  A holomorphic germ \Phi: (C^2, 0) \to (C^3, 0), singular only at the origin,
induces at the links level an immersion of S^3 into S^5. The regular homotopy
type of such immersions are determined by their Smale invariant, defined up to
a sign ambiguity. In this paper we fix a sign of the Smale invariant and we
show that for immersions induced by holomorphic gems the sign-refined Smale
invariant is the negative of the number of cross caps appearing in a generic
perturbation of \Phi. Using the algebraic method we calculate it for some
families of singularities, among others the A-D-E quotient singularities. As a
corollary, we obtain that the regular homotopy classes which admit holomorphic
representatives are exactly those, which have non-positive sign-refined Smale
invariant. This answers a question of Mumford regarding exactly this
correspondence. We also determine the sign ambiguity in the topological
formulae of Hughes-Melvin and Ekholm-Szucs connecting the Smale invariant with
(singular) Seifert surfaces. In the case of holomorphic realizations of Seifert
surfaces, we also determine their involved invariants in terms of holomorhic
geometry.
