A remark on the Hopf invariant for spherical 4-braids

TL;DR

This paper explores the relationship between homotopy groups of the 2-sphere and spherical braids, specifically analyzing the Hopf invariant for spherical 4-braids to understand their isotopy classes.

Contribution

It provides a detailed investigation of the Hopf invariant for spherical 4-braids, extending Wu's approach to the case n=3 for applications in topology.

Findings

Characterization of the Hopf invariant for spherical 4-braids

Connection between homotopy groups and braid isotopy classes

Insights into the topology of spherical braids

Abstract

An approach by J.Wu describes homotopy groups $\pi_{n}(S^2)$ of the standard 2-sphere as isotopy classes of spherical $n+1$--strand Brunnian braids is investigated in the case $n=3$ for applications.