  Model lattices consisting of balls connected by central-force springs provide
much of our understanding of mechanical response and phonon structure of real
materials. Their stability depends critically on their coordination number $z$.
$d$-dimensional lattices with $z=2d$ are at the threshold of mechanical
stability and are isostatic. Lattices with $z<2d$ exhibit zero-frequency
"floppy" modes that provide avenues for lattice collapse. The physics of
systems as diverse as architectural structures, network glasses, randomly
packed spheres, and biopolymer networks is strongly influenced by a nearby
isostatic lattice. We explore elasticity and phonons of a special class of
two-dimensional isostatic lattices constructed by distorting the kagome
lattice. We show that the phonon structure of these lattices, characterized by
vanishing bulk moduli and thus negative Poisson ratios and auxetic elasticity,
depends sensitively on boundary conditions and on the nature of the kagome
distortions. We construct lattices that under free boundary conditions exhibit
surface floppy modes only or a combination of both surface and bulk floppy
modes; and we show that bulk floppy modes present under free boundary
conditions are also present under periodic boundary conditions but that surface
modes are not. In the the long-wavelength limit, the elastic theory of all
these lattices is a conformally invariant field theory with holographic
properties, and the surface waves are Rayleigh waves. We discuss our results in
relation to recent work on jammed systems. Our results highlight the importance
of network architecture in determining floppy-mode structure.
