The virtual flyping theorem
Abstract
We extend the flyping theorem to alternating links in thickened surfaces and alternating virtual links. The proof of the former result uses work of Boden--Karimi to adapt the author's geometric proof of Tait's 1898 flyping conjecture (first proved in 1993 by Menasco--Thistlethwaite), while the proof of the latter involves a diagrammatic correspondence recently introduced by the author in a related paper. In the process, we also extend a classical result of Gordon--Litherland, establishing an isomorphism between their pairing on a spanning surface and the intersection form on a 4-manifold constructed as a double-branched cover using that surface.
Cite
@article{arxiv.2210.03720,
title = {The virtual flyping theorem},
author = {Thomas Kindred},
journal= {arXiv preprint arXiv:2210.03720},
year = {2024}
}
Comments
27 pages, 15 figures, 1 table. There are important changes from version 1, where Theorem 3.10 was false. The overall proof strategy remains the same; changes mirror those from v2 to v3 of arXiv.2008.06490. Also, the diagrammatic correspondence introduced in version 1 is now introduced instead in arXiv.2210.03225, as are "lassos."