  Non-Hermitian singularities are ubiquitous in non-conservative open systems.
These singularities are often points of measure zero in the eigenspectrum of
the system which make them difficult to access without careful engineering.
Despite that, they can remotely induce observable effects when some of the
system's parameters are varied along closed trajectories in the parameter
space. To date, a general formalism for describing this process beyond simple
cases is still lacking. Here, we bridge this gap and develop a general approach
for treating this problem by utilizing the power of permutation operators and
representation theory. This in turn allows us to reveal the following
surprising result which contradicts the common belief in the field: loops that
enclose the same singularities starting from the same initial point and
traveling in the same direction, do not necessarily share the same end outcome.
Interestingly, we find that this equivalence can be formally established only
by invoking the topological notion of homotopy. Our findings are general with
far reaching implications in various fields ranging from photonics and atomic
physics to microwaves and acoustics.
