A parity for 2-colourable links

Rushworth, William

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.

Citation

Rushworth, W. (2021). A parity for 2-colourable links. Osaka Journal of Mathematics, 58(4), 767–801

• Journal Article Type: Article
• Acceptance Date: Jun 3, 2020
• Online Publication Date: Oct 11, 2021
• Publication Date: 2021-10
• Deposit Date: Nov 19, 2022
• Publicly Available Date: Jan 18, 2023
• Journal: Osaka Journal of Mathematics
• Print ISSN: 0030-6126
• Publisher: Osaka University
• Peer Reviewed: Peer Reviewed
• Volume: 58
• Issue: 4
• Pages: 767–801
• Keywords: Geometric Topology
• Public URL: https://hull-repository.worktribe.com/output/4127512
• Publisher URL: https://projecteuclid.org/journals/osaka-journal-of-mathematics/volume-58/issue-4/A-parity-for-2-colourable-links/5174ojm.full