  We construct a geometrically self-similar Cantor set $X$ of genus $2$ in
$\mathbb{R}^3$. This construction is the first for which the local genus is
shown to be $2$ at every point of $X$. As an application, we construct, also
for the first time, a uniformly quasiregular mapping $f:\mathbb{R}^3 \to
\mathbb{R}^3$ for which the Julia set $J(f)$ is a genus $2$ Cantor set.
