  Motivated by Sr$_2$RuO$_4$, edge quasiparticle states are analyzed based on
the self-consistent solution of the Bogolyubov-de Gennes equations for a
topological chiral $p$-wave superconductor. Using a tight-binding model of a
square lattice for the dominant $\gamma$-band we explore the non-trivial
geometry and band structure dependence of the edge states and currents. As a
peculiar finding we show that for high band fillings currents flow in reversed
direction comparing straight and zigzag edges. We give a simple explanation in
terms of the positions of the zero-energy bound states using a quasi-classical
picture. We also show that a Ginzburg-Landau approach can reproduce these
results. Moreover, the band filling dependence of the most stable domain wall
structure is discussed.
