Embedding spaces of split links

We study the homotopy type of the space E (L) of un-parametrised embeddings of a split link L = L1 ... Ln in R3. Our main result is a simple description of the fundamental group, or motion group, of E (L), and we extend this to a description of the motion group of embeddings in S3. The main tool we build is a semi-simplicial space of separating systems, which we show is homotopy equivalent to E (L). This combinatorial object provides a gateway to studying the homotopy type of E (L) via the homotopy type of the spaces E (Li).