  The invariant-comb approach is a method to construct entanglement measures
for multipartite systems of qubits. The essential step is the construction of
an antilinear operator that we call {\em comb} in reference to the {\em
hairy-ball theorem}. An appealing feature of this approach is that for qubits
(or spins 1/2) the combs are automatically invariant under $SL(2,\CC)$, which
implies that the obtained invariants are entanglement monotones by
construction. By asking which property of a state determines whether or not it
is detected by a polynomial $SL(2,\CC)$ invariant we find that it is the
presence of a {\em balanced part} that persists under local unitary
transformations. We present a detailed analysis for the maximally entangled
states detected by such polynomial invariants, which leads to the concept of
{\em irreducibly balanced} states. The latter indicates a tight connection with
SLOCC classifications of qubit entanglement. \\ Combs may also help to define
measures for multipartite entanglement of higher-dimensional subsystems.
However, for higher spins there are many independent combs such that it is
non-trivial to find an invariant one. By restricting the allowed local
operations to rotations of the coordinate system (i.e. again to the
$SL(2,\CC)$) we manage to define a unique extension of the concurrence to
general half-integer spin with an analytic convex-roof expression for mixed
states.
