  We predict the existence of a novel Floquet topological insulator in
three-dimensional two-band systems, the Floquet Hopf insulator, which possesses
two distinct topological invariants. One is the Hopf $\mathbb{Z}$ invariant, a
linking number characterizing the (non-driven) Hopf topological insulator. The
second invariant is an intrinsically Floquet $\mathbb{Z}_2$ invariant, and
represents a condensed matter realization of the topology underlying the Witten
anomaly in particle physics. Both invariants arise from topological defects in
the system's time-evolution, subject to a process in which defects at different
quasienergy exchange even amounts of topological charge. Their contrasting
classifications lead to a measurable physical consequence, namely, an unusual
bulk-boundary correspondence where gapless edge modes are topologically
protected, but may exist at either 0- or $\pi$-quasienergy. Our results
represent a phase of matter beyond the conventional classification of Floquet
topological insulators.
