Leaps-MPS where algebra and topology meet: an investigation in braid group representation theory

Elon University received a $121,965 Project Grant from the National Science Foundation (NSF) under the Mathematical and Physical Sciences program (CFDA 47.049), effective September 1, 2025 through August 31, 2027. This award supports fundamental mathematical research focused on braid group representation theory, specifically investigating the intersection of algebraic and topological methods for encoding and computationally processing knotted structures. The principal investigator will pursue three primary research directions: establishing a topological description of the virtual braid group using mapping class groups and spanning surfaces; classifying link diagrams with bi-orderable complements; and developing modified constructions for braid group representations. The research has applications to quantum computing, molecular biology, and satellite orbital mechanics.

Beyond the core research deliverables, the project incorporates a significant educational outreach component leveraging dance-based visualization and explanation of mathematical concepts to engage and recruit the next generation of mathematicians. This integrated approach aligns with NSF's emphasis on combining intellectual merit with broader impacts, particularly through workforce development and public engagement in mathematical sciences. The award represents a standard project grant within the NSF's commitment to advancing fundamental scientific knowledge and strengthening the nation's mathematical research enterprise.

Abstract

Knots play an important role in the world around US in a much deeper way than just your electronic chords tangling under your desk. DNA molecules are knotted; the paths of satellites form interwoven orbits around earth; quantum computing can be modeled by the braiding of particles. This project focuses on different ways of storing the crossing information and tangled structure of knots in a format that can be computationally processed. This is called a representation- choosing different mathematical ways to represent, or encode, 3 dimensional knotted information. As both a mathematician and dancer, the PI uses dance to help explain her research visually and recruit the next generation of math pioneers. This project focuses on representation of a specific type of knotted structure called a braid. Topologically, a braid is a tangle of n-strands that flow monotonically from the floor to the ceiling. The set of all n-stranded braids can be turned into a group using vertical stacking as the multiplication operation, rendering an algebraic description of braids. To describe a meaningful representation requires constant negotiation between the topological and algebraic descriptions of braids. The research agenda of this project centers on constructing and analyzing representations of braids in three separate directions. Firstly, the PI will find a topological description (in terms of mapping class groups) of an algebraic generalization of the braid group, called the virtual braid group. This will be done by extending a construction on a finite index subgroup of the virtual braid group to the entire group via spanning surfaces. Secondly, using a mapping class group representation of the braid group, the PI will classify when certain link diagrams have a BI-orderable complement. Thirdly, the PI will alter a step in a well known construction for building braid group representations (the long-moody construction) and thereby create a much larger 1-parameter family of representations. 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.- subawards are not planned for this award.