Classification of $2$-component virtual links up to $\Xi$-moves

TL;DR

This paper classifies 2-component virtual links up to -moves by refining invariants like odd writhe and linking numbers, extending previous results on virtual knot invariants.

Contribution

It extends the classification of virtual links up to -moves using refined invariants, building on prior work on virtual knot invariants.

Findings

Classification of 2-component virtual links up to -moves.

Refinement of odd writhe and linking numbers as invariants.

Extension of previous results on virtual knot invariants.

Abstract

The $\Xi$-move is a local move generated by forbidden moves in virtual knot theory. This move was introduced by Taniguchi and the second author, who showed that it characterizes the odd writhe of virtual knots, which is a fundamental invariant defined by Kauffman. In this paper, we extend this result by classifying $2$-component virtual links up to $\Xi$-moves, using refinements of the odd writhe and linking numbers.