Rote's Curious Identity on Self-Stresses: Independence of the Two Points for All Graphs, Existence at Generic Positions, and a Volume Interpretation

In Problem 8 of Oberwolfach Report 13/2015, G. Rote gives a self-stress \(\omega\) of a wheel in general position in the plane with \(\sum_{ij\in E}\omega_{ij}[a\,p_i\,p_j][b\,p_i\,p_j]\)\({}=1\) for all points \(a,b\), and asks whether other graphs with \(n\) vertices and \(2n-2\) edges carry such a self-stress (the triangular prism and \(K_{3,3}\), each with one added edge, are named) and what the identity means. We show that for every graph, every placement of its vertices and every self-stress, the left-hand side does not depend on \(a\) and \(b\). With the brackets replaced by determinants (twice the signed areas), as Rote, Santos and Streinu do, it equals \(N(\omega)\)\({}=\sum\omega_{ij}\det(p_i,p_j)^2\), and all these edge functions, with Rote's line-integral variant, represent one class in the cokernel of the rigidity map, the dual of the stress space. With a projective form of this identity we prove that every graph that is dependent in the generic rigidity matroid of the plane, in particular every graph with \(n\ge2\) vertices and \(2n-2\) edges, carries at generic positions a self-stress satisfying the identity; it is unique exactly when the generic stress space is one-dimensional, which for \(2n-2\) edges means that the graph is generically rigid. The two named graphs get exact certificates. This answers the first question at generic positions, which is how we read the problem's “general position”; “no three points collinear” does not suffice. For a non-crossing drawing, \(N(\omega)\) is six times the volume under the Maxwell–Cremona lifting of \(\omega\), and Rote's line integral is Green's formula for this volume. For the two named graphs the normalised stress is not a monomial in polygon areas, unlike the wheel; sums of products of areas occur, and a closed-formula reading of the first question stays open. The second question is open-ended, and whether these readings are the intended meaning is a judgment. We also correct three points of the printed statement. The case of four points is due to Rote, Santos and Streinu; our identities are elementary and may be folklore, and no printed answer to the questions was found in the sources searched. This is an unrefereed note.