Has the fundamental group of the space of smooth embeddings of $S^1$ into $\mathbb R^3$ been completely computed? If yes, I would really appreciate any reference for the computation of it. Thank you in advance.

Has the fundamental group of the space of smooth embeddings of $S^1$ into $\mathbb R^3$ been completely computed? Say the basepoint is an unknot. Maybe something is known for other components? If yes, I would really appreciate any reference for the computation of it. To be absolutely precise, I am interested whether for every smooth knot there is a non-contractible loop of smooth knots based at it.