CF-moves for virtual links

TL;DR

This paper extends the classification of virtual links under CF-moves by using virtual linking numbers and invariants, providing a comprehensive classification for links with up to three components.

Contribution

It generalizes previous results to classify odd and almost odd virtual links with any number of components and introduces new invariants for 3-component even virtual links.

Findings

Classified odd virtual links with arbitrary components using virtual linking number.

Extended Oikawa's $n$-invariant for broader virtual link types.

Provided a complete classification of 3-component virtual links under CF-moves.

Abstract

Oikawa defined an unknotting operation on virtual knots, called a CF-move, and gave a classification of 2-component virtual links up to CF-moves by the virtual linking number and his $n$-invariant. In particular, it was proved that a CF-move characterizes the information contained in the virtual linking number for 2-component odd virtual links. In this paper, we extend this result by classifying odd virtual links and almost odd virtual links with arbitrary number of components up to CF-moves, using the virtual linking number. Moreover, we extend Oikawa's $n$-invariant and introduce two invariants for 3-component even virtual links. Using these invariants together with the virtual linking number, we classify 3-component even virtual links up to CF-moves. As a result, a classification of 3-component virtual links up to CF-moves is provided.