An Analogy in Geometric Homology: Rigidity and Cofactors on Geometric Graphs
From recent work in two areas of discrete applied geometry, we abstract a common pattern of families of geometric homologies for graphs realized in projective d-space (for static rigidity) or in the projective plane (for bivariate splines). Using distinct algebraic constructions for the local coefficients of the chain complexes (exterior algebra for statics and symmetric algebra for splines), we explore the underlying analogy between the two theories. The analogy starts with isomorphisms for the lowest two members of the families, moves to a conjectured isomorphism for generic realizations in the third family members (static rigidity in 3-space and C 2 1 -cofactors for bivariate splines) and a proven difference with a conjectured injection for all higher family members.
The conjectured isomorphism for static rigidity in 3-space and C 2 1 -cofactors for bivariate splines illustrates fundamental problems in structural rigidity, in polynomial-time algorithms for graph properties which are random polynomial, and in defining ‘freest’ matroids for submodular functions.
Key words and phrasesgeometric homology first-order rigidity static rigidity multivariate splines cofactors exterior algebra symmetric algebra matroids on graphs
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