Is there a sequence of the following operation that change a closed curve with finite number of self-intersections to a simple closed curve?
Also, every self-intersection differs at least $\epsilon$ in distance. The curve never pass though the same point 3 times.
If there is a intersection that locally looks like
a b x c d
we can change it to one of the following
a b = c d a b || c d
If the answer is different on different spaces, I'm interested in $\mathbb{R}^2$.
From some example I tried, it seems one of the move can create 2 closed curves, but not both.
Edit: I don't know how to capture the notion of the curves. I'm wondering about this because I draw some closed curve on a notebook, and figured I can erase a intersection and connect in the above way, and eventually it become a simple closed curve.
Just assume this curve is well behaved enough that I can draw on a notebook.