turkmath.org

In this talk we first go through the basics of two natural generalizations of classical knot theory; the theory of knotoids and virtual knots. Knotoids, introduced by Turaev in 2010, are knotted arcs in surfaces possessing two free ends, and producing a diagrammatic and three dimensional interpretation for this freedom. Virtual knots, introduced by Kaufman in 1998, are knots in thickened surfaces taken up to stabilization relation. We utilize a ‘virtual closure’ to connect these two knotted categories, and we show that there are virtual knots of genus one that are not virtual closures of knotoids in S^2. (joint work with Louis Kauffman).