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A parity for 2-colourable links

Rushworth, William

Authors

William Rushworth



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 Mar 29, 2024
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

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Copyright © 2021 Osaka University and Osaka City University, Departments of Mathematics




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