I can't seem to visualize the process of deforming a circle into the same circle traversed twice.
I read that a positive(ly oriented) circle (centered at the origin) cannot be deformed into another positive circle traversed twice on the punctured plane. I cannot visually confirm this, as I'm not even sure how I could turn a single loop into two of them while not allowing the loop to self intersect etc.
Can someone describe the process of deforming a circle into the same circle traversed twice? Assume the circle lies wholly in a domain. My guess is that you make a big circle, twist at the diameter to make an infinity shape, then fold it back on itself on smoothly. While both circles would be in the same orientation, the intermediate intersects itself and I doubt its a loop.
The problem I'm trying to verify is whether a loop(circle) can be deformed into the same circle in the negative direction twice.
The process is done true scaling the parameter in the parametric equation of the circle.
For example
$$x=r\cos(t)$$
$$y=r\sin(t)$$ $t\in [0,2\pi]$
is traversed counter clockwise.
For traversing twice you extend the domain of $t$ to $[0,4\pi]$ instead of $[0,2\pi]$
Now change $t$ to $-t$ to get
$$x=r\cos(-t)=r\cos(t)$$ $$y=r\sin(-t)=-r\sin(t)$$
Which is traversed clockwise.
Change $t$ to $2t$ to double the speed or to $t/2$ to slow down the motion.