# Virtual concordance and the generalized Alexander polynomial

## TL;DR

This paper links the generalized Alexander polynomial of virtual knots to welded links via the Zh-correspondence, demonstrating its functoriality under concordance and applying it to classify virtual knots by slice genus and sliceness.

## Contribution

It establishes the functoriality of the Zh-map and Tube map under concordance, extending classical link group results to virtual knots, and applies these to classify virtual knots by sliceness.

## Key findings

- The generalized Alexander polynomial vanishes on knots concordant to homologically trivial knots.
- The polynomial vanishes on virtually slice knots.
- Complete calculation of slice genus for certain virtual knots.

## Abstract

We use the Bar-Natan Zh-correspondence to identify the generalized Alexander polynomial of a virtual knot with the Alexander polynomial of a two component welded link. We show that the Zh-map is functorial under concordance, and also that Satoh's Tube map (from welded links to ribbon knotted tori in $S^4$) is functorial under concordance. In addition, we extend classical results of Chen, Milnor, and Hillman on the lower central series of link groups to links in thickened surfaces. Our main result is that the generalized Alexander polynomial vanishes on any knot in a thickened surface which is concordant to a homologically trivial knot. In particular, this shows that it vanishes on virtually slice knots. We apply it to complete the calculation of the slice genus for virtual knots with four crossings and to determine non-sliceness for a number of 5-crossing and 6-crossing virtual knots.