  Three-dimensional magnetic topology is crucial to understanding the explosive
release of magnetic energy in the corona during solar flares. Much attention
has been given to the pre-flare magnetic topology to identify candidate sites
of magnetic reconnection, yet it is unclear how the magnetic reconnection and
its attendant topological changes shape the eruptive structure and how the
topology evolves during the eruption. Here we employed a realistic,
data-constrained magnetohydrodynamic simulation to study the evolving magnetic
topology for an X9.3 eruptive flare that occurred on 2017 September 6. The
simulation successfully reproduces the eruptive features and processes in
unprecedented detail. The numerical results reveal that the pre-flare corona
contains multiple twisted flux systems with different connections, and during
the eruption, these twisted fluxes form a coherent flux rope through
tether-cutting-like magnetic reconnection below the rope. Topological analysis
shows that the rising flux rope is wrapped by a quasi-separatrix layer, which
intersects itself below the rope, forming a topological structure known as
hyperbolic flux tube, where a current sheet develops, triggering the
reconnection. By mapping footpoints of the newly-reconnected field lines, we
are able to reproduce both the spatial location and, for the first time, the
temporal separation of the observed flare ribbons, as well as the dynamic
boundary of the flux rope's feet. Futhermore, the temporal profile of the total
reconnection flux is comparable to the soft X-ray light curve. Such a
sophisticated characterization of the evolving magnetic topology provides
important insight into the eventual understanding and forecast of solar
eruptions.
