Abstract
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 equivalent for finite frameworks in R^d which are well-positioned with respect to P. An edge-labelling determined by the facets of the unit ball and placement of the framework is used to characterise infinitesimal rigidity in R^d in terms of monochrome spanning trees. An analogue of Laman's theorem is obtained for all polyhedral norms on R^2.
Highlights
A bar-joint framework in Rd is a pair (G, p) consisting of a simple undirected graph G = (V (G), E(G)) and a placement p : V (G) → Rd of the vertices such that pv and pw are distinct whenever vw is an edge of G
A well-developed rigidity theory exists in the Euclidean setting for finite bar-joint frameworks, which stems from classical results of Cauchy [6], Maxwell [17], Alexandrov [1] and Laman [14]
Significant progress has been made in topics such as global rigidity [7,8,11] and the rigidity of periodic frameworks [5,16, 20,21] in addition to newly emerging themes such as symmetric frameworks [22] and frameworks supported on surfaces [19]
Summary
Of particular relevance is Laman’s landmark characterisation for generic minimally infinitesimally rigid finite bar-joint frameworks in the Euclidean plane. A study of rigidity with respect to the classical non-Euclidean p norms was initiated in [12] for finite bar-joint frameworks and further developed for infinite bar-joint frameworks in [13] Among these norms the 1 and ∞ norms are simple examples of polyhedral norms and so the results obtained here extend some of the results of [12]. The well-positioned placements of a finite graph are open and dense in the set of all placements, and we show that finite and infinitesimal rigidity are equivalent for these bar-joint frameworks (Theorem 7). Many of the results obtained hold well for both finite and infinite bar-joint frameworks
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