Inner topological structure of Hopf invariant

TL;DR

This paper explores the topological structure of the Hopf invariant, revealing it as a sum of linking and self-linking numbers within knot families, based on $\,phi$-mapping topological current theory.

Contribution

It provides a new expression for the Hopf invariant as a sum of linking and self-linking numbers, deepening understanding of its topological nature.

Findings

Hopf invariant equals the winding number of Gauss mapping.

Hopf invariant expressed as sum of linking and self-linking numbers.

Provides a precise formula for Hopf invariant based on topological current theory.

Abstract

In light of $\phi$-mapping topological current theory, the inner topological structure of Hopf invariant is investigated. It is revealed that Hopf invariant is just the winding number of Gauss mapping. According to the inner structure of topological current, a precise expression for Hopf invariant is also presented. It is the total sum of all the self-linking and all the linking numbers of the knot family.