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October 13, 2025Open Access

On the k-volume rigidity of a simplicial complex in Rᵈ

Key Points

The study establishes that the k-volume rigidity matroid has the same rank as the classical d-rigidity matroid for 2 ≤ k ≤ d-1.
This research highlights a difference in behavior when k equals d, contrasting with findings by Lubetzky and Peled.
A conjectured characterization relates the bases of the k-volume matroid to d-rigidity and a combinatorial Hall condition.
The findings define a new framework in understanding generic rigidity in higher-dimensional simplicial complexes.

Abstract

We define a generic rigidity matroid for k-volumes of a simplicial complex in Rᵈ, and prove that for 2 k d-1 it has the same rank as the classical generic d-rigidity matroid on the same vertex set (namely, the case k=1). This is in contrast with the k=d case, previously studied by Lubetzky and Peled, which presents a different behavior. We conjecture a characterization for the bases of this matroid in terms of d-rigidity of the 1-skeleton of the complex and a combinatorial Hall condition on incidences of edges in k-faces.

On the k-volume rigidity of a simplicial complex in Rᵈ

Key Points

Abstract

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