Georgia Institute of Technology College of Sciences

Virtual knot theory is a variant of classical knot theory in which one allows a new type of crossing called a "virtual" crossing. It was originally developed by Louis Kauffman in order to study the Jones polynomial but has since developed into its own field and has genuine significance in low dimensional topology. One notable interpretation is that virtual knots are equivalent to knots in thickened surfaces. In this talk we'll introduce virtual knots and show why they are a natural extension of classical knots. We will then discuss what virtual knot theory can tell us about the both the classical Jones polynomial and its potential extensions to knots in arbitrary 3-manifolds. An important tool we will use throughout the talk is the knot quandle, a classical knot invariant which is complete up to taking mirror images.