RUI: Knots in Three-Dimensional Manifolds: Quantum Topology, Hyperbolic Geometry, and Applications

Claremont Mckenna College, Claremont CA

Investigators

Abstract

This project is split into two different areas of research concerning a field of mathematics called topology, which studies the properties of objects that remain the same even when they are twisted or deformed continuously. One direction relates to quantum physics, and the other to molecular biology. In 2016, physicists won the Nobel Prize for applying topology to research in condensed matter physics, and the underlying mathematical framework is called a topological quantum field theory (TQFT). The first part of the project focuses on topological constructions from TQFTs and conjectures about them. The PI aims to further advance the basic understanding of the connections between the mathematical and the theoretical physical sides of the subject. This work may be relevant to practical applications, such as the theoretical foundations and development of a topological quantum computer. The second part of the project is about the topology of proteins, which are long and flexible enough to exhibit knotting or linking. It is believed that such topological characteristics affect a protein's functionality, which is governed by its three-dimensional placement. However, little is known about how the proteins fold into a knotted state, and this project analyzes theories of protein folding from a topological viewpoint. In particular, knotted proteins are implicated in neurodegenerative disorders like Parkinson's and are found in bacteria used for bioremediation; a better understanding of the molecular knotting mechanism may lead to novel ways to target topological characteristics which affect specific biological functions. The award also supports undergraduate students participating in this research. Specifically, the research in quantum topology centers around the Kauffman bracket skein algebra of a surface, especially its representations. The skein algebra is related to quantum constructions, such as the Jones polynomial and the Witten-Reshetikhin-Turaev topological quantum field theory, as well as hyperbolic geometric constructions, particularly the SL(2,C)-character variety. The research will explore this relationship, and to exploit it for better understanding other invariants in geometric topology. With similar aims, the project also investigates recent generalizations of the skein algebra that includes arcs. In the second line of research, techniques from topology will be used to analyze evidence from laboratory and computer simulation experiments about knotted proteins, in order to develop new theories for how proteins might fold into a knotted configuration. The theoretical folding pathways can then be compared against widely available structural data in order to identify the most likely folding pathways for specific families of proteins. Thus, while providing valuable insights into folding pathways for all knotted proteins, this research aims to simplify the analysis for molecular biologists studying specific knotted proteins as well. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.