# Configurations of lines in 3-space and rigidity of planar structures

## Orit Raz, Institute for Advanced Studies, Princeton

## November 1, 2016

Let $L=(\ell_1,...,\ell_n)$ be a sequence of $n$ lines in $\mathbb{R}^3$. We
define
the *intersection graph* $G_L = ([n], E)$ of $L$, where $[n] :=
\{1,...,n\}$, and with $(i,j)\in E$ if and only if $i \neq j$ and the
corresponding lines $\ell_i$ and $\ell_j$ intersect, or are parallel
(or
coincide). For a graph $G = ([n], E)$, we say that a sequence $L$
realizes
$G$ if $G\subset G_L$. In the talk I will introduce a
characterization of
those graphs $G$, such that, every (generic) realization of $G$
consists of lines that are either all concurrent or all coplanar.

The result is inspired by connections we have found between
configurations
of lines in 3-space and the classical notion of graph rigidity of
planar
structures. The interaction goes in both directions: Properties
established
in the context of graph rigidity either can be interpreted directly as
properties of line configurations in space, or lead the way to
conjectures
concerning the intersection patterns of lines in space. These general
statements about lines in space, apart from their independent
interest, can
then be used back in the context of graph rigidity to get new
information
in this context. This will be illustrated in the talk.