Topological invariants govern many important physical properties in condensed matter systems. In this work, we obtain the complete set of topological invariants for a family of one-dimensional quasicrystals. The first and best-studied member of the family is the Fibonacci chain, while the successive ones are known in the literature as silver, bronze, etc., and collectively as the metallic mean chains. By considering rational approximants, and by making use of the relationship between these chains and two-dimensional Quantum Hall problems, we write down a gap labeling scheme for finite systems, and extend it to the quasiperiodic limit. We show, by numerical computations on open chains, that the proposed scheme correctly yields the winding numbers of edge states in each of the gaps, in all of the quasicrystals. We observe that their relationship to the two-dimensional model leads, in the strict one-dimensional chains family, to spectra forming a simplified Hofstadter “butterfly” diagram, with the analogs of Landau levels appearing in the asymptotic limit. These topological invariants for noninteracting 1D chains can be expected to remain pertinent for superconducting systems where Majorana or Aharanov bound states occur, as shown in recent studies of interacting or Josephson coupled Fibonacci chains.

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• 1-dimensional systems
• Quasicrystals
• Topological materials