  We characterize bounded simply-connected planar $W^{1,p}$-extension domains
for $1 < p <2$ as those bounded domains $\Omega \subset \mathbb R^2$ for which
any two points $z_1,z_2 \in \mathbb R^2 \setminus \Omega$ can be connected with
a curve $\gamma\subset \mathbb R^2 \setminus \Omega$ satisfying $$\int_{\gamma}
dist(z,\partial \Omega)^{1-p}\, dz \lesssim |z_1-z_2|^{2-p}.$$ Combined with
known results, we obtain the following duality result: a Jordan domain $\Omega
\subset \mathbb R^2$ is a $W^{1,p}$-extension domain, $1 < p < \infty$, if and
only if the complementary domain $\mathbb R^2 \setminus \bar\Omega$ is a
$W^{1,p/(p-1)}$-extension domain.
