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The generalized Alexander polynomial and virtually slice knots
by
Micah Chrisman
The Ohio State University
Coauthors: Hans U. Boden
In 1994, Jaeger, Kauffman, and Saluer (JKS) gave a determinant formulation for the Alexander-Conway polynomial. Their polynomial, which was inspired by the free fermion model in statistical mechanics, can be used to define an invariant of knots in thickened surfaces Σ×[0, 1]. Sawollek further showed that the JKS polynomial is an invariant of virtual knots. Silver-Williams then proved that this is equivalent to the generalized Alexander polynomial of virtual knots. In 2008, Turaev defined a new notion of sliceness for knots in thickened surfaces. A knot K ⊂ Σ×[0, 1] is said to be (virtually) slice if there is a compact connected oriented 3-manifold W and a disc D smoothly embedded in W ×[0, 1] such that ∂W=Σ and ∂D=K. Here we show that the generalized Alexander polynomial is vanishing on all virtually slice knots. To do this, we prove that Bar-Natan's "Zh" correspondence and Satoh's Tube map are both functorial under concordance. As an application, we determine the slice status and slice genus of many low crossing virtual knots. Indeed, a knot K ⊂ Σ×[0, 1] is virtually slice if and only if the corresponding virtual knot is slice in the sense of Kauffman. This project is joint work with H. U. Boden (https://arxiv.org/pdf/1903.08737.pdf).
Date received: April 14, 2019
Copyright © 2019 by the author(s). The author(s) of this work and the organizers of the conference have granted their consent to include this abstract in Topology Atlas. Document # cbpy-02.