  We derive exact results for close-packed dimers on the triangular kagome
lattice (TKL), formed by inserting triangles into the triangles of the kagome
lattice. Because the TKL is a non-bipartite lattice, dimer-dimer correlations
are short-ranged, so that the ground state at the Rokhsar-Kivelson (RK) point
of the corresponding quantum dimer model on the same lattice is a short-ranged
spin liquid. Using the Pfaffian method, we derive an exact form for the free
energy, and we find that the entropy is 1/3 ln2 per site, regardless of the
weights of the bonds. The occupation probability of every bond is 1/4 in the
case of equal weights on every bond. Similar to the case of lattices formed by
corner-sharing triangles (such as the kagome and squagome lattices), we find
that the dimer-dimer correlation function is identically zero beyond a certain
(short) distance. We find in addition that monomers are deconfined on the TKL,
indicating that there is a short-ranged spin liquid phase at the RK point. We
also find exact results for the ground state energy of the classical Heisenberg
model. The ground state can be ferromagnetic, ferrimagnetic, locally coplanar,
or locally canted, depending on the couplings. From the dimer model and the
classical spin model, we derive upper bounds on the ground state energy of the
quantum Heisenberg model on the TKL.
