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

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.