Embedding spaces of split links

TL;DR

This paper characterizes the homotopy type of the space of unparametrised embeddings of split links in 3D space, providing a new combinatorial approach to understanding their fundamental groups and motion groups.

Contribution

It offers a simple description of the fundamental and motion groups of split link embedding spaces, extending to embeddings in the 3-sphere, using a novel semi-simplicial space of separating systems.

Findings

Homotopy equivalence between separating systems and embedding spaces

Explicit description of the fundamental group of embedding spaces

Extension of results to embeddings in the 3-sphere

Abstract

We study the homotopy type of the space $E(L)$ of unparametrised embeddings of a split link $L=L_1\sqcup \ldots \sqcup L_n$ in $\mathbb{R}^3$. 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 $S^3$. 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 $E(L_i)$.