A New Route to the Interpretation of Hopf Invariant

TL;DR

This paper introduces a new theoretical framework linking the Hopf invariant to linking numbers of higher-dimensional knots using $$-mapping topological current theory and topological tensor currents.

Contribution

It presents a novel approach that directly relates Hopf invariant to linking numbers of higher-dimensional submanifolds in Euclidean space.

Findings

Established a relationship between Hopf invariant and linking numbers of higher-dimensional knots.

Introduced a topological tensor current to analyze topological defects.

Provided a new interpretation of Hopf invariant in algebraic topology.

Abstract

We discuss an object from algebraic topology, Hopf invariant, and reinterpret it in terms of the $\phi$-mapping topological current theory. The main purpose in this paper is to present a new theoretical framework which can directly give the relationship between Hopf invariant and the linking numbers of the higher dimensional submanifolds of Euclidean space $R^{2n-1}$. For the sake of this purpose we introduce a topological tensor current which can naturally deduce the $(n-1)$ dimensional topological defect in $R^{2n-1}$ space. If these $(n-1)$ dimensional topological defects are closed oriented submanifolds of $R^{2n-1}$, they are just the $(n-1)$ dimensional knots. The linking number of these knots is well defined. Using the inner structure of the topological tensor current, the relationship between Hopf invariant and the linking numbers of the higher dimensional knots can be constructed.