  It was recently proved that for finitely determined germs $ \Phi: (
\mathbb{C}^2, 0) \to ( \mathbb{C}^3, 0) $ the number $C(\Phi)$ of Whitney
umbrella points and the number $T(\Phi)$ of triple values of a stable
deformation are topological invariants. The proof uses the fact that the
combination $C(\Phi)-3T(\Phi)$ is topological since it equals the linking
invariant of the associated immersion $S^3 \looparrowright S^5$ introduced by
Ekholm and Sz\H{u}cs. We provide a new, direct proof for this equality. We also
clarify the relation between various definitions of the latter invariant.
