# LEAPS-MPS  Where Algebra and Topology Meet: An investigation in Braid Group Representation Theory

> **NSF 01002526DB NSF RESEARCH & RELATED ACTIVIT** · Elon University (NC) · $121,965

## 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

## Key facts

- **NSF award ID:** 2532699
- **Awardee organization:** Elon University (NC)
- **SAM.gov UEI:** SNEKGT9DDUQ6
- **PI:** Nancy C Scherich
- **Primary program:** 01002526DB NSF RESEARCH & RELATED ACTIVIT
- **All programs:** —
- **Estimated total:** $121,965
- **Funds obligated:** $121,965
- **Transaction type:** Standard Grant
- **Period:** 09/01/2025 → 08/31/2027

## Primary source

NSF Award Search: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2532699

## Citation

> US National Science Foundation, Award 2532699, LEAPS-MPS  Where Algebra and Topology Meet: An investigation in Braid Group Representation Theory. Retrieved via AI Analytics 2026-06-07 from https://api.ai-analytics.org/grant/nsf/2532699. Licensed CC0.

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