Representations of the Necklace Braid Group: Topological and Combinatorial Approaches

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.