  The single-wall carbon nanotube (SWNT) can be a one-dimensional topological
insulator, which is characterized by a $\mathbb{Z}$-topological invariant,
winding number. Using the analytical expression for the winding number, we
classify the topology for all possible chiralities of SWNTs in the absence and
presence of a magnetic field, which belongs to the topological categories of
BDI and AIII, respectively. We find that the majority of SWNTs are nontrivial
topological insulators in the absence of a magnetic field. In addition, the
topological phase transition takes place when the band gap is closed by
applying a magnetic field along the tube axis, in all the SWNTs except armchair
nanotubes. The winding number determines the number of edge states localized at
the tube ends by the bulk-edge correspondence, the proof of which is given for
SWNTs in general. This enables the identification of the topology in
experiments.
