Representations of the Necklace Braid Group: Topological and Combinatorial Approaches

Abstract

The necklace braid group ${\mathcal{N}\mathcal{B}}_n$ 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 ${\mathcal{N}\mathcal{B}}_n$, 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 ${\mathcal{N}\mathcal{B}}_n$ 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 ${\mathcal{N}\mathcal{B}}_n$, in contrast to the situation with ${\mathcal{LB}}_n$. We also discuss some directions for future study from categorical and physical perspectives.

30 pages