Is the space of Jordan curves in $\textbf{R}^2$ contractible? In other words, is there a canonical or continuous way to deform each Jordan curve to the unit circle $\textbf{S}^1$. 

If the curves are smooth one can do this via curve shortening flow/rescaling. I think this approach even works for rectifiable curves, by a [paper][1] of Lauer. But I do not know a reference for the general topological case. In the smooth case, is there a way to do this without using flows?


  [1]: https://link.springer.com/article/10.1007/s00039-013-0248-1