The title refers, of course, to Matthew (2:12) ''And being warned in a dream not to return to Herod, they departed to their own country by another way''. To be honest, it is not that specific particular case I'm more interested in.

I'd like to have a reference, or a hint here, for a simple proof of the following fact (intuitive, but not easy to proof, as usual in these matters).

Let $\Gamma$ be a simple Jordan arc in $\mathbb{R}^2$ (a homeomorphic image of the interval $[0,1]$). Then, $\Gamma$ can be included in a simple Jordan loop $ \Sigma $ (a homeomorphic image of $\mathbb{S}^1$).

By the (generalized) Jordan's theorem, we know that $\mathbb{R}^2\setminus\Gamma$ is connected; and being open, it is even connected by piece-wise linear paths. The difficulty is that we need a path connecting the end-points of $\Gamma$; in other words, the question is how to show that $(\mathbb{R}^2\setminus\Gamma)\cup \partial\Gamma $ is path-connected (after that, an injective path could always be extracted).

It seems to me everything would follow easily from this lemma:

Assume that $B(0,2)\setminus \Gamma$ has at least two connected components that meet $B(0,1)$. Then, there are three consecutive points of $\Gamma$, resp. $y_1$, $y_2$, and $y_3$ such that $\|y_2\|=1$, and $\|y_1\|=\|y_3\|=2$

I'm also a bit puzzled by the quantitative aspect of this problem:

Assume that $\Gamma$ is parametrized by a homeomorphism $\gamma:[0,1]\to\Gamma$ with a modulus of continuity $\omega(t)$ (say, a continuous concave function vanishing at $t=0$) and let $\omega_1$ be another modulus of continuity such that $\omega_1(t) > \omega(t)$ for $t > 0$. Is there a Jordan loop $\Sigma$ with parametrization $\gamma_1[0,2 > ]\to\Sigma$ that has modulus of continuity $\omega_1$ ?