  Leveraging topological properties in the response of electromagnetic systems
can greatly enhance their potential. Although the investigation of
singularity-based electromagnetics and non-Hermitian electronics has
considerably increased in recent years in the context of various scattering
anomalies, their topological properties have not been fully assessed. In this
work, we theoretically and experimentally demonstrate that non-Hermitian
perturbations around bound states in the continuum can lead to singularities of
the scattering matrix, which are topologically nontrivial and comply with
charge conservation. The associated scattering matrix poles, zeros and
pole-zero pairs delineate extreme scattering events, including lasing, coherent
perfect absorption, and absorber-lasers. The presented framework enables a
recipe for generation, annihilation and addition of these singularities in
electric circuits, with potential for extreme scattering engineering across a
broad range of the electromagnetic spectrum for sensing, wireless power
transfer, lasing and thermal emission devices.
