Calligraphs and sphere realizations
Matteo Gallet, Georg Grasegger, Niels Lubbes, Josef Schicho

TL;DR
This paper presents a recursive method for counting sphere realizations of minimally rigid graphs, utilizing moduli spaces and graph splitting into calligraphs, with results expressed through triples and quadratic forms.
Contribution
It introduces a novel recursive framework combining moduli space theory and graph splitting into calligraphs for counting realizations on the sphere.
Findings
Counts realizations via triples associated with calligraphs.
Splitting graphs into calligraphs simplifies enumeration.
Realizations correspond to intersections of curves on a blown-up sphere.
Abstract
We introduce a recursive procedure for computing the number of realizations of a minimally rigid graph on the sphere up to rotations. We accomplish this by combining two ingredients. The first is a framework that allows us to think of such realizations as of elements of a moduli space of stable rational curves with marked points. The second is the idea of splitting a minimally rigid graph into two subgraphs, called calligraphs, that admit one degree of freedom and that share only a single edge and a further vertex. This idea has been recently employed for realizations of graphs in the plane up to isometries. The key result is that we can associate to a calligraph a triple of natural numbers with a special property: whenever a minimally rigid graph is split into two calligraphs, the number of realizations of the former equals the product of the two triples of the latter, where this…
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Taxonomy
TopicsAdvanced Materials and Mechanics · Computational Geometry and Mesh Generation · Geometric and Algebraic Topology
