# Classification of virtual string links up to cobordism

## TL;DR

This paper classifies virtual string links up to cobordism and unwelded equivalence using a combinatorial approach, establishing a bijection with a specific abelian group and exploring welded string link cobordism.

## Contribution

It introduces a classification framework for virtual string links up to cobordism and unwelded equivalence, and defines the theory of welded string link cobordism.

## Key findings

- Virtual string links up to cobordism correspond bijectively to ^{n(n-1)}.
- Virtual string links up to unwelded equivalence are classified by these groups.
- Welded string link cobordism is trivial for single-component links.

## Abstract

Cobordism of virtual string links on $n$ strands is a combinatorial generalization of link cobordism. There exists a bijection between virtual string links up to cobordisms and elements of the group $\mathbb{Z}^{n(n-1)}$. This paper also shows that virtual string links up to unwelded equivalence are classified by those groups. Finally, the related theory of welded string link cobordism is defined herein and shown to be trivial for string links with one component.