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Representations of the Necklace Braid Group: Topological and Combinatorial Approaches

Alex Bullivant, Andrew Kimball, Paul Martin, Eric C. Rowell

TL;DR

This work develops a representation-theoretic framework for the necklace braid group $\mathcal{NB}_n$, clarifying its connections to the classical braid group $\mathcal{B}_n$ and the loop braid group $\mathcal{LB}_n$. The authors show that every irreducible $\mathcal{B}_n$ representation extends to $\mathcal{NB}_n$ in a standard way, and they construct several nonstandard extensions for familiar $\mathcal{B}_n$ representations such as the Burau and LKB representations. They also demonstrate that local $\mathcal{B}_n$ representations extend to $\mathcal{NB}_n$ and develop alternative finite-image constructions using Gaussian braided vector spaces and quaternionic algebras, yielding a rich landscape of $\mathcal{NB}_n$ representations. Furthermore, the paper connects these algebraic constructions to topological physics via braided fusion categories and spin chains, and it establishes a precise homomorphism from $\mathcal{NB}_n$ to $\mathcal{LB}_n$, highlighting how $\mathcal{NB}_n$ fits into the broader motion-group framework. Overall, the results expand the repertoire of representations for a 3D motion group and illuminate links between topology, category theory, and quantum physics.

Abstract

The necklace braid group $\mathcal{NB}_n$ is the motion group of the $n+1$ component necklace link $\mathcal{L}_n$ in Euclidean $\mathbb{R}^3$. Here $\mathcal{L}_n$ consists of $n$ pairwise unlinked Euclidean circles each linked to an auxiliary circle. Partially motivated by physical considerations, we study representations of the necklace braid group $\mathcal{NB}_n$, especially those obtained as extensions of representations of the braid group $\mathcal{B}_n$ and the loop braid group $\mathcal{LB}_n$. We show that any irreducible $\mathcal{B}_n$ representation extends to $\mathcal{NB}_n$ in a standard way. We also find some non-standard extensions of several well-known $\mathcal{B}_n$-representations such as the Burau and LKB representations. Moreover, we prove that any local representation of $\mathcal{B}_n$ (i.e. coming from a braided vector space) can be extended to $\mathcal{NB}_n$, in contrast to the situation with $\mathcal{LB}_n$. We also discuss some directions for future study from categorical and physical perspectives.

Representations of the Necklace Braid Group: Topological and Combinatorial Approaches

TL;DR

This work develops a representation-theoretic framework for the necklace braid group , clarifying its connections to the classical braid group and the loop braid group . The authors show that every irreducible representation extends to in a standard way, and they construct several nonstandard extensions for familiar representations such as the Burau and LKB representations. They also demonstrate that local representations extend to and develop alternative finite-image constructions using Gaussian braided vector spaces and quaternionic algebras, yielding a rich landscape of representations. Furthermore, the paper connects these algebraic constructions to topological physics via braided fusion categories and spin chains, and it establishes a precise homomorphism from to , highlighting how fits into the broader motion-group framework. Overall, the results expand the repertoire of representations for a 3D motion group and illuminate links between topology, category theory, and quantum physics.

Abstract

The necklace braid group is the motion group of the component necklace link in Euclidean . Here consists of pairwise unlinked Euclidean circles each linked to an auxiliary circle. Partially motivated by physical considerations, we study representations of the necklace braid group , especially those obtained as extensions of representations of the braid group and the loop braid group . We show that any irreducible representation extends to in a standard way. We also find some non-standard extensions of several well-known -representations such as the Burau and LKB representations. Moreover, we prove that any local representation of (i.e. coming from a braided vector space) can be extended to , in contrast to the situation with . We also discuss some directions for future study from categorical and physical perspectives.

Paper Structure

This paper contains 19 sections, 23 theorems, 51 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Overview of paper
  3. $\mathcal{NB}_n$ representations from $\mathcal{B}_n$ and $\mathcal{LB}_n$ representations
  4. Relationship with $\mathcal{B}_n$
  5. Relationship between $\mathcal{NB}_n$ and $\mathcal{LB}_n$
  6. A group homomorphism
  7. Extensions of Familiar $\mathcal{B}_n$ Representations
  8. Extensions of Standard Representations
  9. Extensions of the Reduced Burau representation
  10. Extensions of the Lawrence-Krammer-Bigelow Representation
  11. Nonstandard Extensions
  12. Local Representations
  13. Gaussian Braided Vector Spaces
  14. Quaternionic Representation
  15. Extending low-dimensional $\mathcal{B}_n$ representations
  16. ...and 4 more sections

Key Result

Theorem 1.1

We have a presentation of a group isomorphic to $\mathcal{NB}_n$ by abstract generators $\sigma_1, \ldots, \sigma_n$, $\tau$ satisfying: Here indices are taken modulo $n$, with $\sigma_{n+1}:=\sigma_1$ and $\sigma_0:=\sigma_n$. ∎

Figures (1)

  • Figure 1: The necklace $\mathcal{L}_n$ as seen from a generic observation point

Theorems & Definitions (50)

  • Theorem 1.1: BB16
  • Lemma 1.2
  • proof
  • Lemma 2.1
  • proof
  • Theorem 2.2
  • proof
  • Remark 2.3
  • Remark 2.4
  • Lemma 2.5
  • ...and 40 more