A parity for 2-colourable links

TL;DR

This paper introduces the 2-colour parity, a new invariant for virtual links that extends Gaussian parity, providing stronger distinctions and new invariants like the 2-colour writhe, which detects sliceness and other properties.

Contribution

The paper defines the 2-colour parity for virtual links, compares it with existing parities, and introduces the 2-colour writhe as a new concordance invariant with applications in link properties.

Findings

2-colour parity extends Gaussian parity to virtual links.

The 2-colour parity is strictly stronger than the Im-Park parity.

The 2-colour writhe obstructs sliceness, amphichirality, and chequerboard colourability.

Abstract

We introduce the 2-colour parity. 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 amphichirality and chequerboard colourability within a concordance class.