On the $k$-volume rigidity of a simplicial complex in $\mathbb{R}^d$
Alan Lew, Eran Nevo, Yuval Peled, Orit E. Raz
TL;DR
This work defines a generic $(k,d)$-volume rigidity matroid ${\cal M}_{k,d}(X)$ for a $k$-dimensional simplicial complex $X$ embedded in ${\mathbb R}^d$ and proves that for all $2\le k\le d-1$, its rank matches the rank of the classical generic $d$-rigidity matroid on the same vertex set, distinguishing this from the $k=d$ case. The main method combines a vertex-addition lemma with a Cayley–Menger-based Jacobian framework: ${\cal B}(X,{\bf p})={\cal C}(X,{\f d})\cdot R(G,{f p})$, enabling an inductive rank analysis via base-case invertibility of ${\cal C}$ (via incidence-matrix results). This yields the exact rank ${\rm rank}{\cal M}_{k,d}(\ abla_{n,k})=dn-{\binom{d+1}{2}}$ for the stated ranges of $(k,d,n)$, and the paper concludes with a conjectural combinatorial characterization of rank in terms of the $1$-skeleton’s rigidity and Hall-type conditions, plus discussion and connections to related rigidity matrices. Practical impact lies in linking higher-volume rigidity of simplicial complexes to classical rigidity theory, enabling unified rank predictions via purely combinatorial and geometric tools.
Abstract
We define a generic rigidity matroid for $k$-volumes of a simplicial complex in $\mathbb{R}^d$, and prove that for $2\leq k \leq d-1$ 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.