I am a condensed matter physicist, and want to understand the hopf link (two circles linked together) from analytic point of view. My question is as follows.

We have two sets of analytic equations, and each set of equations describes a circle in certain space. In that case, how can we tell if the two circle form a hopf link or two trivial disconnected circle? I want to the conditions from analytic geometry point of view.

Could anybody point out some references for this problem?