  Recent advances in numerical relativity have revealed how marginally trapped
surfaces behave when black holes merge. It is now known that interesting
topological features emerge during the merger, and marginally trapped surfaces
can have self-intersections. This paper presents the most detailed study yet of
the physical and geometric aspects of this scenario. For the case of a head-on
collision of non-spinning black holes, we study in detail the world tube formed
by the evolution of marginally trapped surfaces. In the first of this two-part
study, we focus on geometrical properties of the dynamical horizons, i.e. the
world tube traced out by the time evolution of marginally outer trapped
surfaces. We show that even the simple case of a head-on collision of
non-spinning black holes contains a rich variety of geometric and topological
properties and is generally more complex than considered previously in the
literature. The dynamical horizons are shown to have mixed signature and are
not future marginally trapped everywhere. We analyze the area increase of the
marginal surfaces along a sequence which connects the two initially disjoint
horizons with the final common horizon. While the area does increase overall
along this sequence, it is not monotonic. We find short durations of anomalous
area change which, given the connection of area with entropy, might have
interesting physical consequences. We investigate the possible reasons for this
effect and show that it is consistent with existing proofs of the area increase
law.
