Hans Boden, McMaster University - Virtual Knots and Algebraic Concordance

Explore the intricacies of virtual knots and algebraic concordance in this topology seminar talk by Hans Boden from McMaster University. Delve into the mysterious structure of the concordance group of virtual knots, which contains the concordance group of classical knots as a proper subgroup. Learn about the application of the Gordon-Litherland pairing to associate square integral matrices with virtual knots possessing spanning surfaces. Discover new knot invariants, including signatures, LT signatures, and Alexander polynomials, and examine their behavior under virtual concordance. Investigate a novel algebraic concordance group defined using non-orientable spanning surfaces, which serves as a linear approximation to the concordance group of virtual knots. Gain insights into the abelian and infinite rank nature of this group, as well as its 2- and 4-torsion elements. Engage with presented results and open problems in this collaborative work with Homayun Karimi, aimed at advancing our understanding of virtual knot theory and concordance.