Given a linkage \((G,l)\), i.e. a graph \(G\) with Euclidean edge lengths \(l: E(G) \rightarrow R\), its configuration space for dimension \(d\), \(\Phi^d(G,l)\) is the set \(\{ p \in R^d: ||p_i - p_j||^2 = l_{ij}, ij \in E(G)\}\) modulo Euclidean isometries in \(SE(d)\). For a non-edge set \(F\), the Cayley configuration space \(\Omega^d_F(G,l)\) is the set of attainable length vectors for \(F\), i.e. \(<||p_i-p_j||^2: ij \in F, p \in \Phi^d(G,l)>\). For certain non-edge sets \(F\), the configuration space \(\Phi^d(G,l)\) is a branched covering space for which the Cayley configuration space gives a convex base space where the covering map and pre-image are efficiently computable.
This provides an extremely convenient representation of the configuration space \(\Phi^d(G,l)\) especially when it is of low intrinsic dimension living in high ambient dimension \((R^d)^n\). In contrast, a convex Cayley configuration space (base space) \(\Omega^d_F(G,l)\), if it exists, has the same ambient dimension as intrinsic dimension and is moreover convex. I will talk about a recent solution to a 15 year open problem characterizing the pairs \((G,f)\), where \(f\) is a single non-edge, for which for every \(l\), the \(\Omega{\le 3}_f(G,l)\) is convex, namely a single interval. The proof goes via a strong connection to the notion of \(d\)-flattenability of graphs, a minor-closed property.
