Classification of virtual links by arc shift move

TL;DR

This paper introduces the arc shift move as an unknotting operation for certain virtual links, providing a classification method and relating it to the odd writhe, with specific bounds established.

Contribution

It demonstrates that arc shift moves can classify n-component virtual links up to equivalence and connects the arc shift number to the odd writhe, offering new insights.

Findings

Arc shift move acts as an unknotting operation for n-homogeneous proper virtual links.

The arc shift number relates to the odd writhe of virtual links.

Identifies sequences of virtual links with exact upper bounds on arc shift number.

Abstract

In this paper, we establish that the arc shift operation on a $n$-component virtual link diagram acts as an unknotting operation when the virtual link is $n$-homogeneous proper, aiding in the classification of \( n \)-component virtual links up to arc shift equivalence. We explore the connection between the arc shift number and the odd writhe of virtual links which are homogeneous proper. Additionally, we identify sequences of virtual link diagrams \( L_n \) for which the upper bound of the arc shift number is exactly \( n \).