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Can someone suggest an idea for a school project in knot-theory for a 13 year old?

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I appreciate the fact that you know that there is a subject called knot theory at such an young age. But please explain as to what background you have so that the experts here can guide you very nicely. Please explain them as to how much Math you know, and what exactly you expect. I am only telling this so that you receive more useful answers. Dont take me wrong. –  anonymous Dec 8 '10 at 15:07
    
Hi Chandru - this is actually for my cousin who is 13 years old. He has gone through the very basic concepts like invariants, unknot etc. We were looking to focus on some applications and then go deep into it rather than summarize different aspects of the theory. <br>One of the examples we looked at was about distinguishing DNA molecules using knot theory - maths.ed.ac.uk/~s0681349/DNAtalk.pdf –  Prateek Dec 8 '10 at 16:43

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You may wish to take a look at the readable and well-illustrated Why Knot?.

Something to consider about knots is how 'basic' they are -- children will often learn to construct simple knots before learning arithmetic. This demonstrates an ability to manipulate (and distinguish between classes of) closed curves in three dimensions, and amounts to an intuitive theory of topology. We can use this as a staging ground to ask more interesting and difficult questions about knots and their classification.

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Also by Colin Adams. –  Ryan Budney Dec 8 '10 at 18:33

Borromean rings? Can the same thing be done with more than three loops?

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There are some elementary ideas in Colin Adam's book: The Knot Book that could be adapted for investigation.

Another problem to think about would be the number of sides for regions that occur when the same knot (different drawings) is considered embedded in the plane as a 4-valent (degree 4) graph. The number of faces with k sides (knot drawings can have 1-gons, 2-gons, etc.) for such a drawing obeys a relationship that can be derived from Euler's polyhedral formula using the fact that the graph of a projection of a knot is 4-valent.

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