Tait Graphs and Properties of Virtual Knots

Abstract

The primary objective of our research project is to discover relationships between graph theory and knot theory. We are particularly interested in virtual knots, a knot that has three different types of crossings, and how they relate to their Tait graph. A Tait graph is a graph that is associated to a knot. A way to color a Tait graph for virtual knots is established where classical and virtual crossings can be easily identified. We describe how these graphs are invariant under the Reidemeister moves, both normal and virtual. We also establish a way to produce a Tait graph for normal and virtual doubling operators for a Pure Double, Whitehead Double, or Bing Double of a knot given the original Tait graph. Formulas for the number of edges and vertices of a Tait graph for a doubled knot using the number of edges in the original Tait graph are given.