2024/01/17 by Zoe Cooperband, Cooperband, Zoe, Miguel López +3 · 1 citation
Biochemistry, Genetics and Molecular Biology · Computer Science · Mathematics · #Algebraic Topology (math.AT) #Cellular Mechanics and Interactions #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Metric Geometry (math.MG) #Topological and Geometric Data Analysis
paper · pdf · doi:10.48550/arxiv.2401.09392
openalex publication_date 2024/01/17 · openalex created_date 2024/01/19 · openalex updated_date 2026/07/28
This work extends the theory of reciprocal diagrams in graphic statics to frameworks that are invariant under finite group actions by utilizing the homology and representation theory of cellular cosheaves, recent tools from applied algebraic topology. By introducing the structure of an equivariant cellular cosheaf, we prove that pairs of self-stresses and reciprocal diagrams of symmetric frameworks are classified by the irreducible representations of the underlying group. We further derive the symmetry-aligned Euler characteristics of a finite dimensional equivariant chain complex, which for the force cosheaf yields a new formulation of the symmetry-adapted Maxwell counting rule for detecting symmetric self-stresses and kinematic degrees of freedom in frameworks. A freely available program is used to implement the relevant cosheaf homologies and illustrate the theory with examples.