  In this paper we present a sampling framework for RNA structures of fixed
topological genus. We introduce a novel, linear time, uniform sampling
algorithm for RNA structures of fixed topological genus $g$, for arbitrary
$g>0$. Furthermore we develop a linear time sampling algorithm for RNA
structures of fixed topological genus $g$ that are weighted by a simplified,
loop-based energy functional. For this process the partition function of the
energy functional has to be computed once, which has $O(n^2)$ time complexity.
