Topology of non-topological Chern-Simons vortices

The quantized magnetic flux $\Phi=-4\pi N\sk, N=0,\pm1,...$ of non-topological vortices in the non-relativistic Chern-Simons theory is related to the topological degree of the $S^2\to S^2$ mapping defined by lifting the problem to the Riemann spheres. Regular solutions with finite degree only arise for rational functions, whose topological degree, $N$, is the commun number of their zeros and poles on the Riemann sphere, also called their algebraic order.