Spring 2008 courseTopic: Knot Theory
Instructor: Dan Margalit
Take a rope, tangle it up, glue the ends together, and you have what mathematicians call a knot. Peter Tait studied knots formally in the late nineteenth century, motivated by new ideas about matter and the aether developed by Lord Kelvin and James Clerk Maxwell.
One of the basic questions about knots is: how can you tell when two knots are the same? Or, if someone hands you a knotted rope, how can you tell if it is possible to completely disentangle the rope so it looks like the letter "O"? You can explain this problem to a child, but it can be a very difficult problem to solve!
If your rope really is knotted, what is the minimum number of times you would have to pass the knot through itself in order to unknot it?
What is the smallest number of sticks needed to make a tinker toy model of a certain knot?
Is it possible to list all knots in such a way that you never repeat the same one twice?
How do you multiply knots? How can you factor a knot into prime knots?
What is the higher dimensional version of a knot?
Starting with the rigorous mathematical definition of a knot, we will develop increasingly more advanced techniques designed to answer these questions and many more.
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