Virtual concordance and the generalized Alexander polynomial

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.

arXiv comment32 pages, 24 figures

References in corpus (10)

• What is a virtual link?
• On Alexander-Conway Polynomials for Virtual Knots and Links
• Cobordism of knots on surfaces
• Finite Type Invariants of w-Knotted Objects I: w-Knots and the Alexander Polynomial
• Virtual knot groups and almost classical knots
• Alexander invariants for virtual knots
• Polynomial knot and link invariants from the virtual biquandle
• Virtual knot cobordism and bounding the slice genus
• Signature and concordance of virtual knots
• Computations of the slice genus of virtual knots

Cited by in corpus (1)

• Milnor's concordance invariants for knots on surfaces