Bill Thurston writes:

The connected components of the more general configuration space X(a,b,c,d) = {triples of complex numbers {u,v,w} such that a*u + b*v + c*w + d = 0 can be a sphere, torus, double torus, triple torus or quadruple torus.

I assume Bill means triples of UNIT complex numbers (i.e., complex numbers of magnitude 1), since otherwise we'd get 6-2=4 degrees of freedom instead of 3-1=2. But I'm having trouble getting past the end of the first paragraph; I have a hard time picturing "rotating two adjacent bars about the midpoint of the line joining their endpoints" (though I'm sure it's a simple operation and that once I see what it is I'll have a hard time coming up with a better way to describe it!). Are there any pictures (or better yet animations) available on the web? For that matter, is the Thurston-Weeks article available anywhere online? In my previous email, I should have mentioned what happens to the set S_r = {(u,v,w): |u|=|v|=|w|=1, u+v+w=r} when r = 0: it has two S^1 components. And I'm pretty sure that this holds also for S_r when r is sufficiently close to 0. Jim Propp