  The distribution of magnetic flux across the solar photosphere results in a
complex web of coronal magnetic field structures. To understand this
complexity, the magnetic skeleton of the coronal field can be calculated. The
skeleton highlights the separatrix surfaces that divide the field into
topologically distinct regions, allowing open-field regions on the solar
surface to be located. Furthermore, separatrix surfaces and their intersections
(separators) are important likely energy release sites. This paper
investigates, throughout the solar cycle, the nature of coronal magnetic-field
topologies that arise under the potential-field source-surface approximation.
In particular, we characterise the typical global fields at solar maximum and
minimum. Global magnetic fields are extrapolated from observed Kitt Peak and
SOLIS synoptic magnetograms, from Carrington rotations 1645 to 2144, using the
potential-field source-surface model. Hence, variations in the coronal skeleton
are studied over three solar cycles. Key building blocks which make up magnetic
fields are identified and classified according to the nature of their
separatrix surfaces. The magnetic skeleton reveals that, at solar maximum, the
global coronal field involves a multitude of topological structures at all
latitudes. Many open-field regions exist originating anywhere on the
photosphere. At solar minimum, the coronal topology is heavily influenced by
the solar dipole. A strong dipole results in a simple large-scale structure
involving just two large polar open-field regions, but, at short radial
distances between plus or minus 60 deg latitude, the small-scale topology is
complex. If the solar dipole if weak, as in the recent minimum, then the
low-latitude quiet-sun magnetic fields may be globally significant enough to
create many disconnected open-field regions at low latitudes, in addition to
the two polar open-field regions.
