An Elliptic Curve Governing Hopf Linking in an $A_4$-Symmetric Tensegrity

Abstract

We study in detail an A4-symmetric tensegrity appearing in Connelly's catalog. The realizable configurations form a one-parameter family that can be parametrized by points on the elliptic curve with Cremona label 30a2. The curve has only twelve rational points, among which only one corresponds to a stable tensegrity configuration whose cable framework forms a cuboctahedron. From a topological viewpoint, however, the underlying pair of the strut triangles preserves a Hopf link structure throughout the entire interval 0 < \omega_1 < 1 of the stress parameter.