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The recently elucidated structure of the delafossite YCuO$_{2.5}$ reveals aCu-O network with nearly independent $\Delta$ chains having differentinteractions between the $s=1/2$ spins. Motivated by this result, we study the$\Delta$ chain for various ratios $J_{\rm bb}/J_{\rm bv}$ of the base-base andbase-vertex interactions. By exact diagonalization and extrapolation, we showthat the elementary excitation spectrum, which (within numerical error) is thesame for total spins $S_{\rm tot}=0$ and 1, has a gap only in the interval$0.4874(1) \leq J_{\rm bb}/J_{\rm bv} \leq 1.53(1)$. The gap is dispersionlessfor $J_{\rm bb}/J_{\rm bv}=1$, but has increasing $k$-dependence as $J_{\rmbb}/J_{\rm bv}$ moves away from unity, related to the instability of dimers inthe ground state.Comment: 4 pages, 6 figures (revtex twocolumn

Quantum topological excitations: from the sawtooth lattice to the Heisenberg chain