Differential topology method of identifying 3D null points in solar vector magnetic fields

Employing the Poincaré index of isolated null-point in a vector field, we work out a mathematical method of searching 3D null-points in coronal magnetic fields. After introducing the theory of differential topology, we test the method by using the analytical model of Brown & Priest. The location of null-point identified by our method coincides precisely with the analytical solution. Finally we apply the method to search 3D null-points in coronal magnetic fields reconstructed based on observed magnetograms of two active regions (NOAA AR 10488 and AR 10720). We find that the 3D null-points seem to be a key element to the magnetic topology associated with flare occurrence.