Suppose you have $\gamma(t):[0,1]\rightarrow \mathbb{C}$ simple piecewise smooth, $\gamma(0) = 0$ and $\gamma(1)=1$. Does there exist $\eta:[0,1]\rightarrow \mathbb{C}$, another simple piecewise smooth curve, with $\eta(0) = 0$ and $\eta(1)$ a point on the unit circle, such that $\gamma$ and $\eta$ only intersect at the origin?
My attempt: I tried approaching this constructively, but I couldn't seem to get my hands on anything useful. The best alternative I could come up with was trying to use the Riemann mapping theorem to find a conformal map $f: \mathbb{D}\setminus\gamma \rightarrow \mathbb{D}$. If $f$ extends continuously to the boundary, then I could use a chord $A$ which joins $f(0)$ to $f(-1)$, and then let $\eta = f^{-1}(A)$. But, even then, I'm not sure if I can guarantee that the resulting $\eta$ has nonzero one-sided derivatives at the endpoints. So my strategy only works in nice situations (as far as I know).
Is there a general topological approach to this? Can my Riemann mapping strategy be made to work, or is it hopeless?