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

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