Re: [math-fun] simple curves on punctured planes
Fred wrote: << On 10/24/10, Dan Asimov <dasimov@earthlink.net> wrote: << [Most of this was written before several posts that explain much of what I've described below, but for clarity I'm leaving this unedited.] Let C be a simple closed curve in R^3 - {p,q,r}, where #{p,q,r} = 3.
Should read " R^2 - {p,q,r} "?
Yes, indeed. Thank you for pointing that out. << ... In the case of the simple closed curve of the painting, it's fun to let the exterior "ooze into" the curve, by shading that in. What remains is a very skinny and folded snake with one eye at each end (namely, the two punctures in its interior). Once this becomes clear, it's easy to imagine how a sequence of Dehn twists might result in unfolding the snake. The inverse of the corresponding word is, I'm guessing, what was painted on the wall. (But the resolution of the photo is too low for me to be able to read that word.)
a b c' b' a' b' a b c b' a' b c' b' a' b' a b c' b' a' b a b c b' --- the last "a" being a trifle dubious. WFL
Wow, when I magnify that picture on my computer I can't make head or tail of those letters. But if there are three of them, it's clear that my guess (that it was a word in the usual two Dehn twists that generate that group) was wrong. --Dan Those who sleep faster are more rested sooner.
I'll hazard a guess that the curve in the picture is supposed to be the homotopic sum of loops around the individual punctures, and that the Word is the formula for the curve. Perhaps A is a simple counterclockwise loop around the first puncture, etc. If so, the transciption has a couple of problems: For the winding number of the curve to be 0 around each puncture, the total exponent of A should be 0 (it is), and ditto for B and C. (Nope: There are extra B' and C'.) IIRC, a contractable loop would simplify to Identity when A,B,C are interpreted in the (non-commutative) free group on three generators. I'm uneasy about this interpretation, since the rules for homotopy of curves allow the curve to cross itself, while the puzzle at hand forbids this. Rich --------------- Quoting Dan Asimov <dasimov@earthlink.net>:
Fred wrote:
<< On 10/24/10, Dan Asimov <dasimov@earthlink.net> wrote: << [Most of this was written before several posts that explain much of what I've described below, but for clarity I'm leaving this unedited.]
Let C be a simple closed curve in R^3 - {p,q,r}, where #{p,q,r} = 3.
Should read " R^2 - {p,q,r} "?
Yes, indeed. Thank you for pointing that out.
<< ... In the case of the simple closed curve of the painting, it's fun to let the exterior "ooze into" the curve, by shading that in. What remains is a very skinny and folded snake with one eye at each end (namely, the two punctures in its interior). Once this becomes clear, it's easy to imagine how a sequence of Dehn twists might result in unfolding the snake. The inverse of the corresponding word is, I'm guessing, what was painted on the wall. (But the resolution of the photo is too low for me to be able to read that word.)
a b c' b' a' b' a b c b' a' b c' b' a' b' a b c' b' a' b a b c b'
--- the last "a" being a trifle dubious. WFL
Wow, when I magnify that picture on my computer I can't make head or tail of those letters.
But if there are three of them, it's clear that my guess (that it was a word in the usual two Dehn twists that generate that group) was wrong.
--Dan
Those who sleep faster are more rested sooner.
