Solar flares – the most prominent manifestation of solar activity – typically manifest themselves as a single or a set of luminous arcs (magnetic flux tubes) rooted in regions of opposite polarity in the photosphere. However, a careful analysis of archival data from the Hinode satellite reveals occasional surprising cases of flaring arcs whose footpoints belong to regions of the same polarity or to areas without any appreciable magnetic field. Despite the counterintuitive nature of this phenomenon, it can be reasonably interpreted in the framework of the so-called ‘topological model’ of magnetic reconnection, whereby a magnetic null point is formed owing to a specific superposition of influences from remote sources rather than by local current systems. As a result, the energy release propagates along the separator of a flipping two-dome structure rather than along a fixed magnetic field line. Therefore, the luminous arc no longer needs to be associated immediately with the magnetic sources. Here, we report observational cases of the above-mentioned type and provide a theoretical model and numerical simulations.