The rigidity of periodic frameworks as graphs on a fixed torus

TL;DR

This paper investigates the infinitesimal rigidity of periodic frameworks modeled as graphs on a fixed torus, establishing fundamental definitions, results, and necessary conditions for their rigidity in a fixed-dimensional setting.

Contribution

It introduces a gain graph framework for periodic structures and derives new necessary conditions for their generic rigidity on a fixed torus.

Findings

Fundamental definitions for periodic frameworks on a torus.

Results on infinitesimal rigidity of these frameworks.

Necessary conditions for generic rigidity in fixed dimensions.

Abstract

We define periodic frameworks as graphs on the torus, using the language of gain graphs. We present some fundamental definitions and results about the infinitesimal rigidity of graphs on a torus of fixed size and shape, and find necessary conditions for the generic rigidity of pe- riodic frameworks on a d-dimensional fixed torus.