A homotopy classification of two-component spatial graphs up to neighborhood equivalence

TL;DR

This paper provides a complete classification of two-component spatial graphs up to neighborhood homotopy using linking matrices, advancing understanding of their topological equivalences.

Contribution

It introduces a homotopy classification method for 2-component spatial graphs based on elementary divisors of linking matrices, extending to handlebody-links.

Findings

Classification based on linking matrix elementary divisors

Complete characterization of 2-component spatial graphs

Extension to handlebody-link homotopy

Abstract

A neighborhood homotopy is an equivalence relation on spatial graphs which is generated by crossing changes on the same component and neighborhood equivalence. We give a complete classification of all 2-component spatial graphs up to neighborhood homotopy by the elementary divisor of a linking matrix with respect to the first homology group of each of the connected components. This also leads a kind of homotopy classification of 2-component handlebody-links.