Inductive constructions for frameworks on a two-dimensional fixed torus

TL;DR

This paper establishes necessary and sufficient conditions for the minimal rigidity of frameworks on a fixed two-dimensional torus and introduces inductive constructions for building such frameworks from smaller components.

Contribution

It extends classical rigidity results to periodic frameworks on a fixed torus using inductive constructions and gain graphs.

Findings

Characterization of generic minimal rigidity on a fixed torus

Development of inductive construction methods for periodic frameworks

Extension of Laman and Henneberg results to torus frameworks

Abstract

An infinite periodic framework in the plane can be represented as a framework on a torus, using a $\mathbb Z^2$-labelled gain graph. We find necessary and sufficient conditions for the generic minimal rigidity of frameworks on the two-dimensional fixed torus $\mathcal T_0^2$. It is also shown that every minimally rigid periodic orbit framework on $\mathcal T_0^2$ can be constructed from smaller frameworks through a series of inductive constructions. These are fixed torus adapted versions of the results of Laman and Henneberg respectively for finite frameworks in the plane. The proofs involve the development of inductive constructions for $\mathbb Z^2$-labelled graphs.