The $d$-dimensional realisation number of a rigid graph

TL;DR

This paper introduces new tools for calculating the number of realisations of rigid graphs in any dimension, establishing lower bounds and proving conjectures related to graph operations and realisation counts.

Contribution

It provides novel methods linking subgraph inclusion to realisation number divisibility and offers lower bounds for realisation numbers under specific graph operations across all dimensions.

Findings

Every triangulated sphere with n vertices has at least 2^{n-4} realisations in 3D.

Proved that subgraph inclusion implies divisibility of realisation numbers.

Solved conjectures on how graph operations affect realisation counts.

Abstract

Determining the number of (complex) realisations of a rigid graph for a specific choice of edge lengths is a fundamental problem in discrete geometry. In this article we provide two new tools for determining realisation numbers in arbitrary dimensions: (i) we prove that subgraph inclusion translates to realisation number divisibility; and (ii) we provide lower bounds on realisation numbers under specific graph operations in all dimensions. We use these methods to prove that every triangulated sphere with $n$ vertices has at least $2^{n-4}$ edge-length equivalent realisations in 3-dimensions, extending a 2-dimensional result of Jackson and Owen in the case of planar graphs. Additionally, our tools solve a family of conjectures set by Grasegger regarding how 1-extensions, X-replacements, and V-replacements affect realisation numbers.