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

Cross-caps, triple points and a linking invariant for finitely determined germs