  Over the past few years, topological insulators have taken center stage in
solid state physics. The desire to tune the topological invariants of the bulk
and thus control the number of edge states has steered theorists and
experimentalists towards periodically driving parameters of these systems. In
such periodically driven setups, by varying the drive sequence the effective
(Floquet) Hamiltonian can be engineered to be topological: then, the principle
of bulk--boundary correspondence guarantees the existence of robust edge
states. It has also been realized, however, that periodically driven systems
can host edge states not predicted by the Floquet Hamiltonian. The exploration
of such edge states, and the corresponding topological phases unique to
periodically driven systems, has only recently begun. We contribute to this
goal by identifying the bulk topological invariants of periodically driven
one-dimensional lattice Hamiltonians with chiral symmetry. We find simple
closed expressions for these invariants, as winding numbers of blocks of the
unitary operator corresponding to a part of the time evolution, and ways to
tune these invariants using sublattice shifts. We illustrate our ideas on the
periodically driven Su-Schrieffer-Heeger model, which we map to a discrete time
quantum walk, allowing theoretical results about either of these systems to be
applied to the other. Our work helps interpret the results of recent
simulations where a large number of Floquet Majorana fermions in periodically
driven superconductors have been found, and of recent experiments on discrete
time quantum walks.
