Geometric reason for the unique regular homotopy coincidence among A–D–E immersions g_G ∘ p

Ascertain a geometric explanation for why, among the immersions g_G ∘ p: S^3 → R^4 obtained by composing Kinjo’s immersion g_G: M(G) → R^4 associated with Dynkin diagrams of types A–D–E and the universal covering map p: S^3 → M(G), only the immersions corresponding to D_{17} and E_8 are regularly homotopic.

Background

The paper computes Smale invariants for the family of immersions g_G ∘ p: S^3 → R^4, where G ranges over Dynkin diagrams of types A, D, and E, g_G is constructed via plumbing copies of the Whitney 2-sphere according to G, and p is the universal covering S^3 → M(G).

From these computations the authors deduce that among all A–D–E cases, exactly two immersions—those for D_{17} and E_8—are regularly homotopic, a phenomenon for which they do not have a geometric explanation.

This leaves an explicit unresolved question seeking a geometric reason or structural mechanism that explains the observed coincidence of regular homotopy classes for D_{17} and E_8, but not for other A–D–E types.