Related papers: Generic infinitesimal rigidity for rotational grou…

A bar-joint framework $(G,p)$ is the combination of a graph $G$ and a map $p$ assigning positions, in some space, to the vertices of $G$. The framework is rigid if every edge-length-preserving continuous motion of the vertices arises from…

A result due in its various parts to Hendrickson, Connelly, and Jackson and Jord\'an, provides a purely combinatorial characterisation of global rigidity for generic bar-joint frameworks in $\mathbb{R}^2$. The analogous conditions are known…

For a complex projective space the inertia group, the homotopy inertia group and the concordance inertia group are isomorphic. In complex dimension 4n+1, these groups are related to computations in stable cohomotopy. Using stable homotopy…

We present a rigorous study of framework rigidity in general finite dimensional normed spaces from the perspective of Lie group actions on smooth manifolds. As an application, we prove an extension of Asimow and Roth's 1978/9 result…

This article is concerned with the rigidity properties of geometric realizations of incidence geometries of rank two as points and lines in the Euclidean plane; we care about the distance being preserved among collinear points. We discuss…

A framework, which is a (possibly infinite) graph with a realization of its vertices in the plane, is called flexible if it can be continuously deformed while preserving the edge lengths. We focus on flexibility of frameworks in which…

A framework (a straight-line embedding of a graph into a normed space allowing edges to cross) is globally rigid if any other framework with the same edge lengths with respect to the chosen norm is an isometric copy. We investigate global…

This paper deals with certain results on the number of smooth structures on quaternionic projective spaces, obtained through the computation of inertia group and its analogues, which in turn are computed using techniques from stable…

This work considers the problem of estimating the unscaled relative positions of a multi-robot team in a common reference frame from bearing-only measurements. Each robot has access to a relative bearing measurement taken from the local…

We introduce and study asymptotically rigid mapping class groups of certain infinite graphs. We determine their finiteness properties and show that these depend on the number of ends of the underlying graph. In a special case where the…

In this paper a special group of bijective maps of a normed plane, called the group of general rotations, is introduced; it contains the isometry group as a subgroup. The concept of general rotations leads to the notion of flexible motions…

We construct infinite periodic versions of the stress matrix and establish sufficient conditions for periodic tensegrity frameworks to be globally rigid in $\mathbb{R}^d$ in the cases when the lattice is either fixed, fully flexible, or…

A rigidity theory is developed for bar-joint frameworks in linear matrix spaces endowed with a unitarily invariant norm. Analogues of Maxwell's counting criteria are obtained and minimally rigid matrix frameworks are shown to belong to the…

The combinatorial characterization of generic rigidity for bar-joint frameworks in dimensions $d \ge 3$ has been a long-standing open problem in discrete geometry. While the two-dimensional case was resolved in 1927 by Pollaczek-Geiringer…

A bar-joint framework $(G,p)$ in a (non-Euclidean) real normed plane $X$ is the combination of a finite, simple graph $G$ and a placement $p$ of the vertices in $X$. A framework $(G,p)$ is globally rigid in $X$ if every other framework…

We characterise finite and infinitesimal rigidity for bar-joint frameworks in R^d with respect to polyhedral norms (i.e. norms with closed unit ball P a convex d-dimensional polytope). Infinitesimal and continuous rigidity are shown to be…

Let P be a locally finite disk pattern on the complex plane C whose combinatorics are described by the one-skeleton G of a triangulation of the open topological disk and whose dihedral angles are equal to a function \Theta:E\to [0,\pi/2] on…

A bar-joint framework $(G,p)$ in the Euclidean space $\mathbb{E}^d$ is globally rigid if it is the unique realisation, up to rigid congruences, of $G$ in $\mathbb{E}^d$ with the edge lengths of $(G,p)$. Building on key results of…

We characterise rigid graphs for cylindrical normed spaces $Z=X\oplus_\infty \mathbb{R}$ where $X$ is a finite dimensional real normed linear space and $Z$ is endowed with the product norm. In particular, we obtain purely combinatorial…

We consider the space of all configurations of finitely many (potentially nested) circles in the plane. We prove that this space is aspherical, and compute the fundamental group of each of its connected components. It turns out these…