A growing number of applications in areas as diverse as materials science and geophysical fluid flows display robustness with respect to defects that may be modeled as topological. In particular, interface Hamiltonians describe an observable and practically useful transport asymmetry at the one-dimensional interface between two two-dimensional half spaces ‘in different topologies’ that is immune to large classes of (locally) arbitrarily large perturbations. This talks considers such perturbed interface Hamiltonian operators and presents which types of topological invariants we can assign to them, what these invariants are robust against, and which physical observables are associated to these non-trivial topologies.