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