Linear embeddings of a graph in R^3 up to rigid isotopy
E.Glushak
Let $G$ be a graph (i.e. a finite 1-dimensional simplicial space). An
embedding $f:G \rightarrow \mathbb{R}^{3}$ is called a \emph{linear} one, if
the restriction of the map $f$ on any edge of the graph $G$ is a linear map.
Let $L(G)$ be a topological space of linear embeddings $G \rightarrow
\mathbb{R}^{3}$ endowed with a compact-open topology; two embeddings
$f_0,f_1 \in L(G)$ are \emph{rigid isotopic}, if there is a path $F:I
\rightarrow L(G)$, such that $F(0)=f_0$ and $F(1)=f_1$. Our main problem is
to classify linear embeddings $G \rightarrow \mathbb{R}^{3}$ up to rigid
isotopy.
In the first part of the talk for all graphs with five or less vertices we
give a solution of the problem. In the second part of the talk we give some
lower and upper bounds of the number of rigid isotopic classes of linear
embeddings $G \rightarrow \mathbb{R}^{3}$.