  We demonstrate general classifications of Riemann surface topology generated
by multiple arbitrary-order exceptional points of quasi-stationary states. Our
studies reveal all possible product permutations of holonomy matrices that
describe a stroboscopic encircling of 2nd order exceptional points. The
permutations turn out to be categorized into a finite number of classes
according to the topological structures of the Riemann surfaces. We further
show that the permutation classes can be derived from combinations of cyclic
building blocks associated with higher-order exceptional points. Our results
are verified by an effective non-Hermitian Hamiltonian founded on generic
Jordan forms and then examined in physical systems of desymmetrized optical
microcavities.

Optics;Classical Physics;Quantum Physics