Permuted Petal Projections

In our talk, we will present the idea of a multicrossing projection of a knot and the specific projection called the petal projection. A petal projection is a projection of the knot containing a single crossing. Imagine that we have such a diagram where we do not know which strand is on top of which. What are the possible knots that such a projection might represent? When we choose a particular ordering of strands, we find that the knots created are an interesting family called torus knots. We examine how petal diagrams can be simplified to result in a standard diagram of a knot in the torus family. While much of the work that we will discuss in this talk was done by Colin Adams and student collaborators in the SMALL REU, we will also describe our current research related to their work.