

Inventing an algebraic knot theory for eight year olds (III) - theaeolist
http://researchblogs.cs.bham.ac.uk/thelablunch/2015/05/inventing-an-algebraic-knot-theory-for-eight-year-olds-iii/

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mrcactu5
While at Cambridge, Thomas Fink wrote "The 85 Ways to Tie a Tie: The Science
and Aesthetics of Tie Knots" (2001)

[http://www.amazon.com/The-85-Ways-Tie-
Aesthetics/dp/18411556...](http://www.amazon.com/The-85-Ways-Tie-
Aesthetics/dp/1841155683)

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](http://www.tcm.phy.cam.ac.uk/~tmf20/85ways.shtml)
[https://en.wikipedia.org/wiki/The_85_Ways_to_Tie_a_Tie](https://en.wikipedia.org/wiki/The_85_Ways_to_Tie_a_Tie)

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oldbuzzard
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](http://en.wikipedia.org/wiki/Christopher_Zeeman)
[2][http://richannel.org/christmas-
lectures/1978/1978-christophe...](http://richannel.org/christmas-
lectures/1978/1978-christopher-zeeman)

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j2kun
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?

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danghica
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.

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j2kun
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.

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ul5255
Why is the left-most portion in the picture:
[http://researchblogs.cs.bham.ac.uk/thelablunch/files/2015/05...](http://researchblogs.cs.bham.ac.uk/thelablunch/files/2015/05/overhandalg.png)

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?

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danghica
Fixed, thanks!

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pfortuny
Eight year olds is a bit of an overstretch, after reading the notation and
descriptions...

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ColinWright
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.

~~~
zeidrich
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.

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octatoan
Is the unknot (as in, a loop) representable in this notation?

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drdeca
Would it be C◦C* ?

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danghica
Yes!

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ColinWright
Not loading for me - has it collapsed under the load already?

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TuringTest
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...](https://webcache.googleusercontent.com/search?q=cache:ovysh3wHivMJ:researchblogs.cs.bham.ac.uk/thelablunch/2015/05/inventing-
an-algebraic-knot-theory-for-eight-year-olds-iii/+&cd=1&hl=es&ct=clnk)

~~~
ColinWright
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._

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amelius
Anybody got the formula for tying a tie?

