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

TL;DR

This paper studies the classification of immersions of 3-spheres in 5-space, introducing new invariants to distinguish regular homotopy classes and analyzing their properties and implications.

Contribution

It constructs two new invariants, J and St, for generic immersions of 3-spheres in 5-space, and demonstrates their independence and completeness among first order invariants.

Findings

J and St are independent first order invariants.

Any first order invariant is a linear combination of J and St.

Restrictions on the topology of self intersections are derived.

Abstract

Regular homotopy classes of immersions of a 3-sphere in 5-space 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. 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 of 3-spheres in 4-space are obtained. Using them, we find restrictions on the topology of self intersections.