Counting of freedom in symmetric frameworks
Louis Theran, Aalto University


Bar-joint frameworks are mechanical structures made of fixed-length bars, connected by universal joints with full rotational freedom.  The allowed motions preserve the lengths and connectivity of the bars.  The rigidity question for bar-joint frameworks asks if all the allowed motions are rigid body motions.  For frameworks with generic geometry in dimension 2, a theorem of Laman classifies the minimally rigid combinatorial types (i.e., the graph that has as its edges the bars).  Laman’s result is proved by making rigorous a degree of freedom heuristic that goes back to Maxwell.  I’ll discuss similar types of results for symmetric frameworks (which are far from generic), with a focus on the appropriate analogue of Maxwell counting.