  A fluid particle changes its dynamics from diffusive to oscillatory as the
system density increases up to the melting density. Hence, the notion of the
Frenkel line was introduced to demarcate the fluid region into rigid and
nonrigid liquid subregions based on the collective particle dynamics. In this
work, we apply a topological framework to locate the Frenkel lines of the
soft-sphere and the hard-sphere models relying on the system configurations.
The topological characteristics of the ideal gas and the maximally random
jammed state are first analyzed, then the classification scheme designed in our
earlier work is applied. The classification result shows that the fraction of
solid-like atoms increases from zero to one in the rigid liquid region. The
dependence of the solid-like fraction on the bulk density is understood based
on the theory of fluid polyamorphism. The percolation behavior of solid-like
clusters is described based on the fraction of solid-like molecules in an
integrated manner. The crossover densities are obtained by examining the
percolation of solid-like clusters. The resultant crossover densities of
soft-sphere fluids converge to that of hard-sphere fluid. Hence, the
topological method successfully highlights the generality of the Frenkel line.
