Topology inside NC${}^1$

We show that ACC${}^0$ is precisely what can be computed with constant-width circuits of polynomial size and polylogarithmic genus. This extends a characterization given by Hansen, showing that planar constant-width circuits also characterize ACC${}^0$. Thus polylogarithmic genus provides no additional computational power in this model. We consider other generalizations of planarity, including crossing number and thickness. We show that constant-width circuits of polynomial size and thickness two already suffice to capture all of NC${}^1$.