  We study the homotopy type of the space $\mathcal{E}(L)$ of unparametrised
embeddings of a split link $L=L_1\sqcup \ldots \sqcup L_n$ in $\mathbb{R}^3$.
Inspired by work of Brendle and Hatcher, we introduce a semi-simplicial space
of separating systems and show that this is homotopy equivalent to
$\mathcal{E}(L)$. This combinatorial object provides a gateway to studying the
homotopy type of $\mathcal{E}(L)$ via the homotopy type of the spaces
$\mathcal{E}(L_i)$. We apply this tool to find a simple description of the
fundamental group, or motion group, of $\mathcal{E}(L)$, and extend this to a
description of the motion group of embeddings in $S^3$.
