The virtual flyping theorem

TL;DR

This paper extends the flyping theorem to alternating links in thickened surfaces and virtual links, using geometric and diagrammatic methods, and also generalizes a classical Gordon--Litherland result.

Contribution

It introduces new extensions of the flyping theorem to broader classes of links and generalizes a classical pairing isomorphism, combining geometric and diagrammatic approaches.

Findings

Extended flyping theorem to alternating links in thickened surfaces.

Extended flyping theorem to alternating virtual links.

Generalized Gordon--Litherland pairing isomorphism.

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.