We consider a Bose-Hubbard trimer, i.e. an ultracold Bose gas populating three quantum states. The latter can be either di erent sites of a triple-well potential or three internal states of the atoms. The bosons can tunnel between di erent states with variable tunnelling strength between two of them. This will allow us to study; i) di erent geometrical con gurations, i.e. from a closed triangle to three aligned wells and ii) a triangular con guration with a -phase, i.e. by setting one of the tunnellings negative. By solving the corresponding three-site Bose-Hubbard Hamiltonian we obtain the ground state of the system as a function of the trap topology. We characterise the di erent ground states by means of the coherence and entanglement properties. For small repulsive interactions, fragmented condensates are found for the -phase case. These are found to be robust against small variations of the tunnelling in the small interaction regime. A low-energy e ective many-body Hamiltonian restricted to the degenerate manifold provides a compelling description of the -phase degeneration and explains the low-energy spectrum as excitations of discrete semi uxon states.