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On the same page as the photograph there occurs the following explanation, upon which nobody has previously remarked. << Later Thurston wrote to add: “The letters refer to a word in the free group on three generators, which is the fundamental group of the plane minus the 3 points. If you imagine 3 ‘branch cuts’ going vertically from the three spots, and label them a, b, and c, then as you trace out the word starting from the left inside (I believe) it will trace out the given word, where a
[Previous posting inadvertantly truncated by dodgy PDF character code.] On the same page as the photograph there occurs the following explanation, upon which nobody has previously remarked. << Later Thurston wrote to add: “The letters refer to a word in the free group on three generators, which is the fundamental group of the plane minus the 3 points. If you imagine 3 ‘branch cuts’ going vertically from the three spots, and label them a, b, and c, then as you trace out the word starting from the left inside (I believe) it will trace out the given word, where a' designates a^(-1), etc. >> Apparently ‘branch cuts’ run upwards from 'spots' (puncture points); generators a,b,c represent crossing a cut from left to right; the curve runs clockwise from its (inside left) starting point. With this interpretation, the transcribed word is correct: a b c' b' a' b' a b c b' a' b c' b' a' b' a b c' b' a' b a b c b' [When all else fails, read the instructions. It sometimes works. Eventually.] The subsequent paragraph attempts motivate the formula in terms of what is called a 'braiding' construction. I am unable to cast more light on this at present, at least until re-reading said paragraph several times more, and possibly resorting to the assistance of pencil and paper. Fred Lunnon On 10/28/10, rcs@xmission.com <rcs@xmission.com> wrote:
I'll hazard a guess that the curve in the picture is supposed to be the homotopic sum of loops around the individual punctures, and that the Word is the formula for the curve. Perhaps A is a simple counterclockwise loop around the first puncture, etc. If so, the transciption has a couple of problems: For the winding number of the curve to be 0 around each puncture, the total exponent of A should be 0 (it is), and ditto for B and C. (Nope: There are extra B' and C'.) IIRC, a contractable loop would simplify to Identity when A,B,C are interpreted in the (non-commutative) free group on three generators. I'm uneasy about this interpretation, since the rules for homotopy of curves allow the curve to cross itself, while the puzzle at hand forbids this.
Rich
--------------- Quoting Dan Asimov <dasimov@earthlink.net>:
Fred wrote:
<< On 10/24/10, Dan Asimov <dasimov@earthlink.net> wrote: << [Most of this was written before several posts that explain much of what I've described below, but for clarity I'm leaving this unedited.]
Let C be a simple closed curve in R^3 - {p,q,r}, where #{p,q,r} = 3.
Should read " R^2 - {p,q,r} "?
Yes, indeed. Thank you for pointing that out.
<< ... In the case of the simple closed curve of the painting, it's fun to let the exterior "ooze into" the curve, by shading that in. What remains is a very skinny and folded snake with one eye at each end (namely, the two punctures in its interior). Once this becomes clear, it's easy to imagine how a sequence of Dehn twists might result in unfolding the snake. The inverse of the corresponding word is, I'm guessing, what was painted on the wall. (But the resolution of the photo is too low for me to be able to read that word.)
a b c' b' a' b' a b c b' a' b c' b' a' b' a b c' b' a' b a b c b'
--- the last "a" being a trifle dubious. WFL
Wow, when I magnify that picture on my computer I can't make head or tail of those letters.
But if there are three of them, it's clear that my guess (that it was a word in the usual two Dehn twists that generate that group) was wrong.
--Dan
Those who sleep faster are more rested sooner.
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
The subsequent paragraph seems no motivation after all: we know the 'how', but remain in the dark about the 'why'. In particular, the relation between fundamental group and cut-line-crossing interpretation of a,b,c is not further explained. "Start with ... a circle enclosing say the right two" --- the left-hand puncture lies also outside the final curve, but middle and right have been exchanged during 'braiding'. "you'll get the given curve after a few passes" --- 5 passes, to be precise; the permuted punctures and crossing counts proceed thus: L M R, M L R, M R L, R M L, R L M, L R M; 0 1 1, 1 2 1, 1 3 2, 3 5 2, 3 8 5, 8 13 5. WFL On 10/28/10, Fred lunnon <fred.lunnon@gmail.com> wrote:
... On the same page as the photograph there occurs the following explanation, upon which nobody has previously remarked.
<< Later Thurston wrote to add: “The letters refer to a word in the free group on three generators, which is the fundamental group of the plane minus the 3 points. If you imagine 3 ‘branch cuts’ going vertically from the three spots, and label them a, b, and c, then as you trace out the word starting from the left inside (I believe) it will trace out the given word, where a' designates a^(-1), etc. >>
Apparently ‘branch cuts’ run upwards from 'spots' (puncture points); generators a,b,c represent crossing a cut from left to right; the curve runs clockwise from its (inside left) starting point. With this interpretation, the transcribed word is correct: a b c' b' a' b' a b c b' a' b c' b' a' b' a b c' b' a' b a b c b' [When all else fails, read the instructions. It sometimes works. Eventually.]
The subsequent paragraph attempts motivate the formula in terms of what is called a 'braiding' construction. I am unable to cast more light on this at present, at least until re-reading said paragraph several times more, and possibly resorting to the assistance of pencil and paper.
Fred Lunnon
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