We introduce the 2-parity colour. It is a theory of parity for a large class of virtual links, defined using the interaction between orientations of the link components and a certain type of colouring. The 2-colour parity is an extension of the Gaussian parity, to which it reduces on virtual knots. We show that the 2-colour parity descends to a parity on free links. We compare the 2-colour parity to other parity theories of virtual links, focusing on a theory due to Im and Park. The 2-colour parity yields a strictly stronger invariant than the Im-Park parity.
We introduce an invariant, the 2-colour writhe, that takes the form of a string of integers. The 2-colour writhe is a concordance invariant, and so obstructs sliceness. It is also an obstruction to ($\pm$)-amphichirality and chequerboard colourability within a concordance class.
We thank Hans Boden and Andrew Nicas for their encouragement and numerous helpful conversations and comments. We thank the referee for their comments and careful reading of the paper.
"A parity for 2-colourable links." Osaka J. Math. 58 (4) 767 - 801, October 2021.