Topological Methods for the Analysis of Solar Magnetic Fields

The solar coronal magnetic field is anchored to a complex distribution of photospheric flux consisting of sunspots and magnetic elements. Coronal activity such as flares, eruptions and general heating is often attributed to the manner in which the coronal field responds to photospheric motions. A number of powerful techniques have been developed to characterize the response of the coronal field by describing its topology. According to such analyses, activity will be concentrated around topological features in the coronal field such as separatrices, null points or bald patches. Such topological properties are insensitive to the detailed geometry of the magnetic field and thereby create an analytic tool powerful and robust enough to be useful on complex observations with limited resolution. This article reviews those topological techniques, their developments and applications to observations.