On Sep 11, 2010, at 11:13 PM, James Propp wrote:

Let me be more concrete about this, and phrase this as a puzzle: What is the topology of the set of triples of unit complex numbers (u,v,w) satisfying u+v+w = some fixed complex number z?

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'm using the language Conway advocates, double torus rather than 2-holed torus or surface of genus 2, because people easily get confused about punctures or disks removed, especially when you have things like a 3-holed twice-punctured torus]. It's independent of the order of {a,b,c,d}: there are maps between configuration spaces obtained by rotating two adjacent bars about the midpoint of the line joining their endpoints. You can treat the line between the pair of endpoints as a bar, and mod out by the group of rigid motions. One way to compute the configuration space is to first remove the pin in the middle. Each side gives you a torus. The missing pin defines a map of each torus to the plane. The configuration space for the original linkage is the fiber product of the two tori over the map to the plane. The image of each torus is an annulus (or a disk, in degenerate cases like the one you've named); for most points on the first torus that map to the intersection of the images, there are exactly two points on the other half, but along the boundary of the second annulus, the two become one, so. With this picture, it's easy to work out the configuration space. If the intersection of images is a disk, then the configuration space is a sphere. I won't go through all the stages, but the most complicated topology (the quadruple torus) is where the annuli have central holes narrower than the annuli, and the image is topologically a disk minus two holes. For most paths through the parameter space, the evolution follows level sets of a Morse function. If just one of the coeffients is changing, you can think of it a s level sets of a function on the configuration space when you remove that bar. For the partiuclar case at hand though, the singularities are degenerate, corresponding to points where the norm of the constant is an odd integer. At one time, at Princeton, I had a computer program that showed how these surfaces evolve as you change distances between anchor points and bar lengths. These days computers are fast enough that it's easy to do something like that with Mathematica, using the functions Manipulate together with ContourPlot3D Over the years, people have studied these and related examples quite a lot. Kevin Walker, as an undergraduate at Princeton, wrote a senior thesis extending some of the above to higher dimensions, using more bars in a row. I proved the relatively easy theorem that every manifold can be realized as a component of the configuration space for some linkage---it was in response to something Atiyah said at lunch at the Institute for Advanced Study, that configuration spaces always seemed to end up being spheres and toruses. He was wrong. There's also a bit about linkages in the 1989 Scientific American article I wrote with Jeff Weeks. Bill