Rigidity of symmetric frameworks in normed spaces

TL;DR

This paper establishes a combinatorial rigidity theory for symmetric frameworks in various normed spaces, including new characterisations for specific symmetries and spaces, expanding understanding of rigidity beyond Euclidean settings.

Contribution

It introduces a new combinatorial framework for symmetric rigidity in normed spaces, including Maxwell-type counts and a Henneberg-type construction for gain-tight graphs.

Findings

Matroidal Maxwell-type sparsity counts identified for rotational symmetry in normed spaces.

Complete combinatorial characterisations for half-turn rotation in $oldsymbol{ ext{l}^1}$ and $oldsymbol{ ext{l}^ ext{infty}}$-planes.

Development of a new inductive construction for gain-tight graphs.

Abstract

We develop a combinatorial rigidity theory for symmetric bar-joint frameworks in a general finite dimensional normed space. In the case of rotational symmetry, matroidal Maxwell-type sparsity counts are identified for a large class of $d$-dimensional normed spaces (including all $\ell^p$ spaces with $p\not=2$). Complete combinatorial characterisations are obtained for half-turn rotation in the $\ell^1$ and $\ell^\infty$-plane. As a key tool, a new Henneberg-type inductive construction is developed for the matroidal class of $(2,2,0)$-gain-tight graphs.