[math-fun] simple curves on punctured planes
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
I have just drawn one that is so simple it must be the simplest. I guess I could run it over to your house if you want to stay up a few more minutes. I don't know the proof for two holes, but it sounds like a corollary of some fixed-point theorem. On Fri, Oct 22, 2010 at 10:40 PM, James Propp <jpropp@cs.uml.edu> 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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Hi Jim, This might be the painting you mentioned: http://math.berkeley.edu/publications/newsletter/2002/specialinterest.html A larger, black-and-white photo is available at the bottom of this PDF file: http://www.math.tamu.edu/~manshel/Riverside/m10B/Jordan-curve-diagram.pdf The article says it was painted in the fall of 1971 by Dennis Sullivan and Bill Thurston. Tom
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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Aaaaaand the only reason I was able to come up with an "example" so fast is that I completely missed the requirement that the curve be "simple", that is, non-self-crossing. So never mind. On Fri, Oct 22, 2010 at 11:09 PM, Tom Karzes <karzes@sonic.net> wrote:
Hi Jim,
This might be the painting you mentioned:
http://math.berkeley.edu/publications/newsletter/2002/specialinterest.html
A larger, black-and-white photo is available at the bottom of this PDF file:
The article says it was painted in the fall of 1971 by Dennis Sullivan and Bill Thurston.
Tom
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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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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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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I withdraw the remark about typos; it's just that the style is terse, and requires several passes to parse correctly ... WFL On 10/23/10, Fred lunnon <fred.lunnon@gmail.com> wrote:
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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Sorry for the terse style --- I should have taken more time to explain it. Bill On Oct 23, 2010, at 8:41 PM, Fred lunnon wrote:
I withdraw the remark about typos; it's just that the style is terse, and requires several passes to parse correctly ... WFL
On 10/23/10, Fred lunnon <fred.lunnon@gmail.com> wrote:
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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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. On Sat, Oct 23, 2010 at 9:58 PM, Bill Thurston <wpthurston@gmail.com> wrote:
Sorry for the terse style --- I should have taken more time to explain it. Bill On Oct 23, 2010, at 8:41 PM, Fred lunnon wrote:
I withdraw the remark about typos; it's just that the style is terse, and requires several passes to parse correctly ... WFL
On 10/23/10, Fred lunnon <fred.lunnon@gmail.com> wrote:
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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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
participants (7)
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Allan Wechsler -
Bill Thurston -
Bill Thurston -
Fred lunnon -
Gareth McCaughan -
James Propp -
Tom Karzes