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We determine the functional behavior near the discrete rotational symmetryaxis of discrete vortices of the nonlinear Schr\"odinger equation. We show thatthese solutions present a central phase singularity whose charge is restrictedby symmetry arguments. Consequently, we demonstrate that the existence ofhigh-charged discrete vortices is related to the presence of other off-axisphase singularities, whose positions and charges are also restricted bysymmetry arguments. To illustrate our theoretical results, we offer twonumerical examples of high-charged discrete vortices in photonic crystal fibersshowing hexagonal discrete rotational invariance.

Symmetry, winding number and topological charge of vortex solitons in discrete-symmetry media