Advanced Geometry Summary

Monday - Friday, July 3 - 7, 2006

With suddenly reduced numbers (our DIPD associates have bid us farewell) we continued to learn and explore the notions of Knot theory presented in the Undergraduate lecture. Among the topics covered this week were tangles, Conway's notation, orientable surfaces, and genus. We have been making an attempt to keep current in the Knot Book as well. Our "work" as it were, has been largely focused on gaining understanding and insights into the material presented. This week we slowly made the transition to thinking about how we might present some of this material as teachers. (We are morphing ourselves from students to teachers, though with some difficulty and reluctance).

Monday, as students, we were somewhat overwhelmed with some notation involving tangles. We did, however, learn of a free piece of software called KnotPlot. It is sort of a Sketchpad for knots. We have a lot to learn about how to use it. Monday as teachers we experienced some mild panic about what it was that we could do for a "product". Our first inspiration began with the observation that the single edge of a mobius strip formed a trefoil knot. We began to explore the possibility of cutting mobius strips into halves and thirds and see what sort of knots were created. After several objects were created, we tried to make a table of what we had created. We quickly came to the conclusion that this was not leading us to a product.

Wednesday, as students, we untangled the mystery of tangles and Conway's notation. We experimented with two ropes and became competent in constructing tangles. we also learned that equivalent tangles have equivalent continued fractions. Wednesday, as teachers, we contemplated how we might use tangles and/or continued fractions. Our panic intensified mildly and we agreed to look at years past to see what some "products" might be. Meanwhile we practiced making tangles and then unmaking them by following the Conway notation in reverse for knots believed to be inverses.

Thursday, as students, we learned about algebraic tangles, orientable surfaces and Siefert Spheres. In particular we learned a fairly simple algorithm to construct a surface with any particular knot as a boundary! Furthermore, every knot has such a surface! Thursday, as teachers, our heads were spinning and our panic was beginning to look like a tropical storm heading for warm water. Henri saved us with a nicely crafted problem. As students we went deeper into the ideas of surfaces and genus. In particular we learned that the genus number of composed knots is equal to the sum of the genus numbers (cool proof)!. We also learned that every knot has a corresponding non-orientable surface.

As teachers we had an interesting discussion about what would constitute mathematical content. (What is math? In what context do you put the rather abstract material that we are playing with? What is the essence of "that which can be learned" through studying knot theory? Good questions.) There are some great mathematical ideas that can be teased out including invariance, abstraction, generalization, classification,... We concluded with the realization that we had two more meeting before we reporting out. We will investigate the ideas presented by Henri and look to se if we can construct a lesson that draws out the connection between surfaces and knots.

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