On the homotopy type of the spaces of spherical knots in $R^n$

We study the spaces of embeddings $S^m\hookrightarrow R^n$ and those of long embeddings $R^m\hookrightarrow R^n$, i.e. embeddings of a fixed behavior outside a compact set. More precisely we look at the homotopy fiber of the inclusion of these spaces to the spaces of immersions. We find a natural fiber sequence relating these spaces. We also compare the $L_{\mathrm{\infty }}$-algebras of diagrams that encode their rational homotopy type, when the codimension $n-m\ge 3$.