Question. Is there a continuous curve in the plane that has a non-unique loop-erasure?
Here is the definition of a loop-erasure. A continuous curve $Y:[c,d]\to\mathbb R^2$ is a loop-erasure of a curve $X:[a,b]\to\mathbb R^2$ if there exists an increasing and right-continuous function $w:[c,d]\to [a,b]$ such that:
- $Y(t)=X(w(t)), \forall t$,
- For every $T$, the image of the curve $Y(t), c\leq t\leq T$ does not intersect the image of the curve $X(s), w(T)<s\leq b$.
Note that if $w$ has a jump at time $t$, then one should have $X(w(t^-))=X(w(t))$. This corresponds to erasing a loop in $X$.
Update. Two loop-erasures are equivalent if they have the same image.