Title: Counting Realisations for Rigid Graphs

Speaker: Sean Dewar, University of Bristol

Date and time: 2pm (New York time), Tuesday, February 13, 2024

Place: On Zoom (details provided on the seminar mailing list)

A graph is $d$-rigid if for any generic positioning of its vertices in $d$-dimensional Euclidean space, there are finitely many other realisations of the graph in $d$-dimensional Euclidean space (modulo isometries) with the same length edges. Combinatorial characterisations for $1$-rigidity (i.e., connectivity) and $2$-rigidity are known, but it is currently an open problem for $d > 2$. My talk will be a survey of results involving a variation of this problem: given a $d$-rigid graph with a generic realisation, how many other realisations of the graph exist with the same length edges (now allowing for complex realisations also)? For example, given the $2$-rigid graph formed by gluing two triangles at an edge, every generic realisation in the plane has exactly one other edge-length equivalent realisation that is formed by flipping one of the triangles. I will also cover a recent result from Georg Grasegger (RICAM) and I that determines the relationship between the Euclidean and non-Euclidean (e.g., spherical, hyperbolic) variants of the problem.