On Sunday 24 October 2010 17:03:02 Allan Wechsler wrote:
OK, having put my foot in my mouth earlier in this conversation with a false alarm, I have realized that I still have a spare foot.
Having inspected the illustration carefully, I agree that the curve is not contractible. But I do not understand the claim that removing any of the impediments allows the curve to contract; in fact, I think the claim is wrong.
The leftmost of the three punctures in the illustration is outside the curve. The other two points are inside the curve. Surely removing an impediment outside the curve can have no effect on contractibility. And removing one of the two impediments inside the curve leaves the other one to prevent contraction.
I agree. That picture is, unless I've goofed, a mere obfuscation of this one (note: monospaced font required): +-----. ,-----+ | \ / | | O \ O / O | | `-------' | +-------------------------+ where the three holes are, in order, the middle, left and right ones from the original picture. I suppose I should show my working. Let "A,B C,D E,F" represent a diagram with three holes, of which the leftmost has A lines above and B below, etc. (They are to be joined up in the obvious way.) Then we have: original diagram: 8,8 13,3 5,5 move middle hole to left: 3,3 8,2 5,5 move middle hole to right:3,3 5,1 2,2 move right hole to left: 2,2 1,3 1,1 move middle hole to left: 1,1 0,2 1,1 QED. -- g