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This sketch represents an example of ghost symmetry, as the following observations demonstrate. First, there is a "constellation", consisting of 8 numbered points. The small unmarked point in the center represents the origin in R2. The three thick lines colored blue, green, and red and passing through the origin represent 1-dimensional subspaces. (There are an additional 5 unmarked points. These have a special significance as explained below.) Next, if one projects the set of points orthogonally to one of these three subspaces, then one obtains a constellation having two-fold symmetry. Thus, while the 2-dimensional constellation of 8 points has no apparent symmetry, there are three 1-dimensional projections which each have two-fold symmetry. The purpose of the thin lines is to assist in showing that the configuration has these three ghost symmetries. Notice that there are 8 thin lines of each color, each line has precisely one point of the constellation on that line, and each point lies on a line from each of these three families. One can think of these lines as the "lines of projection", as they are parallel to the kernels of the projection maps. The assembly of 8 points has ghost symmetries because this configuration of 24 lines has two-fold rotational symmetry. Associated to this constellation of points is a graph:
The 8 vertices of this graph correspond to the points of the constellation. One draws an edge colored by C between vertices V and W if projection onto the line colored by C yields a ghost symmetry which interchanges the images of V and W. For example, there is a blue edge between 2 and 6 because the blue lines passing through the points marked 2 and 6 lie at the same distance to the origin. Thus, projecting points 2 and 6 onto the blue subspace yields images which are the same distance to the origin. This graph is trivalent and 3-edge colored. Thus, every vertex has precisely 3 edges connecting to it, one of each color. There are 12 edges and each color yields a 1-factor (also known as a "perfect matching") of the graph. This type of graph is often called "Tait-colored". The Theorem. Given a constellation of points in the plane which has three ghost symmetries, one may always associate such a Tait-colored graph. However, the converse is not automatic. That is, given a Tait-colored graph representing a triple of permutations corresponding to three desired ghost symmetries, one cannot always construct a corresponding constellation. The prescribability theorem, stated below, completely characterizes these graphs. Before stating the main theorem, it is necessary to give some terminology. A "fixed-point-free (FPF) involution" on a set S is a permutation f of order 2 on S for which f(x)≠x for all x. The blue edges in the graph above correspond to the FPF involution (1,3)(2,6)(4,7)(5,8), and so on. In graph-theoretic terminology, an FPF involution coincides with a 1-factor. FPF involutions arise for the simple reason that a two-fold symmetry of a line is necessarily fixed-point free on the non-zero points on the line. One may now state: Theorem. Sketch of Proof. If one knows some things about convex polytopes, then one may see a similarity to Steinitz's theorem. This theorem states that a graph is the 1-skeleton of a convex 3-dimensional polytope if and only if it is simple, planar, and 3-connected. In fact, the proof of the theorem stated above is similar to several well-known proofs of Steinitz's theorem. One of the basic techniques in the proof is to use the fact that one may perturb such a configuration in an interesting way. Consider perturbing those five unmarked points lying in the the three subspaces colored blue, green and red. If one perturbs these slightly, then one will notice that some of the points and lines in the sketch move. For example, if one perturbs the point on the blue subspace corresponding to point 4, then the result is a perturbation of points 4, 5, 6, and 7 and the corresponding lines. Notice that points 4, 5, 6, and 7 are the vertices of the cycle in the graph. As it turns out, there is a similar type of perturbation for every cycle in the graph, and this allows one enough wiggle room in order to exploit 3-connectivity and wrestle out a proof by induction. The details are given below in a preprint.
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