  This article investigates phonons and elastic response in randomly diluted
lattices constructed by combining (via the addition of next-nearest bonds) a
twisted kagome lattice, with bulk modulus $B=0$ and shear modulus $G>0$, with
either a generalized untwisted kagome lattice with $B>0$ and $G>0$ or with a
honeycomb lattice with $B>0$ and $G=0$. These lattices exhibit jamming-like
critical end-points at which $B$, $G$, or both $B$ and $G$ jump discontinuously
from zero while the remaining moduli (if any) begin to grow continuously from
zero. Pairs of these jamming points are joined by lines of continuous rigidity
percolation transitions at which both $B$ and $G$ begin to grow continuously
from zero. The Poisson ratio and $G/B$ can be continuously tuned throughout
their physical range via random dilution in a manner analogous to "tuning by
pruning" in random jammed lattices. These lattices can be produced with modern
techniques, such as 3D printing, for constructing metamaterials.
