Rank characterization via rigidity-matroid-independent edge sets

Characterize the rank of the (k,d)-volume rigidity matroid of a (k-dimensional simplicial complex X on n vertices by proving that, for d and 1kd-1, it equals the maximum matching number of the bipartite incidence graph between the k-simplices of X and an edge set EX_1 that is independent in the standard d-rigidity matroid.

Background

The paper defines the (k,d)-volume rigidity matroid \mathcal{M}_{k,d}(X) from the generic Jacobian of the squared k-volume map on the k-simplices of a simplicial complex X. Via the chain rule, its rigidity matrix factors through the standard rigidity matrix of the 1-skeleton and a Jacobian describing the dependence of simplex volumes on squared edge lengths.

The authors conjecture that the rank is determined by optimizing the matching number of the incidence graph H_{E,X_k}, where E ranges over edge sets independent in the standard d-rigidity matroid. This would provide a general rank formula extending the paper's complete-complex rank theorem and relating volume rigidity to both geometric rigidity and a combinatorial Hall-type condition.

References

However, we propose the following conjecture, which gives a characterization for the rank of the $(k,d)$-volume rigidity matroid of a complex in terms of the standard $d$-rigidity matroid of its $1$-skeleton.

— On the $k$-volume rigidity of a simplicial complex in $\mathbb{R}^d$  (2503.01665 - Lew et al., 3 Mar 2025) in Section 4, Discussion, Conjecture 1 (labeled \ref{conj})