 year olds, and I think that's where the disconnect is. The idea is you can decompose the knot into some elementary relationships, and then you can do some operations on those sets of relationships, and define and learn about the structure of the knot.

This sort of thing is really cool for kids I think. Because it takes something that's complex and unknowable and breaks it down very very simply, (is it straight, is it crossed, is it looping back?), and then lets them see and explain such a complex system simply. That's really rewarding, and it doesn't require any special knowledge.

You can teach a few terms (in context) and operators, and variable substitution, and kids can pick up on this easily enough, as long as you don't require too much background. But ultimately, you're essentially doing what kids that age excel at doing, which is taking something complex and classifying it and breaking it down into simple parts to understand it. Whether it's parts of knots, or good and evil, or elements, or friends/enemies, or pokemon types or whatever else, kids that age love to make sense of their world through abstraction, simplification and categorization.

I think we can easily think kids are less capable than they truly are because of this lack of body of knowledge. So they have difficulty with division, obviously they can't integrate, most of this must be beyond their grasp. But this project while it might seem a bit complex to an adults eyes, is actually really simple, can you identify 3 knot parts visually, separate them, and write them down with a letter corresponding to the part?

An eight year old might have an easier time than an adult, because the adult might be stuck trying to figure out what K◦K’:(m, n’) really means relative to their own understanding of those symbols, while the kid might just get told that ◦ means putting them together, and the first number in the brackets means the number of ends on the left, and the second number means the number of ends on the right, and in this context that's all you really need to know.

Yes!

It's got 5 points - maybe nobody's commenting on it because very few are succeeding in loading it.

Thanks for the cache link.

Edit: it seems to be back.