Classical results for alternating virtual links

TL;DR

This paper extends classical knot theory results to virtual links, establishing criteria for split links and properties of Alexander polynomials, and discusses Tait conjectures in the virtual setting.

Contribution

It introduces new theorems for virtual links, including split link criteria and polynomial properties, expanding classical knot theory to virtual and welded links.

Findings

An alternating virtual link is split iff visibly split.

The Alexander polynomial of almost classical alternating virtual links is alternating.

Tait's second conjecture does not hold for alternating welded links.

Abstract

We extend some classical results of Bankwitz, Crowell, and Murasugi to the setting of virtual links. For instance, we show that an alternating virtual link is split if and only if it is visibly split, and that the Alexander polynomial of any almost classical alternating virtual link is alternating. The first result is a consequence of an inequality relating the link determinant and crossing number for any non-split alternating virtual link. The second is a consequence of the matrix-tree theorem of Bott and Mayberry. We extend the first result to semi-alternating virtual links. We discuss the Tait conjectures for virtual and welded links and note that Tait's second conjecture is not true for alternating welded links.