Knot theory is the study of the path components of $\operatorname{Emb}(S^1 , \mathbb{R}^3)$. I know sometimes in geometric topology, these function spaces have very complicated path components, but relatively simple higher homotopy. Are the homotopy types of the path components of any particular knots known? And more generally, do we know the higher homotopy groups of $\operatorname{Emb}(S^1 , \mathbb{R}^n)$?
I imagine this is probably approached through embedding calculus, so I offer a meagre first step: if $n>3$ the codimension of the spaces is enough that the embedding tower converges. Moreover, it is well known that the first stage of the embedding tower is immersions, which in our case is equivalent to formal immersions. If I am not mistaken, this space $\operatorname{fimm}(S^1, \mathbb{R}^n)$ is equivalent to $\Omega_{free} S^{n-1}$ because both $S^1$ and $\mathbb{R}^n$ are parallelizable and the latter contractible.
Could anyone outline the known work or provide references?
Tyrone: Plenty is known about the case you suggest. The space of long knots just removes some ambiguity.
Remember that there is a fiber sequence $\text{Emb}_*(S^1, S^3) \to \text{Emb}(S^1, S^3) \to (TS^3 \setminus 0)$, the last map sending $\gamma$ to $\gamma'(0)$ and the first is the space of embeddings with prescribed values for $\gamma(0)$ and the derivative there.
The first space is seen to be equivalent to the space of long knots in $\Bbb R^3$ by a standard fiber sequence relating the space of embeddings with specified first derivative to the space of embeddings with specified germ. So a full understanding of the space of long knots gives rise to a full understanding of the space of all knots. The former is more natural and easier to compute in terms of.
To my recollection Budney has completely finished these calculations (perhaps in the paper you cite).