New Directions in Algebra, Geometry and Topology Special Lecture
Novel measures of complexity of open curves in 3-space

Dr. Eleni Panagiotou

Arizona State University

Abstract

Filamentous material may exhibit structure dependent material properties and function that depends on their entanglement. Even though intuitively entanglement is often understood in terms of knotting or linking, many of the filamentous systems in the natural world are not mathematical knots or links.  In this talk we will introduce a novel framework in knot theory that can characterize the complexity of (collections of) open curves in 3-space in general. This leads to novel metrics of entanglement of open curves in 3-space that generalize classical topological invariants, like, for example, the Jones polynomial and Vassiliev invariants. For open curves, these are continuous functions of the curve coordinates and tend to topological invariants of classical knots and links when the endpoints of the curves tend to coincide. We will apply our methods to proteins and we will show that these enable us to create a new framework for understanding protein folding, which is validated by experimental data.  When applied to the SARS-CoV-2 spike protein, we see that the local native topological landscape can predict residues where mutations can have an important impact on protein structure and possibly in viral transmissibility.  These methods thus not only open a new mathematical direction in knot theory, but can also help us understand polymer and biopolymer function and material properties in many contexts with the goal of their prediction and design.