Self $C_2$-equivalence of two-component links and invariants of link maps

TL;DR

This paper leverages invariants of link maps in four-dimensional space to classify two-component links in three-dimensional space up to certain homotopies, providing new proofs without relying on Clasper Theory.

Contribution

It offers a novel proof of the Nakanishi-Ohyama classification of two-component links using link map invariants, extending results to string links without Clasper Theory.

Findings

Classifies two-component links up to $ riangle$-link homotopy.

Extends classification to string links.

Provides proofs without Clasper Theory.

Abstract

We use Kirk's invariant of link maps $S^2\sqcup S^2\to S^4$ and its variations due to Koschorke and Kirk-Livingston to deduce results about classical links. Namely, we give a new proof of the Nakanishi-Ohyama classification of two-component links in $S^3$ up to $\Delta$-link homotopy. We also prove its version for string links, which is due (in a slightly different form) to Fleming-Yasuhara. The proofs do not use Clasper Theory.