School project in knot theory

Question

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

Thanks

Answer 1

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.

Answer 2

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

Answer 3

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.