  This paper addresses reconnection of vortex tubes, with particular focus on
the topology of the vortex lines (field lines of the vorticity). This analysis
of vortex line topology reveals previously undiscovered features of the
reconnection process, such as the generation of many small flux rings, formed
when reconnection occurs in multiple locations in the vortex sheet between the
tubes. Consideration of three-dimensional reconnection principles leads to a
robust measurement of the reconnection rate, even once instabilities break the
symmetry. It also allows us to identify internal reconnection of vortex lines
within the individual vortex tubes. Finally, the introduction of a third vortex
tube is shown to render the vortex reconnection process fully
three-dimensional, leading to a fundamental change in the topological structure
of the process. An additional interesting feature is the generation of
vorticity null points.
