  In this paper, we demonstrate the existence of topological states in a new
collective dynamics model. This individual-based model (IBM) describes
self-propelled rigid bodies moving with constant speed and adjusting their
rigid-body attitude to that of their neighbors. In previous works, a
macroscopic model has been derived from this IBM in a suitable scaling limit.
In the present work, we exhibit explicit solutions of the macroscopic model
characterized by a non-trivial topology. We show that these solutions are well
approximated by the IBM during a certain time but then the IBM transitions
towards topologically trivial states. Using a set of appropriately defined
topological indicators, we reveal that the breakage of the non-trivial topology
requires the system to go through a phase of maximal disorder. We also show
that similar but topologically trivial initial conditions result in markedly
different dynamics, suggesting that topology plays a key role in the dynamics
of this system.
