2023/06/23 by Southgate, Jack
#52C25 #Combinatorics (math.CO) #FOS: Mathematics
paper · doi:10.48550/arxiv.2306.13560
Maxwell introduced a necessary minimum number of edges in terms of the number of vertices required for a graph to yield a Euclidean rigid generic framework in ℝ3, this count was generalised to ℝd, for all d≥ 1. In this paper, we give the analogous minimum number of k-simplices, for all 0≤ k≤ d, required for a pure d-dimensional simplicial complex to yield a volume rigid generic framework in ℝd, for all d≥ 1. In order to do so, we prove some basic facts about the volume rigidity matroid and use exterior algebraic shifting, a recently added tool to the study of volume rigidity. We later prove a volume rigidity Vertex Removal Lemma and use our count to strengthen the statement.