# 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.