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

The $Xi$-move is a local move which refines the usual forbidden moves in virtual knot theory. This move was introduced by Taniguchi and the second author, who showed that it characterizes the information contained by the odd writhe of virtual knots, 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.