A geometric formula for Haefliger knots
Haefliger showed that the group of isotopy classes of smooth \((4k-1)\)-knots in \(S^{6k}\) is infinite cyclic, and gave a simple, explicit construction of a generator. He defined an invariant \(\Omega(F)\in\mathbb{Z}\) determining the isomorphism in terms of the cohomology of a submanifold of \(S^{6k}\) with boundary \(F\). This paper gives a new formula for \(\Omega(F)\), using immersions rather than embeddings. Haefliger's work is used to show that every such knot \(F\) extends to a generic immersion \(\widetilde{F}\) of a compact orientable \(4k\)-manifold \(V\) with boundary \(S^{4k-1}\). The new formulation of the invariant involves double and triple point data of \(\widetilde{F}\), the \(k\)th normal Pontrjagin number of \(V\cup{D^{4k}}\) and a Hopf invariant. The linking number of \textit{A. Szűcs} [Comment. Math. Helv. 61, 360-369 (1986; Zbl 0602.57021)], interpreted via the Hopf invariant, plays a key role.
scientific article; zbMATH DE number 2117936