Differential 3-knots in 5-space with and without self-intersections

Abstract

Regular homotopy classes of immersions S 3→ R 5 constitute an infinite cyclic group. The classes containing embeddings form a subgroup of index 24. The obstruction for a generic immersion to be regularly homotopic to an embedding is described in terms of geometric invariants of its self-intersection. Geometric properties of self-intersections are used to construct two invariants J and St of generic immersions which are analogous to Arnold's invariants of plane curves [1]. We prove that J and St are independent first-order invariants and that any first-order invariant is a linear combination of these. As by-products, some invariants of immersions S 3→ R 4 are obtained. Using them, we find restrictions on the topology of self-intersections.