We seek to extend invariants such as the knot signature, the signature functions, and the Fox-Milnor condition on the Alexander polynomial to virtual knots and use them to study virtual knot concordance. We also hope to construct the algebraic concordance group for virtual knots and identify its structure. One of the motivating questions is whether the concordance group of long virtual knots is commutative, and one of the goals of this project is to resolve that question.
Impact
In joint work with M. Nagel, we show that two classical knots are virtually concordant if and only if they are classically concordant. This result holds in the smooth and topological categories. Many classical concordance invariants have been extended to virtual knots, and they have been used them to determine sliceness, the slice genus, and unknotting numbers for virtual knots.
Student Experience
This project involves the former MSc student Robin Gaudreau, who spent 12 months as a visiting student at University of Geneva for her last year of graduate study at McMaster (during the time this work was completed). It also involves a collaboration with a PhD student at University of Regensburg, Dr. Matthias Nagel, who since earned his Ph.D. and now holds a Britton Postdoctoral Fellowship in Math & Stats at McMaster University.
