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.
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