On Brunnian-type links and the link invariants given by homotopy groups of spheres

TL;DR

This paper introduces homotopy groups of spheres as new link invariants for Brunnian-type links, revealing differences from classical invariants and connecting to geometric Massey products.

Contribution

It develops a novel approach to link invariants using homotopy groups of spheres, extending the detection capabilities beyond Milnor invariants.

Findings

Homotopy groups distinguish certain links undetectable by Milnor invariants.

Higher homotopy-group invariants can identify complex link structures.

All homotopy groups of spheres relate to geometric Massey products on links.

Abstract

We introduce the (general) homotopy groups of spheres as link invariants for Brunnian-type links through the investigations on the intersection subgroup of the normal closures of the meridians of strongly nonsplittable links. The homotopy groups measure the difference between the intersection subgroup and symmetric commutator subgroup of the normal closures of the meridians and give the invariants of the links obtained in this way. Moreover the higher homotopy-group invariants can produce some links that could not be detected by the Milnor invariants. Furthermore all homotopy groups of spheres can be obtained from the geometric Massey products on links.