Primeness of alternating virtual links

TL;DR

This paper introduces a new method using lassos to relate cellular link diagrams on surfaces to virtual links, and extends classical primeness results to alternating links in thickened surfaces.

Contribution

It establishes a new correspondence between cellular diagrams and virtual links, and extends Menasco's classical primeness result to a stricter notion for alternating links.

Findings

A new correspondence between cellular diagrams and virtual links.

Classical primeness results extended to a stricter sense for alternating links.

Method to determine primeness of virtual links by inspection.

Abstract

Using a new tool called lassos, we establish a new correspondence between cellular link {diagrams} on closed surfaces and equivalence classes of virtual link {diagrams}. This is analogous to a well-known correspondence among the links represented by these diagrams, but with a crucial subtlety. We explain how, under these correspondences, the traditional notion of primeness for virtual links is stricter than the one for links in thickened surfaces. We extend a classical result of Menasco by proving that an alternating link in a thickened surface is prime in the stricter sense unless it is ``obviously" composite. (Adams et al and Howie--Purcell previously extended Menasco's result for the other notion of primeness.) We describe, given an alternating virtual link diagram, how to determine by inspection whether the virtual link it represents is prime in either sense.