Jason Cantarella (University of Georgia): Radius of gyration of molecules in phantom network theory and beyond

We are interested in studying the geometry of molecular networks in the James-Guth-Flory model of phantom network theory. In this talk, we focus on the radius of gyration, as it’s both mathematically tractable and chemically significant. We will discuss new results on subdivided networks in phantom network theory, including a new exact formula for the mean radius of gyration of a subdivided network. Our formula uses a new permutation averaging method which applies to other models of random knots. We will then discuss the statistics of radius of gyration in other settings, such as polygons of a fixed knot type, where we find preliminary numerical evidence supporting the “topological swelling” hypothesis.