Planar Linkages and Algebraic Sets

An abstract linkage is a one dimensional simplicial complex L with a positive number `(vw) assigned to each edge vw. A planar realization of an abstract linkage (L, `) is a mapping φ from the vertices of L to C so that |φ(v) − φ(w)| = `(vw) for all edges vw. We will investigate the topology of the space of planar realizations of linkages. You may think of an abstract linkage as an ideal mechanical device consisting of a bunch of stiff rods (the edges) with length given by ` and sometimes attached at their ends by rotating joints. A planar realization is some way of placing this linkage in the plane. If L is a finite graph, we let V(L) denote the set of vertices of L and let E(L) denote the set of edges of L. We will often wish to fix some of the vertices of a linkage whenever we take a planar realization. So we say that a linkage L is a foursome (L, `, V, μ) where (L, `) is an abstract linkage, V ⊂ V(L) is a subset of its vertices, and μ : V → C. So V is the set of fixed vertices and μ tells where to fix them. The configuration space of realizations is defined by: