  Much of our understanding of vibrational excitations and elasticity is based
upon analysis of frames consisting of sites connected by bonds occupied by
central-force springs, the stability of which depends on the average number of
neighbors per site $z$. When $z<z_c \approx 2d$, where $d$ is the spatial
dimension, frames are unstable with respect to internal deformations. This
pedagogical review focuses on properties of frames with $z$ at or near $z_c$,
which model systems like randomly packed spheres near jamming and network
glasses. Using an index theorem, $N_0 - N_S = dN - N_B$ relating the number of
sites, $N$, and number of bonds, $N_B$, to the number, $N_0$, of modes of zero
energy and the number, $N_S$, of states of self stress, in which springs can be
under positive or negative tension while forces on sites remain zero, it
explores the properties of periodic square, kagome, and related lattices for
which $z=z_c$ and the relation between states of self stress and zero modes in
periodic lattices to the surface zero modes of finite free lattices (with free
boundary conditions). It shows how modifications to the periodic kagome lattice
can eliminate all but trivial translational zero modes and create topologically
distinct classes, analogous to those of topological insulators, with protected
zero modes at free boundaries and at interfaces between different topological
classes.
