The material appears fascinating --- but beware unfortunate typos ... WFL On 10/23/10, Bill Thurston <wpt4@cornell.edu> wrote:
A simple curve in the plane is the boundary of a disk. If the disk has no points in its interior, the curve is contractible. The curve has winding number 1 or -1 about any point in the interior of the disk.
The point is that in the plane with 3 or more punctures, there can be very complicated simple curves, even though they automatically have small winding numbers (0, 1 or -1) about all the punctures.
Bill On Oct 22, 2010, at 10:40 PM, James Propp wrote:
What's the simplest example of a simple closed curve in the triply-punctured plane that has winding number 0 around each of the punctures but is not contractible? (Back when I was in grad school in Berkeley in the '80s, there was a painting of one such curve on the wall, along with the associated word in the fundamental group of the surface.) Also, what's the simplest way to prove that no analogous curve exists for the doubly-punctured plane?
Jim Propp
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