  Integer-valued topological indices, characterizing nonlocal properties of
quantum states of matter, are known to directly predict robust physical
properties of equilibrium systems. The Chern number, e.g., determines the
quantized Hall conductivity of an insulator. Using fermionic atoms in a
periodically driven optical lattice, here we demonstrate experimentally that
the Chern number determines also the far-from-equilibrium dynamics of a quantum
system. Following the proposal of ref. [Wang et al., Phys. Rev. Lett. 118,
185701 (2017)] and extending it to Floquet systems, we measure the linking
number that characterizes the trajectories of momentum-space vortices emerging
after a strong quench. We observe that it directly corresponds to the
ground-state Chern number. This one-to-one relation between a dynamical and a
static topological index allows us to experimentally map out the phase diagram
of our system. Furthermore, we measure the instantaneous Chern number and show
that it remains zero under the unitary dynamics.
