  We present a topological characterization of time-periodically driven
two-band models in 2+1 dimensions as Hopf insulators. The intrinsic periodicity
of the Floquet system with respect to both time and the underlying
two-dimensional momentum space constitutes a map from a three dimensional torus
to the Bloch sphere. As a result, we find that the driven system can be
understood by appealing to a Hopf map that is directly constructed from the
micromotion of the drive. Previously found winding numbers are shown to
correspond to Hopf invariants, which are associated with linking numbers
describing the topology of knots in three dimensions. Moreover, after being
cast as a Hopf insulator, not only the Chern numbers, but also the winding
numbers of the Floquet topological insulator become accessible in experiments
as linking numbers. We exploit this description to propose a feasible scheme
for measuring the complete set of their Floquet topological invariants in
optical lattices.
