Calligraphs and sphere realizations

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 product is specified by a fixed quadratic form. These triples and quadratic form codify the fact that we express realizations as intersections of two curves on the blowup of a sphere along two pairs of complex conjugate points.