Configuration Spaces of Linkages

This paper studies the configuration space of all possible positions of a linkage in R n. For example, it shows that for every compact algebraic set, there is a linkage whose configuration space is analytically isomorphic to a finite number of copies of the algebraic set. If flexible edges are allowed, any compact set given by polynomial equalities and inequalities is the configuration space of a linkage. This paper also studies semiconfiguration spaces of all possible positions of a finite number of points on a linkage. For example any compact semialgebraic set is such a semiconfiguration space. Loosely speaking, a linkage is an ideal mechanical device consisting of a bunch of stiff rods sometimes attached at their ends by rotating joints. A realization of a linkage in R n is some way of placing this linkage in R n. The configuration space for a linkage is the space of all such realizations, which can be determined by looking at all possible positions of the ends of all the rods. A semiconfiguration space of a linkage is the space of all possible positions of only some of ends of the rods, we ignore the other ends. For example, what figure does a particular point on the linkage trace out? In this paper we will give characterizations of configuration spaces and semicon-figuration spaces of linkages as well as of cabled linkages, for n ≥ 3. (In a cabled linkage you also can attach flexible cables between rods.) The characterizations of these spaces for n = 2, planar linkages, was studied in [3], [1], and [2]. The results for n ≥ 3 turn out to be analogous to the n = 2 results, but in a couple of places the proofs are different. In particular, we can completely characterize semiconfig-uration spaces, we can characterize configuration spaces of cabled linkages up to analytic isomorphism, and we can characterize configuration spaces of linkages up to analytically trivial finite covers. Let us now define linkages more precisely. Suppose L is a finite one dimensional simplicial complex, in other words, a finite set V(L) of vertices and a finite set E(L) of edges between certain pairs of vertices. An abstract linkage is a finite one dimensional simplicial complex L with a mapping ℓ : E(L) → (0, ∞). You should think of ℓ as giving the length of each edge. A realization of an abstract linkage (L, …