On flexes associated with higher-order flexible bar-joint frameworks
Georg Nawratil
TL;DR
The paper tackles the problem of defining and computing higher-order flexes for bar-joint frameworks by reformulating flexion/rigidity orders within an algebraic-geometry framework. It leverages Newton-Puiseux (Puiseux) expansions and projection techniques to characterize branches of configuration spaces and to obtain minimal parametrizations corresponding to flexes, while incorporating reality by taking real parts of these parametrizations. A removal procedure for isostatic frameworks is developed to identify feasible higher-order flexes and is extended via a pencil of constraints to enumerate irreducible branches, addressing prior dilemmas such as cusp-related higher-order rigidity. The approach provides a rigorous, computable method for identifying finite $(k,n)$-flexes and determining their real realizations, with potential extensions through tropical geometry to improve computational efficiency.
Abstract
The famous example of the double-Watt mechanism given by Connelly and Servatius raises some problems concerning the classical definitions of higher-order flexibility and rigidity, respectively. Recently, the author was able to give a proper redefinition of the flexion/rigidity order for bar-joint frameworks, but the question for the flexes associated with higher-order flexible structures remained open. In this paper we properly define these flexes based on the theory of algebraic curves and demonstrate their computation by means of Puiseux series. The presented algebraic approach also allows to take reality issues into account.