Homotopy type of embeddings of circle in the plane What is the homotopy type of the space of (topological) emdeddings of $S^1$ in $\mathbb R^2$? 
My conjecture: This space deformation retracts to $S^1\sqcup S^1$, and a retraction in each of orientation preserving and reversing components is given by  "rotation number" in $S^1$. (I don't have an exact proof or even an exact definition of the conjectured retraction map)
 A: For $C^2$-embeddings you can follow weighted (length preserving) mean curvature flow until you have a circle, then move the center to 0, then use a homothethy to go to the unit circle, then use the contraction in $\{\phi\in Diff(S^1):\phi(1)=1\}$ to get arc-length parametrization, so you end up with a circle. The initial point of the parametrization is then the $S^1$ for your orientation. 
Now, how to get from topological embedding to $C^2$-embedding? Use convolution with a smooth bump function? Does this destroy topological embedding?
Added:
Mean curvature flow preserves embeddings; see http://www.ams.org/journals/bull/2015-52-02/S0273-0979-2015-01468-0/S0273-0979-2015-01468-0.pdf
About John Pardons remark: Embedded simple closed curves (without parametrisation!) are diffeomorphic to $PSL(2,\mathbb R)\backslash Diff(S^1)$ and this diffeomorphism goes down to $H^{3/2}$-embeddings and corresponds to a subgroup of quasi symmetric homeomorphisms of the circle. The idea is to take a Riemann mapping $\phi_{int}$ from the unit disk to the interior of $\Gamma$ (= the embedded circle) which is unique up to a Möbius transformation, and 
a Riemann mapping $\phi_{ext}$ from the exterior of the disk to the exterior of $\Gamma$, fixing $\infty$ and the derivative at $\infty$, so that it becomes unique. Then
$\phi_{ext}^{-1}\circ \phi_{int}$ is in $PSL(2,\mathbb R)\backslash Diff(S^1)$ which is contractible. 
See Takhtaja and Teo, Memoirs of the AMS, or see the following papers for smooth embeddings. 


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*E. Sharon and D. Mumford. 2d-shape analysis using conformal mapping. In Proc. IEEE Conf. Computer Vision and Pattern Recognition, pages 350–357, 2004.

*E. Sharon and D. Mumford. 2d-shape analysis using conformal mapping. International Journal of Computer
Vision, 70:55–75, 2006.
