TITLE: Fundamental group of the space of smooth embeddings of $S^1$ into $\mathbb R^3$
QUESTION [7 upvotes]: 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.

REPLY [4 votes]: Your question is a bit ambiguous, but if you really are interested in the fundamental group with the base point the unknot, then you only care about the homotopy type of the component of the space of embeddings containing the unknot.  This was computed entirely in Hatcher's classical paper
Hatcher, Allen E.,
A proof of the Smale conjecture, Diff(S3)≃O(4),
Ann. of Math. (2) 117 (1983), no. 3, 553–607.
See the the 15th equivalent form of the Smale conjecture on the last page of text before the bibliography.  Budney's work generalizes this (and I assume probably depends on it?).  A different kind of generalization that might interest you is in
Brendle, Tara E., Hatcher, Allen,
Configuration spaces of rings and wickets,
Comment. Math. Helv. 88 (2013), no. 1, 131–162.
which computes the homotopy type of the component of the space of embeddings of k disjoint circles containing the unlink.