We study topological properties of large-scale structure in a set of s
cale-free N-body simulations using the genus and percolation curves as
topological characteristics. Our results show that as gravitational c
lustering advances, the density field shows an increasingly pronounced
departure from Gaussian reflected in the changing shape of the percol
ation curve as well as the changing amplitude and shape of the genus c
urve. Both genus and percolation curves differentiate between the conn
ectedness of overdense and underdense regions if plotted against the d
ensity. When plotted against the filling factor, the percolation curve
alone retains this property. The genus curve shows a pronounced decre
ase in amplitude caused by phase correlations in the nonlinear regime.
Both genus and percolation curves provide complementary probes of lar
ge-scale structure topology, and can be used to discriminate between m
odels of structure formation and the analysis of observational data, s
uch as galaxy catalogs and microwave background radiation maps.