  Hopf insulators are intriguing three-dimensional topological insulators
characterized by an integer topological invariant. They originate from the
mathematical theory of Hopf fibration and epitomize the deep connection between
knot theory and topological phases of matter, which distinguishes them from
other classes of topological insulators. Here, we implement a model Hamiltonian
for Hopf insulators in a solid-state quantum simulator and report the first
experimental observation of their topological properties, including fascinating
topological links associated with the Hopf fibration and the integer-valued
topological invariant obtained from a direct tomographic measurement. Our
observation of topological links and Hopf fibration in a quantum simulator
opens the door to probe rich topological properties of Hopf insulators in
experiments. The quantum simulation and probing methods are also applicable to
the study of other intricate three-dimensional topological model Hamiltonians.
