  We investigate the connection between the concept of affine balancedness
(a-balancedness) introduced in [Phys. Rev A. {\bf 85}, 032112 (2012)] and
polynomial local SU invariants and the appearance of topological phases
respectively. It is found that different types of a-balancedness correspond to
different types of local SU invariants analogously to how different types of
balancedness as defined in [New J. Phys. {\bf 12}, 075025 (2010)] correspond to
different types of local SL invariants. These different types of SU invariants
distinguish between states exhibiting different topological phases. In the case
of three qubits the different kinds of topological phases are fully
distinguished by the three-tangle together with one more invariant. Using this
we present a qualitative classification scheme based on balancedness of a
state. While balancedness and local SL invariants of bidegree $(2n,0)$ classify
the SL-semistable states [New J. Phys. {\bf 12}, 075025 (2010), Phys. Rev. A
{\bf 83} 052330 (2011)], a-balancedness and local SU invariants of bidegree
$(2n-m,m)$ gives a more fine grained classification. In this scheme the
a-balanced states form a bridge from the genuine entanglement of balanced
states, invariant under the SL-group, towards the entanglement of unbalanced
states characterized by U invariants of bidegree $(n,n)$. As a by-product we
obtain generalizations to the W-state, states that are entangled, but contain
only globally distributed entanglement of parts of the system.
