Can every closed curve be modified in the following way to produce a simple closed curve?

Question

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.

Answer 1

I think I got this.

Parametrize the curve as f(t) = (x,y). Pick any intersection, mark 4 local points a,b,c,d, such that it looks like

a c
 x
b d

and $f(t_a)=a, f(t_b)=b, f(t_c)=c, f(t_d)=d$. such that $t_a<t_b<t_c<t_d$

I claim the following would not make two closed curves.

a c
 =
b d

The a,b,c,d divide the curve into 4 pieces. $ab, bc, cd, da$. Where $xy$ is the piece of curve $\{f(t)|t\in[t_x,t_y]\}$, where $f(t_x) = x$ and $f(t_y)=y$.

This movement only change two pieces. $ab$ to $ac$, $cd$ to $bd$. $bc$, $ad$ are not changes. Therefore if we travel though a, we will get $acbd$,