The embedding space of a Hopf link

We study the unparametrised smooth embedding space of a Hopf link in ${\mathbb{R}}^3$, and prove that it is homotopy equivalent to the closed 3-manifold $S^3\mathrm{/}{\mathbb{Q}}_8$. As an intermediate step in the proof, we show that the inclusion of the subspace of round embeddings is a homotopy equivalence. We provide analogous results for the unparametrised smooth embedding space of a Hopf link in $S^3$, which we show is homotopy equivalent to $\mathbb{R}{P}^2\times \mathbb{R}{P}^2$.