The integer quantum Hall effect sparked enduring interest in topological materials and their properties. However, there is no systematic method to create topological materials at will, and it is often unclear which quantity reveals their Chern or winding numbers. I will discuss conditions for a Hamiltonian to acquire nontrivial topology and show how engineered potentials representing defects or textures can lead to topological features. Using graphene, we calculate them, e.g. protected zero-energy quantum states and fractional charges. We introduce an innovative approach to extract topological numbers via a interference effect seen in STM wavefront dislocations. Finally, I will describe how topology alters basic physical phenomena like Friedel oscillations and Coulomb interactions leading to local magnetic moments, emphasizing the connection to many-body entanglement between spatially localized topological states.