Topology Seminar- THE VIRTUAL KNOT CONCORDANCE GROUP IS NOT ABELIAN

The next talk in the Greater Washington Topology Seminar is on Friday, October 30 from 1pm to 2pm EDT and will be held virtually via Zoom. 

1. October 2, 2020: Louis H. Kauffman (University of Illinois at Chicago and Novosibirsk State University)

2. October 9, 2020: Charles Frohman (University of Iowa)

3. October 16, 2020: Thang T. Q. Lê (Georgia Institute of Technology)

4. October 23, 2020: Pierrick Bousseau (ETH Zurich)

5. October 30, 2020: Micah Chrisman (Ohio State University)

6. November 13, 2020: Boštjan Gabrovšek (University of Ljubljana)

7. November 20, 2020: Razvan Gelca (Texas Tech University)

8. December 4, 2020: Helen Wong (Claremont McKenna College) 

Speaker: Micah Chrisman

Title: THE VIRTUAL KNOT CONCORDANCE GROUP IS NOT ABELIAN

Abstract: The concordances classes of knots in the 3-sphere form an abelian group C under the connected sum operation. Boden and Nagel proved that C embeds as a subgroup into the center of the long virtual knot concordance group VC. While the structure of the classical knot concordance group has been extensively studied in both the smooth and topological categories, little is known about the structure of VC. Virtual knots can be realized as knots in thickened surfaces Σ × I, where Σ is closed and oriented. As not all knots in Σ × I are homologically trivial, the usual tools of classical knot concordance (e.g. signature functions, the Arf invariant, and the algebraic concordance group), cannot be directly generalized to all virtual knots. Here we construct some concordance invariants of virtual knots that are instead closely related to Milnor’s concordance invariants of classical multi-component links. The invariants are derived from the lower central series of an extension of the group of a virtual knot. Using these extended Milnor invariants, we will give new examples of non-slice virtual knots having trivial writhe polynomial, generalized Alexander polynomial, graded genus, Rasmussen invariant, and parity projection. Furthermore, we will show that in contrast to the classical knot concordance group, the virtual knot concordance group is not abelian.

The Zoom information is listed below (it is the same as last time): 

Topic: GW Topology Seminar 

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Meeting ID: 951 840 9059

Password: The last name of the Fields Medalist famous for his work on von Neumann algebras and knot polynomials; first letter capitalized.