  We explore the phase diagram and the low-energy physics of three Heisenberg
antiferromagnets which, like the kagome lattice, are networks of corner-sharing
triangles but contain two sets of inequivalent short-distance resonance loops.
We use a combination of exact diagonalization, analytical strong-coupling
theories, and resonating valence bond approaches, and scan through the ratio of
the two inequivalent exchange couplings. In one limit, the lattices effectively
become bipartite, while at the opposite limit heavily frustrated nets emerge.
In between, competing tunneling processes result in short-ranged spin
correlations, a manifold of low-lying singlets (which can be understood as
localized bound states of magnetic excitations), and the stabilization of
valence bond crystals with resonating building blocks.
