We use the Bar-Natan Ж-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 Ж-map is functorial under concordance, and also that Satoh’s Tube map (from welded links to ribbon knotted tori in S4) 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 virtually 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.

中文翻译：

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虚拟索引和广义亚历山大多项式

我们使用 Bar-Natan Ж-correspondence 来识别虚拟结的广义 Alexander 多项式与两组件焊接连接的 Alexander 多项式。我们证明了 Ж-map 在一致性下是功能性的，而且 Satoh 的 Tube map（从焊接链接到带状打结的 tori 在小号4) 是符合下的函项。此外，我们将 Chen、Milnor 和 Hillman 的经典结果在较低的中央系列链接组上扩展到加厚表面中的链接。我们的主要结果是广义亚历山大多项式在加厚表面中的任何结上消失，这实际上与同调平凡结一致。特别是，这表明它在几乎切片结上消失。我们应用它来完成对具有四个交叉的虚拟结的切片属的计算，并确定多个 5 交叉和 6 交叉虚拟结的非切片性。