With the cylindrical equal area (CEA) projection data from the Helioseismic and Magnetic Imager (HMI) onboard the Solar Dynamics Observatory (SDO), we reconstructed the three-dimensional (3D) magnetic fields in the corona, using a non-linear force-free field (NLFFF) extrapolation method every 12 minutes during five days, to calculate the squashing degree factor Q in the volume. The results show that this AR has an hyperbolic flux tube (HFT) configuration, a typical topology of quadrupole, which is stable even during the two large flares (M6.6 and X2.2 class flares).

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.