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

TL;DR

This paper explores an A4-symmetric tensegrity structure parametrized by an elliptic curve, revealing a unique stable configuration and a persistent Hopf link topology across stress variations.

Contribution

It identifies a specific elliptic curve governing the configurations and highlights the topological invariance of the Hopf link in the tensegrity.

Findings

Only one rational point corresponds to a stable configuration.

Configurations form a one-parameter family on the elliptic curve 30a2.

The Hopf link structure persists across the entire stress parameter interval.

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.