Exotic resonances due to $\eta$ exchange

The meson $X(3872)$ and several related states appear to be, at least in part, hadronic molecules in which a heavy flavored meson (such as $D^0$) is bound to another heavy meson (such as $\bar D^{*0}$). Although not the only effect contributing to the binding, pion exchange seems to play a crucial role in generating the longest-range force between constituents. Mesons without $u$ and $d$ light quarks (such as $D_s$) cannot exchange pions, but under suitable circumstances can bind as a result of $\eta$ exchange. Channels in which this mechanism is possible are identified, and suggestions are made for searches for the corresponding molecular states, including a manifestly exotic baryonic $\Lambda_c \bar D_s^*$ resonance decaying into $J/\psi~\Lambda$.