Frameworks with forced symmetry II: Orientation-preserving crystallographic groups

TL;DR

This paper characterizes minimally rigid planar frameworks with orientation-preserving crystallographic symmetry, extending previous methods to include groups generated by translations and rotations, using matroid theory and submodular functions.

Contribution

It introduces a combinatorial characterization for such frameworks and extends prior work to more complex symmetry groups using novel matroid-based methods.

Findings

Provides a new combinatorial characterization of rigid frameworks with crystallographic symmetry.

Extends previous results to groups generated by translations and rotations.

Uses a new family of matroids and submodular functions to prove main theorems.

Abstract

We give a combinatorial characterization of minimally rigid planar frameworks with orientation-preserving crystallographic symmetry, under the constraint of forced symmetry. The main theorems are proved by extending the methods of the first paper in this sequence from groups generated by a single rotation to groups generated by translations and rotations. The proofs make use of a new family of matroids defined on crystallographic groups and associated submodular functions.