Inventing an algebraic knot theory for eight year olds (III)(researchblogs.cs.bham.ac.uk)
researchblogs.cs.bham.ac.uk
Inventing an algebraic knot theory for eight year olds (III)
http://researchblogs.cs.bham.ac.uk/thelablunch/2015/05/inventing-an-algebraic-knot-theory-for-eight-year-olds-iii/
8 comments
DS8 and I have recently been enjoying Christopher Zeeman's[1] Royal Institution Christmas Lectures[2]. The first one on linking and knotting has the same flavor as the article and could be a good intro for parents doing this at home.
[1]http://en.wikipedia.org/wiki/Christopher_Zeeman [2]http://richannel.org/christmas-lectures/1978/1978-christophe...
[1]http://en.wikipedia.org/wiki/Christopher_Zeeman [2]http://richannel.org/christmas-lectures/1978/1978-christophe...
Submitting in the comment box on the blog failed with "Could not open socket," so I'll post my question here.
Now that you have a notation (even if it may not be complete, you've raised the question in exercise 5), I am curious about computation.
1. Given two knots that are describable in this notation, is there an algorithm that decides if they are equivalent?
2. If so, what is the known (time/space) complexity of this problem?
Now that you have a notation (even if it may not be complete, you've raised the question in exercise 5), I am curious about computation.
1. Given two knots that are describable in this notation, is there an algorithm that decides if they are equivalent?
2. If so, what is the known (time/space) complexity of this problem?
From a browse on wikipedia it looks like knot equivalence is decidable. If I'm reading [1] correctly however, it looks like strong bounds on the complexity of knot equivalence in general haven't been found.
[1] http://arxiv.org/abs/math/9712269
[1] http://arxiv.org/abs/math/9712269
Good questions! But remember this is a theory that I am developing along as we go with a bunch of kids. We may end up with a lot of open problems.
For (1) I think the answer is 'yes' because the knots are finite structures. For (2) I don't know.
For (1) I think the answer is 'yes' because the knots are finite structures. For (2) I don't know.
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.
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.
Why is the left-most portion in the picture: http://researchblogs.cs.bham.ac.uk/thelablunch/files/2015/05...
I x C: (3,3) and not I x C: (1,3) ?
Similarily the right-most one should be C* x I: (3,1) no?
I x C: (3,3) and not I x C: (1,3) ?
Similarily the right-most one should be C* x I: (3,1) no?
Fixed, thanks!
Eight year olds is a bit of an overstretch, after reading the notation and descriptions...
You need to read parts I and II, and in my personal experience, 7 and 8 year olds are capable of far more than people expect, provided it's presented in a way that allows them to explore. They come up with some amazingly sophisticated ideas, and the notation is easily within their grasp.
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.
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.
Agreed, they're probably too old for this. Maybe six year olds would still have the fluid intelligence for this sort of abstract mathematics.
[deleted]
Not loading for me - has it collapsed under the load already?
Unlikely. It doesn't seem to be a popular link, as nobody is commenting on it.
The text is still available under Google's cache:
https://webcache.googleusercontent.com/search?q=cache:ovysh3...
The text is still available under Google's cache:
https://webcache.googleusercontent.com/search?q=cache:ovysh3...
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.
Thanks for the cache link.
Edit: it seems to be back.
Anybody got the formula for tying a tie?
http://www.amazon.com/The-85-Ways-Tie-Aesthetics/dp/18411556...
He connects his model of neckties to statistaical mechanics and invents a few new styles.
http://www.tcm.phy.cam.ac.uk/~tmf20/85ways.shtml https://en.wikipedia.org/wiki/The_85_Ways_to_Tie_a_Tie