Bounds on Embeddings of Triangulations of Spheres

2023/01/11 by Southgate, Jack
#52C25 #Combinatorics (math.CO) #FOS: Mathematics #Metric Geometry (math.MG)

paper · doi:10.48550/arxiv.2301.04394

Abstract

Borcea and Streinu showed that the upper bound of the number of congruence classes of a minimally d-volume rigid (d+1)-uniform hypergraph on n vertices in ℝ^d increases exponentially in n and d. We show that this result also holds for triangulations of \mathbbS² in ℝ², and then find a geometrically motivated bound linear in n for bipyramids. By the methods used to deduce this bound, we show that, in general, global d-volume rigidity in ℝ^d is not a generic property of a (d+1)-uniform hypergraph.

Related

• Volume rigidity and algebraic shifting
• Braced triangulations and rigidity
• New upper bounds for the number of embeddings of minimally rigid graphs
• On Spheres with k Points Inside
• Rigidity with few locations