Flexible circuits and $d$-dimensional rigidity

Abstract: A framework is a geometric realisation of a graph in Euclidean $d$-space. Edges of the graph correspond to bars of the framework and vertices correspond to joints with full rotational freedom. The framework is rigid if every edge-length-preserving continuous deformation of the vertices arises from isometries of $d$-space. Generically, rigidity is a rank condition on an associated rigidity matrix and hence is a property of the graph which can be described by the corresponding row matroid. Characterising which graphs are generically rigid is solved in dimension $1$ and $2$. However determining an analogous characterisation when $d\geq 3$ is a long standing open problem, and the existence of non-rigid (i.e. flexible) circuits is a major contributing factor to why this problem is so difficult. We begin a study of flexible circuits by characterising the flexible circuits in $d$-dimensions which have at most $d+6$ vertices. This is joint work with Georg Grasegger, Hakan Guler and Bill Jackson.

combinatorics

Audience: researchers in the topic

( paper | slides | video )

Virtual seminar on algebraic matroids and rigidity theory

Series comments: The COVID-19 pandemic is forcing us all to stay home, foregoing conferences and departmental seminars for the next few months. This weekly virtual seminar is an attempt to patch that departmental-seminar-sized void in our lives until it is safe to resume our more traditional forms of professional networking. Since geographic location matters a lot less for a virtual seminar than for an in-person seminar, this virtual seminar will be defined purely by its mathematical theme, algebraic matroids and rigidity theory, and not any particular department nor region.